رياضيات

 ١١٥‫א‬‫א‬
 
 ١١٥
 J‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 ١١٥‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
،   ‫א‬   ،     ‫א‬ ‫א‬،  ‫א‬

  ‫א‬ ‫א‬ ‫א‬ ‫א‬‫א‬  ‫א‬ ‫א‬  ‫א‬ ‫א‬ 

 W
‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
‫א‬ ‫א‬  ‫א‬‫א‬‫א‬‫א‬
 ‫؛‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 K ‫א‬‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

  ‫א‬        ،‫א‬ ‫א‬‫א‬  
‫א‬  ‫א‬ ‫א‬ ‫א‬ ‫א‬ ‫א‬   ‫א‬    ، 
‫א‬،‫א‬‫א‬‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬
‫א‬‫א‬‫א‬،‫א‬
 K‫א‬‫א‬،‫א‬
 ? ٤١١٥ ?‫א‬‫א‬ 

‫א‬‫א‬ ‫א‬     ‫א‬ ‫א‬  ? ‫א‬ ‫א‬  
 K‫א‬‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ،‫א‬ ،‫א‬‫א‬‫א‬‫א‬
 K‫א‬‫א‬‫א‬‫א‬‫א‬
 ،‫א‬‫א‬‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 

 K‫א‬
 ١١٥‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 ‫א‬
 
‫א‬
‫א‬
٢
‫א‬W‫א‬‫א‬
 ٢
 ‫א‬‫א‬
٣
‫א‬‫א‬
٤
 ‫א‬‫א‬‫א‬‫א‬‫א‬
٩
‫א‬‫א‬‫א‬‫א‬
14
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
١٧
‫א‬‫א‬
18
‫א‬‫א‬‫א‬‫א‬‫א‬
 20
 ‫א‬‫א‬‫א‬‫א‬‫א‬
22

26
‫א‬‫א‬‫א‬‫א‬W‫א‬‫א‬
26
‫א‬‫א‬‫א‬‫א‬‫א‬
 27
 ‫א‬‫א‬‫א‬
 31
 ‫א‬‫א‬‫א‬
 32
 ‫א‬‫א‬‫א‬
 33
 ‫א‬‫א‬
 34
 ‫א‬‫א‬
 37
 ‫א‬‫א‬‫א‬‫א‬
 40
 ‫א‬
 43
 ‫א‬‫א‬
 45
 
 50
‫א‬‫א‬‫א‬‫א‬‫א‬W‫א‬‫א‬
 ١١٥‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 51
 
 52
 ‫א‬
 53
 ‫א‬‫א‬‫א‬‫א‬‫א‬
 54
 ‫א‬‫א‬‫א‬
 60
 NAND‫א‬‫א‬‫א‬
 63
 
 67
‫א‬W‫א‬‫א‬‫א‬
 69
 
 70
 ‫א‬E‫א‬F‫א‬‫א‬‫א‬
 71
 ‫א‬‫א‬
 72
 ‫א‬‫א‬
74
 ‫א‬‫א‬‫א‬
77
 ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
78
 ‫א‬
79
 ‫א‬L‫א‬‫א‬‫א‬‫א‬
 81
 ‫א‬‫א‬
82
 ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 83
 ‫א‬‫א‬
84
 ‫א‬‫א‬
84
 ‫א‬
85
 ‫א‬L‫א‬‫א‬
 87
 
 89
‫א‬‫א‬
 
 

 ١١٥‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
K‫א‬‫א‬‫א‬‫א‬، ‫א‬،‫א‬‫א‬
‫א‬‫א‬    W
‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
‫א‬  ‫א‬   _‫א‬ ‫א‬  _ ‫א‬  K‫א‬ ‫א‬‫א‬
  ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
    ‫א‬ ‫א‬ ‫א‬   ‫א‬  ‫א‬    ‫א‬ ‫א‬
 K‫א‬‫א‬‫א‬
 W‫א‬‫א‬‫א‬
K‫א‬‫א‬‫א‬‫א‬ 
K‫א‬‫א‬‫א‬‫א‬‫א‬ 
K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ 
K‫א‬‫א‬‫א‬ 
  _‫א‬_‫א‬‫א‬
 W‫א‬‫א‬‫א‬
‫א‬  ‫א‬ ‫א‬  ‫א‬  ‫א‬ ‫א‬  ‫א‬  ‫א‬ ‫א‬
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 ‫א‬  ‫א‬‫א‬  ‫א‬ ‫א‬ ‫א‬   ‫א‬ ‫א‬
 K
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 
‫א‬‫א‬
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 
‫א‬‫א‬
 ‫א‬
0
 
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
‫א‬‫א‬‫א‬‫א‬W‫א‬‫א‬
 K‫א‬‫א‬‫א‬،‫א‬‫א‬
 
 W‫א‬‫א‬‫א‬
K‫א‬‫א‬‫א‬ 
K‫א‬‫א‬‫א‬‫א‬‫א‬ 
K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ 
K‫א‬‫א‬‫א‬‫א‬‫א‬ 
K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ 
K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ 
 
 
 
1
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
◌ّ ‫א‬
‫א‬‫א‬،‫א‬‫א‬‫א‬
،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 K‫א‬‫א‬‫א‬ 
‫א‬   ‫א‬ ‫א‬ ‫א‬    ١٢٣ ‫א‬ W‫א‬  
 ‫א‬‫א‬‫א‬K
 K١١١١٠١١‫א‬‫א‬
‫א‬ ‫א‬   ‫א‬ ‫א‬   ‫א‬ ‫א‬ 
‫א‬‫א‬‫( ?א‬ASCII) ‫א‬‫א‬‫א‬
K?The American Standard Code for Information Interchange‫א‬
 
 W (Decimal System) ‫א‬‫א‬K١
0, 1, 2, 3, 4, 5, 6, W‫א‬‫א‬
   ‫א‬ 7, 8, 9
 ‫א‬ ،9 ‫ و‬0    ‫א‬    
 K‫א‬‫א‬‫א‬‫א‬،‫א‬
‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬
 K‫א‬
 
9
 K4.5‫א‬‫א‬ ‫א‬‫א‬W١
2
 K‫א‬‫א‬‫א‬
      ‫א‬    ‫א‬ ‫א‬     ‫א‬
 W‫א‬‫א‬‫א‬‫א‬‫א‬،
1  (10) 0 , 10  (10) 1 , 100  (10) 2 , 1000  (10) 3 , ......
 W‫א‬‫א‬‫א‬‫א‬‫א‬
  0.1  (10) 1 , 0.01  (10) -2 , 0.001  (10) -3 , 0.0001  (10) -4 , ......
2
 
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
‫א‬ 9‫א‬ 0‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ W
 K‫א‬١٠‫א‬
 
 W٢
374  3  10 2  7  101  4  10 0
  0.95  9  10 1  5  10 2
374.95  3  10 2  7  101  4  10 0  9  10 1  5  10 2
 
 
 W(Binary System)‫א‬‫א‬K٢
‫א‬  1  ٠ ‫א‬   ٢      ‫א‬ ‫א‬
‫א‬‫א‬?‫?א‬??‫א‬‫א‬
 KE2F2‫א‬‫א‬??
 
 W‫א‬‫א‬‫א‬‫א‬?‫א‬‫?א‬‫א‬‫א‬
 1  2 0 , 2  21 , 4  2 2 , 8  2 3 , 16  2 4 , 32  2 5 , ......
 
 W‫א‬‫א‬?‫א‬‫?א‬‫א‬‫א‬‫א‬‫א‬
  0.5  2 1 , 0.25  2 -2 , 0.125  2 3 , 0.0625  2 -4 , ......
 
 ١٠‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  
 K٢‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬
 
 
‫א‬‫א‬ ٢٠  0‫א‬‫א‬‫א‬‫א‬‫א‬W٣
 W‫א‬‫א‬
 
3
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 ‫א‬‫א‬
 ‫א‬‫א‬
0
٠
١
1
١٠
2
١١
3
١٠٠
4
١٠١
5
١١٠
6
١١١
1000
1001
1010
1011
1100
1101
1110
1111
١٠٠٠٠
7
8
9
10
11
12
13
14
15
١٦
 ١٠٠٠١
 ١٧
 ١٠٠١٠
 ١٨
 ١٠٠١١
 ١٩
 ١٠١٠٠
 ٢٠
 
 
 W‫א‬‫א‬‫א‬‫א‬‫א‬K٣
 
 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬K١{٣
‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬
 K‫א‬‫א‬،٢‫א‬
 
 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W٤
4
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
  (1) 110101
,
(2) 0.1101
,
(3) 101.111
 
 W‫א‬
(1) 110101  (1  2 0 )  (0  21 )  (1  2 2 )  (0  2 3 )  (1  2 4 )  (1  2 5 )
 
 (1  1)  (0  2)  (1  4)  (0  8)  (1  16)  (1  32)
 1  0  4  0  16  32  53
 
(2) 0.1101  (1  2 -1 )  (1  2 2 )  (0  2 -3 )  (1  2 4 )
 (1  0.5)  (1  0.25)  (0  0.125)  (1  0.0625)
 0.5  0.25  0  0.0625  0.8125
(3) 101.111  (1  2 0 )  (0  21 )  (1  2 2 )  (1  2 1 )  (1  2 2 )  (1  2 3 )
 (1  1)  (0  2)  (1  4)  (1  0.5)  (1  0.25)  (1  0.125)
 
 1  0  4  0.5  0.25  0.125  5.875
 
 
 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬K٢{٣
‫؛‬‫א‬‫א‬‫א‬‫א‬
 W‫א‬‫א‬‫א‬‫א‬،
 
 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬K١{٢{٣
 K‫א‬ ‫א‬‫א‬‫א‬‫א‬2‫א‬E١
 K‫א‬ ‫א‬‫א‬‫א‬‫א‬2‫א‬‫א‬E٢
 K٠‫א‬E٣
 
 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W٥
  (1) 12 , (2) 19 , (3) 45
5
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 W‫א‬
 
 
 W‫א‬‫א‬‫א‬2E١
12
 0‫א‬  6 
2
6
 0‫א‬  3
2
3
1‫א‬  1
2
1
 1‫א‬  0
2
 
KE1210  1100 2 F1100W‫א‬‫א‬12
 
 W‫א‬‫א‬‫א‬2E٢
19
1‫א‬  9
2
9
 1‫א‬  4
2
4
 0‫א‬  2
2
2
 0‫א‬  1
2
1
 1‫א‬  0
2
 
 KE1910  10011 2 F10011W‫א‬‫א‬19
 
 W‫א‬‫א‬‫א‬2E٣
6
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
45
 22
2
22
 0‫א‬  11
2
11
 1‫א‬  5
2
5
 1‫א‬  2
2
2
 0‫א‬  1
2
1
 1‫א‬  0
2
 1‫א‬
 
 KE 4510  101101 2 F101101W‫א‬‫א‬45
 
 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬K٢{٢{٣
‫א‬ ‫א‬   ،       2  ‫א‬  E١
 K،‫א‬‫א‬‫א‬ ‫א‬‫א‬
 2‫א‬‫א‬‫א‬‫א‬E٢
 K،‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬،
 K0‫א‬‫א‬E٣
 
 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W٦
  (1) 0.6875
,
(2) 0.3125
 W‫א‬
 ‫א‬ ‫א‬    ‫א‬ ‫א‬  2    E1
 1‫א‬‫א‬ 0.6875  2  1.375
 0‫א‬‫א‬ 0.375  2  0.75
 1‫א‬‫א‬ 0.75  2  1.5
7
 W‫א‬
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 1‫א‬‫א‬ 0.5  2  1.0
 
 K٠‫א‬‫א‬
 
 KE 0.687510  0.1011 2 F0.1011W‫א‬‫א‬0.6875‫א‬
 
 ‫א‬ ‫א‬    ‫א‬ ‫א‬  2    E2
 0‫א‬‫א‬ 0.3125  2  0.625
 W‫א‬
 1‫א‬‫א‬ 0.625  2  1.25
 0‫א‬‫א‬ 0.25  2  0.5
 1‫א‬‫א‬ 0.5  2  1.0
 
 K٠‫א‬‫א‬
 KE 0.312510  0.01012 F0.0101W‫א‬‫א‬0.3125‫א‬
 
 ‫؟‬‫א‬‫א‬‫א‬‫א‬91.15625‫א‬‫א‬W٧
 W‫א‬
 W2‫א‬‫א‬
91
1‫א‬  45
2
45
 1‫א‬  22
2
22
 0‫א‬  11
2
11
 1‫א‬  5
2
5
 1‫א‬  2
2
8
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
2
 0‫א‬  1
2
1
 1‫א‬  0
2
 KE 9110  1011011 2 F١٠١١٠١١W‫א‬‫א‬٩١
 
 W2‫א‬‫א‬ 
 0‫א‬‫א‬ 0.15625  2  0.3125
 0‫א‬‫א‬ 0.3125  2  0.625
 1‫א‬‫א‬ 0.625  2  1.25
 0‫א‬‫א‬ 0.25  2  0.5
 1‫א‬‫א‬ 0.5  2  1.0
 F 0.00101 W‫א‬ ‫א‬   0.15625 ‫א‬ 
KE 0.15625 10  0.00101 2
 
 1011011.00101W‫א‬‫א‬91.15625‫א‬‫א‬
 KE 91.15625 10  1011011 .00101 2 F
 
 W‫א‬‫א‬‫א‬‫א‬K٤
 W‫א‬‫א‬‫א‬K١{٤
?‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
‫א‬‫א‬‫א‬9‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬‫א‬K?‫א‬‫א‬‫א‬،‫א‬
 W‫א‬‫א‬
000
0 1 1
1 0 1
1  1  0 , with a carry of 1 to the next column
1  1  1  1 , , with a carry of 1 to the next column
9
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 W‫א‬‫א‬‫א‬‫א‬W٨
  (1) 10  11 , (2) 11  11 , (3) 110  100 , (4) 111  101
 W‫א‬
 
1
1 0
 1 1
(1)
1 0 1
 10  11  101W
 
1
1
1 1
 1 1
(2)
1 1 0
 11  11  110 W
1
1 1 0
 1 0 0
(3)
1 0 1 0
 110  100  1010 W
1
1
1
1 1 1
 1 0 1
( 4)
1 1 0 0
 111  101  1100 W
 
 
 W‫א‬‫א‬‫א‬K٢{٤
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬
 K
10
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W٩
 1101 , 1010.01 , 0.010101 , 10000
 W‫א‬
 
 10000  1101  1010.01  0.010101
 
‫א‬‫א‬‫א‬‫א‬W
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬
 W‫א‬
000
1 0 1
 
11  0
0  1  1 , , with a borrow of 1 from the next column
 
 
 W‫א‬‫א‬‫א‬‫א‬W١٠
  (1) 11  10 , (2) 101  11
1 1
 W‫א‬
1 0
0 1
(1)
 11  10  1W
 
0
1
1 0 1
1 1
( 2)
0 1 0
 101  11  10 W
 
11
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 W‫א‬‫א‬‫א‬K٣{٤
‫א‬  ‫א‬  ‫א‬ ‫א‬    ‫א‬ ‫א‬  ‫א‬‫א‬ 
 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 
 
00  0
0 1  0
1 0  0
11  1
 
 W‫א‬‫א‬‫א‬‫א‬W١١
  (1) 11  11 , (2) 101  111 , (3) 1001  1011
1 1

 
1 1
1 1
(1)

 W‫א‬
1 1
1 0 0 1
 
 11  11  1001 W
 
1 0 1

1 1 1
1 0 1
( 2)

 
1 0 1
1 0 1
1 0 0 0 1 1
 
12
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 101  111  100011 W
 
 
1 0 0 1

1 0 1 1
1 0 0 1
(3)

 
1 0 0 1
0 0 0 0
1 0 0 1
1 1 0 0 0 1 1
 
 1001  1011  1100011 W
 
 W‫א‬‫א‬‫א‬K٤{٤
 K‫א‬‫א‬‫א‬‫א‬‫א‬
 
 W‫א‬‫א‬‫א‬‫א‬W١٢
  (1) 110  11 , (2) 110  10
 W‫א‬
10
11 110
11
000
 110  11  10 W
 
13
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
11
10 110
10
010
 
10
00
 110  10  11 W
 
 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬K٥
0 ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ n ‫א‬W١
 K 2 n  1
0‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬8‫א‬W١٣
 K655350‫א‬‫א‬‫א‬‫א‬16‫א‬ K255
 
 ‫؟‬‫א‬‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 K
 W J‫א‬‫א‬.1.5
 1‫א‬ 0 ‫א‬‫א‬ ‫א‬‫א‬
 K‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬
 
8‫א‬ J‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W١٤
  (1) 19 , (2)  19 W
 W‫א‬
 K10011W‫א‬‫א‬19‫א‬٥‫א‬2‫א‬E1
 K00010011W8‫א‬ J‫א‬‫א‬
 K10010011W J‫א‬‫א‬-19E2
 
W‫א‬‫א‬ J‫א‬‫א‬‫א‬‫א‬‫א‬W١٥
14
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
  (1) 10010 , (2) 00101
 
 
W‫א‬
(1) 10010  (10)  (1 21  0  2 0 )  2
(2) 00101   (101)   (1 2 2  0  21  1 2 0 )  4  0  1  5
 
 
‫א‬‫א‬‫א‬ J‫א‬‫א‬ n ‫א‬W٢
( 2 n 1  1)  ( 2 n 1  1) ‫א‬
 
‫א‬‫א‬‫א‬ J‫א‬‫א‬8‫א‬W١٦
 K127 -127 ‫א‬
K32767 -32767‫א‬‫א‬‫א‬‫א‬16‫א‬
 
 W1‫א‬‫א‬.2.5
   J ‫א‬ ‫א‬    ‫א‬ ‫א‬‫א‬  ‫א‬ ‫א‬ 
K‫א‬10،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 
W8‫א‬1‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W17
(1) 19 , (2)  19
 W‫א‬
 J‫א‬‫א‬ 19‫א‬ 14‫א‬ 1‫א‬E١
 K1‫א‬‫א‬‫א‬00010011W8‫א‬
00010011،1‫א‬‫א‬-19‫א‬E٢
 K11101100W1
W‫א‬‫א‬1‫א‬‫א‬‫א‬‫א‬‫א‬W18
  (1) 00010111 , (2) 11101000
15
 W‫א‬
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
(1) 00010111   (10111)  1 2 4  0  2 3  1 2 2  1 21  1 2 0
 16  0  4  2  1  23
 
(2) 11101000  (00010111)  (10111)
 (2 4  2 2  21  2 0 )  (16  4  2  1)  23
 
 
‫א‬‫א‬‫א‬‫א‬1‫א‬‫א‬ n ‫א‬W٣
n 1
n 1


(
2

1
)
(

2
 1) 
 
 
‫א‬‫א‬‫א‬1‫א‬‫א‬8‫א‬W19
 K127 -127 ‫א‬
 K32767 -32767‫א‬‫א‬‫א‬‫א‬16‫א‬
 
 W2‫א‬‫א‬.3.5
   J ‫א‬ ‫א‬    ‫א‬ ‫א‬‫א‬  ‫א‬ ‫א‬ 
‫א‬10،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 K‫א‬‫א‬ 1
 
W8‫א‬2‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W20
  (1) 19 , (2)  19
 W‫א‬
 J‫א‬‫א‬ 19‫א‬ 14‫א‬ 1‫א‬E١
 K2‫א‬‫א‬‫א‬00010011W8‫א‬
 
00010011،2‫א‬‫א‬-19‫א‬E٢
K11101101W111101100W1
 
16
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 WE8‫א‬F‫א‬‫א‬‫א‬‫א‬
2 0 , 21 , 2 2 , 2 3 , 2 4 , 2 5 , 2 6 ,  2 7
 
W‫א‬‫א‬2‫א‬‫א‬‫א‬‫א‬‫א‬W21
  (1) 01010110 , (2) 10101010
 W‫א‬
(1) 01010110  0  ( 2) 7  1  2 6  0  2 5  1  2 4
 0  2 3  1  2 2  1  21  0  2 0
 2 6  2 4  2 2  21  64  16  4  2  86
 
(2) 10101010  1  (2) 7  0  2 6  1  2 5  0  2 4
 1  2 3  0  2 2  1  21  0  2 0
 (2) 7  2 5  2 3  21  128  32  8  2  86
 
‫א‬‫א‬‫א‬‫א‬2‫א‬‫א‬ n ‫א‬W٤
( 2 n 1  1)  ( 2 n 1 ) 
 
‫א‬‫א‬‫א‬2‫א‬‫א‬8‫א‬W22
 K127 -128 ‫א‬
 K32767 -32768‫א‬‫א‬‫א‬‫א‬16‫א‬
 
‫א‬2‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬ ‫א‬
‫א‬1‫א‬‫א‬
W
  00000000 , 11111111
W J‫א‬
17
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
  00000000 , 10000000
 
 W(Hexadecimal System)‫א‬‫א‬K6
 W ‫א‬‫א‬‫א‬‫א‬
  0 , 1 ,2 , 3 , 4, 5, 6 , 7 , 8 , 9 , A , B , C , D , E , F
 KE16F16‫א‬‫א‬
 
 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W‫א‬
 
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W23
 W‫א‬‫א‬
 ‫א‬
 0
 1
2
3
 4
5
6
7
8
9






A
B
C
D
E
F
 ‫א‬
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
 ‫א‬
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
 
 
 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬K7
 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬.1.7
18
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
   ‫א‬ ‫א‬ ‫א‬  ‫א‬  ‫א‬  ‫א‬‫א‬ 
 2‫א‬? 16 2‫א‬‫א‬ ‫א‬ ‫א‬ ‫א‬
‫א‬‫א‬‫א‬ 16‫א‬ 16‫א‬ 2‫א‬
 K?‫א‬‫א‬
 
 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W24
  (1) 9719 , (2) 0.78125
 
 W‫א‬
 W‫א‬‫א‬‫א‬16E١
9719
 607 
 7‫א‬
16
607
 37
 15‫א‬
16
37
5‫א‬  2
16
2
 2‫א‬  0
16
 
  25F 716 W‫א‬‫א‬9710
  (9710 )10  (15 F 7)16 W
 
 ‫א‬ ‫א‬    ‫א‬ ‫א‬  16    E2
 12‫א‬‫א‬ 0.78125  16  12.5
 8‫א‬‫א‬ 0.5  16  8.0
 W‫א‬
 
  0.C 816 W‫א‬‫א‬0.78125
  (0.78125 )10  (0.C 8)16 W
19
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬.2.7
‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 WK162‫א‬‫א‬
 
 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W25
  39.B816
 W‫א‬
39.B816  (3  161 )  (9  16 0 )  (11  16 1 )  (8  16 2 )
 
 
 (3  16)  (9  1)  (11  0.0625)  (8  0.00390625)
 48  9  0.6875  0.03125  57.71875
 
57.71875W‫א‬‫א‬ 39.B816 
  (39.B8)16  (57.71875 )10 W
 
 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬.3.7
 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 K4 ‫א‬‫א‬‫א‬E١
 K‫א‬ ‫א‬‫א‬4‫א‬E٢
K‫א‬‫א‬‫א‬‫א‬‫א‬E٣
 
W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W26
  (1) 1100101000 10111 , (2) 1111110000 1101001
20
W‫א‬
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
(1) 1100101000 10111  0110010100 010111
 0110 0101 0001 0111  (6517)16
 
(2) 1111110000 1101001  0001111110000 1101001
 0001 1111 1000 0110 1001  (1F 869)16
 
 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬.4.7
 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 K4‫א‬‫א‬‫א‬E١
 K‫א‬‫א‬E٢
 
W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W27
  (1) 10 A416
 
, ( 2) CF 8 E16
 
(1) (10 A4)16  0001 0000 1010 0100  1000010100100
W‫א‬
(2) (CF 8 E )16  1100 1111 1000 1110  1100111110 001110
 WW28
  (59.125)10  (111011 .001) 2  (3B.2)16
 
 
 
 
 
 
 
 
21
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 
 
 
 
 
 
 
 
 
 
 
 ‫א‬W1
W10‫א‬‫א‬‫א‬‫א‬
  (1) 10 , (2) 100 , (3) 10000 , (4) 0,001
 ‫؟‬٤‫א‬W2
W‫א‬‫א‬‫א‬‫א‬W3
  (1) 11 , (2) 100 , (3) 1110 , (4) 100101
W‫א‬‫א‬‫א‬‫א‬W4
  (1) 11 .01 , (2) 10.01 , (3) 0.1110 , (4) 1001.011
‫؟‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W5
  (1) 17 , (2) 35 , (3) 49 , (4) 68 , (5) 82 , (6) 205
W‫א‬‫א‬‫א‬‫א‬W6
  (1) 10 , (2) 17 , (3) 63 , (4) 628 , (5) 125 , (6) 186
W‫א‬‫א‬‫א‬‫א‬W7
  (1) 0.32 , (2) 0.246 , (3) 0.0981 , (4) 12.34
W‫א‬‫א‬‫א‬W8
  (1) 10  11 , (2) 110  10 , (3) 111  111 , (4) 1010  1011
W‫א‬‫א‬‫א‬W9
22
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
  (1) 11  10 , (2) 110  10 , (3) 111  101 , (4) 1110  1011
W‫א‬‫א‬‫א‬ W10
  (1) 10  11 , (2) 110  10 , (3) 111  111 , (4) 1010  1011
W‫א‬‫א‬‫א‬ W11
  (1) 100  10 , (2) 1001  11 , (3) 1100  100
E‫א‬‫א‬‫א‬F8‫א‬‫א‬‫א‬‫א‬
 ‫א‬W12
W‫א‬‫א‬‫א‬‫א‬
  (1)  29 , (2)  85 , (3) 100 , (4)  123 , (5)  99 , (6) 57
W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W13
  (1) 110011 , (2) 10110 , (3) 1100010 , (4) 010100101
 
W‫א‬‫א‬‫א‬‫א‬W14
  (1) 5516 , (2) A7516 , (3) 8 B 5 D16 , (4)1CF 316
W‫א‬‫א‬‫א‬‫א‬W15
  (1) 110011 , (2) 10110 , (3) 1100010 , (4) 010100101
W‫א‬‫א‬‫א‬‫א‬W16
  (1) 5516 , (2) A7516 , (3) 8 B 5 D16 , (4)1CF 316
W‫א‬‫א‬‫א‬‫א‬W17
  (1) 16 , (2) 377 , (3) 2784 , (4) 1024
 
 
 
 
 
 
 
 
 
23
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 
 
 
 
 
 
‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
24
 

 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 
 
 
 
 
 
 
 
‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
25
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
‫א‬،‫א‬‫א‬‫א‬‫א‬W‫א‬‫א‬
 K‫א‬‫א‬
 
 W‫א‬‫א‬‫א‬W‫א‬‫א‬‫א‬
K‫א‬‫א‬‫א‬‫א‬‫א‬ 
K‫א‬‫א‬‫א‬‫א‬ 
K‫א‬‫א‬‫א‬‫א‬‫א‬ 
K‫א‬‫א‬‫א‬‫א‬‫א‬‫ א‬
 K‫א‬‫א‬‫א‬‫א‬ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
26
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 ‫א‬‫א‬‫א‬‫א‬
 
 WK1
   ‫א‬       ‫א‬ ‫א‬ 
  ‫א‬  ،‫א‬         ‫א‬ ‫؛‬ ‫א‬
‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬ ‫؛‬
 K‫א‬E‫א‬‫א‬F‫א‬‫א‬
K‫א‬‫א‬‫א‬‫א‬
 (True : T ) ‫א‬‫א‬‫א‬‫א‬‫א‬
 K‫א‬‫א‬K ( False : F )
 
 W‫א‬‫א‬‫א‬‫א‬‫א‬K2
‫א‬‫א‬
 W‫א‬‫א‬
 K‫א‬
 W١
K‫א‬‫א‬ Ea
180  ‫א‬‫א‬‫א‬‫א‬ Eb
4  2  1 Ec
2
x  16 W x  3 Ed
‫؟‬‫א‬ Ee
K‫א‬ Ef
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
    ‫א‬ ‫א‬  K ‫א‬  ‫א‬  ،
 K‫א‬ ،‫א‬
‫א‬‫א‬‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬K‫א‬
K
27
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 W٢
 Ea
 4 cm  ABCD ‫ א‬Eb
‫א‬‫ א‬Ec
‫א‬ ‫א‬ 
  ، ‫א‬   ‫א‬ ‫א‬  ‫א‬ ‫א‬ ‫א‬
K????W‫א‬‫א‬‫א‬‫א‬‫א‬
‫?א‬ W‫א‬ ‫א‬  ? 4 cm  ?  ? ABCD  ‫?א‬ W‫א‬ ‫א‬
 K?‫?א‬?
 
‫א‬‫א‬‫א‬‫א‬‫א‬
 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬ ‫א‬ ‫א‬
‫א‬ ‫א‬‫א‬   p , q , r , s , ....  ‫א‬ ‫א‬ ‫א‬  
 K‫א‬‫א‬
 ‫א‬‫א‬‫א‬K3
W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 W "and " ??W(conjunction)‫א‬‫א‬

 W " or" ??W (disjunction)‫א‬‫א‬

W " not " ?KKKK?W(negation) ‫א‬‫א‬

 

  p  q ‫א‬.1.3
‫א‬‫א‬
  p  q W‫א‬‫א‬KE "and " F??‫א‬‫א‬
‫א‬‫א‬‫א‬ p  q ‫א‬K? q  p ?
q  p ‫א‬ p  q ‫א‬K q  p 
 K‫א‬‫א‬ p  q ،‫א‬
28
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 ‫א‬       p  q  ‫א‬   
 K 1 ‫א‬‫א‬
 
p  q   q   p  ‫א‬  ‫א‬  ‫א‬ ‫א‬   
  q   p  ‫א‬   ‫א‬  ‫א‬ ‫א‬   ،
 K‫א‬ p  q
 
p
q
 pq
T
T
T
T
F
F
F
T
F
F
F
F
  1 ‫א‬
 
 (F ) ‫א‬ (T ) ‫א‬‫א‬‫א‬ 
‫א‬_‫א‬  p  q ، q  p ‫א‬‫א‬‫א‬
 K  q  p _
 
 W‫א‬‫א‬‫א‬‫א‬W٣
‫א‬‫א‬‫א‬‫א‬‫ א‬E١
4  2  7 ‫א‬‫ א‬E٢
2‫א‬6‫א‬ E٣
 06‫א‬ E٤
 
  ،  ‫א‬ ‫א‬‫א‬   ‫א‬ ‫א‬  ‫א‬ ‫א‬ 
 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 
 
29
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
  p  q ‫א‬K٢{٣
‫א‬‫א‬‫א‬
p  q W‫א‬‫א‬‫א‬KE " or" F??‫א‬
 
‫א‬‫א‬‫א‬ p  q ‫א‬K? q  p 
 p ‫א‬‫א‬‫א‬
 ‫א‬ p  q ‫א‬K q  p 
 q  p ‫א‬ p  q ، q 
 K‫א‬
‫א‬‫א‬ p  q ‫א‬
 K 2 ‫א‬
 p ‫א‬  ‫א‬ p  q ‫א‬‫א‬
  K‫א‬ q 
 p 
q
 pq
T
T
T
T
F
T
F
T
T
F
F
F
 ٢‫א‬
 
 W‫א‬‫א‬‫א‬‫א‬W٤
‫א‬‫א‬‫ א‬E١
4  2  7 ‫א‬‫ א‬E٢
2‫א‬6‫א‬ E٣
06‫א‬ E٤
  ‫؛‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 K‫א‬‫א‬‫א‬‫א‬
30
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 K‫א‬??L
‫א‬، q  p ? q  p ?‫א‬
 K? q  p ?‫א‬??K‫א‬
 
‫א‬  ‫א‬    ،     q    p   
 K
‫א‬??‫א‬ L2  L1 ‫?א‬‫א‬
 KE‫א‬F
K‫א‬‫א‬‫א‬??‫א‬‫א‬
 K? q | p ? ‫א‬ p  q
 
  p ‫א‬.3.3
K‫א‬‫א‬
 p ‫א‬‫א‬‫א‬‫א‬،‫א‬?KKKKKKK?‫א‬
 K p  p W
‫א‬ p  p ‫א‬، p  p 
 W‫א‬ p  p 
p
 p
T
F
F
T
 
 
 
 
 W‫א‬‫א‬‫א‬W٥
‫ א‬E١
‫א‬‫א‬ E٢
‫א‬ E٣
K?‫?א‬?‫א‬?W‫א‬‫א‬
 K?‫א‬‫א‬??‫א‬‫א‬?W‫א‬‫א‬
31
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
W‫א‬?‫א‬?W‫א‬‫א‬
 K‫א‬?‫א‬?
 
 
، q  p ‫א‬‫א‬‫א‬‫א‬
‫א‬‫א‬‫א‬
K‫א‬ ‫א‬‫א‬‫א‬‫א‬
‫א‬ p , q , r ‫א‬ 
 W‫א‬‫א‬
 p  
q
r
T
T
T
T
T
F
T
F
T
T
F
F
F
T
T
F
T
F
F
F
T
F
F
F
 
 
 WE‫א‬F‫א‬‫א‬‫א‬K٤
‫א‬ ‫א‬ ‫א‬‫א‬      ‫א‬  P ( p, q, r ,....)  
KE F  ,  ,
‫א‬‫א‬‫א‬E‫א‬‫א‬F p, q, r ,....
،‫א‬‫א‬‫א‬  ‫א‬‫א‬‫א‬
 K‫א‬‫א‬ ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 W‫א‬‫א‬‫א‬‫א‬‫א‬
32
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
   ( p  q ) W‫א‬‫א‬‫א‬W٦
 
 
 W‫א‬
 p q  q  p  q ( p  q )
T T F
 F
T F T
 T
F T F
 F
F F T
 F
 T
 F
 T
 T
 W‫א‬
‫א‬  ‫א‬ ‫א‬  
‫א‬، q  p
  ‫א‬ ‫א‬  
   K‫א‬‫א‬  (F )   (T )
‫א‬‫א‬  ‫א‬ ‫א‬ ‫א‬  
 ‫א‬      ‫א‬   K‫א‬ ‫א‬   ‫א‬
‫א‬ ‫א‬K‫א‬  ,  ,
‫א‬‫א‬‫א‬‫א‬‫א‬
 K‫א‬‫א‬‫א‬‫א‬
   ( p  q )   ‫א‬ ‫א‬      
‫א‬ ‫א‬  ‫א‬ ‫א‬  ،‫א‬ ‫א‬ q   p  ‫א‬ 
 W‫א‬‫א‬‫א‬K‫א‬
p
q ( p  q )
T
T
T
F
F
T
F
F
T
F
T
T
 
 
 
 
 
 
 
 W‫א‬‫א‬‫א‬K٥
‫א‬‫א‬،‫א‬ (T ) ‫א‬‫א‬‫א‬ 
‫א‬‫א‬‫א‬،‫א‬‫א‬ ‫א‬E‫א‬F
 K ‫א‬
33
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
‫א‬‫א‬F (F )    ‫א‬‫א‬
 K‫א‬E (F ) 
 
  a ) p  ( p  q ) ‫א‬ b) ( p  q )  ( p  q ) W  W٧
 W‫א‬
 p  q  p  p  q  p  ( p  q)
‫א‬ (a
T T F
T F F
،‫א‬ (T )  ‫א‬
 T
 T
F T T  T
F F T  F
p  ( p  q ) W‫א‬
 T
 T
W‫א‬
T

 T
  p  ( p  q)  T
 
 E‫א‬‫א‬  F
 
 W‫א‬‫א‬ (b
pq
 ( p  q)  ( p  q)
 
 T
 T
F
F
F
F
 
 T
 F
F
T
F
F
 p q
 pq  pq
T T
 T
T F
 F
F T
 F
F F
 F
 
 
 
 
،‫א‬، (F )  ( p  q )  ( p  q ) W
  ( p  q )  ( p  q )  F W
 
 W‫א‬‫א‬K٦
‫א‬   ‫א‬ E  F       
  P ( p, q,...)  Q ( p, q,...) W‫א‬‫א‬،
 KE‫א‬‫א‬  F
34
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
  p  q  p  q W‫א‬‫א‬E‫א‬F‫א‬‫א‬W٨
W‫א‬W‫א‬
 p q  p  q
T
T  T
T
F  F
F
T  F
F
F  F
  p q
pq
 T T
 T F
 F T
F

 T
 T
 T
 F F
p
q
pq
F
F
F
F
T
T
T
F
T
T
T
T
 
 
 
 
 
 
 W p  q  p  q ‫א‬‫א‬
 pq  pq
 
 W‫א‬‫א‬K٧
‫א‬‫א‬‫א‬‫א‬، p, q, r ‫א‬‫א‬‫א‬‫א‬‫א‬
 K‫א‬‫א‬‫א‬‫א‬
 ‫א‬‫א‬‫א‬E١
 

a) p  q  q  p
 ‫א‬‫א‬E٥
 
b) p  q  q  p
a) p  T  T
b) p  F  p
c) p  F  F
d) p  T  p
 
 ‫א‬‫א‬E٢
 
a) p  (q  r )  ( p  q)  r
a ) p  (q  r )  ( p  q )  ( p  r )
 ‫א‬‫א‬E٦

b) p  ( q  r )  ( p  q )  r
 ‫א‬‫א‬E٣



a) p  p  T
c) T  F
d) F  T
 ‫א‬E٧
 p p
b) p  ( q  r )  ( p  q )  ( p  r )
 ‫א‬‫א‬E٤
 
a) p  p  p
b) p  p  F

 ‫א‬E٨
 
b) p  p  p
35
a) ( p  q)  p  q
b) ( p  q )  p  q
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 W٩
‫א‬‫א‬E‫א‬F (2a ) ‫א‬
 K‫א‬‫א‬
 p  ( q  r ) ‫א‬‫א‬ ‫א‬ ( p  q )  r 
  ( p  q )  r  p  ( q  r ) W،‫א‬‫א‬
 
r
pq
( p  q)  r
qr
p  (q  r )
T
T
T
T
T
T
T
T
F
T
T
T
T
 
T
F
T
T
T
T
T
 
T
F
F
T
T
F
T
F
T
T
T
T
T
T
 
F
T
F
T
T
T
T
F
F
T
F
T
T
T
F
F
F
F
F
F
F
p
q
T
 
 
 
 
 
 
 E‫א‬F ( p  q )  p  p W‫א‬‫א‬W١٠
 p  q  p  q  ( p  q)  p
T
T  T
T
F  F
F
T  F
F
F  F
T
T
F
F
 
W‫א‬
E p F‫א‬‫א‬
W‫א‬،E ( p  q )  p F‫א‬
  ( p  q)  p  p
K‫א‬‫א‬‫א‬
  ( p  q )  p  p ‫א‬
 
K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 K‫א‬‫א‬‫א‬‫א‬
 
36

 
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 W‫א‬‫א‬‫א‬‫א‬W١١
  a ) ( p  q)  ( p  q)  p


b) ( p  q )  ( p  q )  q  T
 
 W‫א‬
 ‫א‬‫א‬
  ( a
E٨ a F  
 ‫א‬
( p  q)  ( p  q)  ( p  q)  ( p  q)
 p  ( q  q )  
E٣ b F‫א‬
E٦ a F‫א‬
  p  T 
E٥ d F‫א‬
    p 
 
 
  (b
 ‫א‬‫א‬
( p  q)  ( p  q) q  q  ( p  p) q
 q  F  q  
  q  q  
 ‫א‬
E٣ a F‫א‬
E٦ b F‫א‬
E٥ b F‫א‬
  T   E٦ a F‫א‬
 
 
‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 W‫א‬‫א‬
 
 W‫א‬W١٢
  ( p  q)  ( p  q  r )  ( p  q)
 
 W‫א‬
37
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 ‫א‬‫א‬
 ‫א‬
 ( p  q )  [ p  ( q  r )]  ( p  q )
 ( p  q )  { p  [( q  r )  q ]}
E٣ b F‫א‬
E١٠F‫א‬ 
 ( p  q)  ( p  q)
E٣ b F‫א‬ 
 p  (q  q) 
E٦ a F‫א‬ 
 p T

E٨ a F 
 E٥ d F ‫א‬
 p
 
 
 K‫א‬‫א‬‫א‬‫א‬K٨
 W‫א‬‫ א‬.1.8
‫א‬ q  p  (if p then q ) ? q  p ‫א‬?‫א‬‫א‬
 K p  q W،
 K? q ‫א‬ p ?? q  p ? p  q ‫א‬‫א‬
 W‫א‬‫א‬
 p q  p  q
T T  T
T F  F
F T  T
F F  T
 p    q   ‫א‬  ‫א‬  p  q    ‫א‬
 K ‫א‬‫א‬،
‫א‬ ‫א‬      ‫א‬ ‫א‬ 
‫א‬‫א‬‫א‬‫א‬،‫א‬
‫א‬‫א‬K‫א‬
 p ? q  p ‫א‬?
‫א‬‫א‬‫א‬، q  p ‫א‬‫א‬
 W
 
 W‫א‬‫א‬‫א‬W١٣
  q W‫א‬‫א‬‫א‬، p W‫א‬
38
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
‫א‬‫א‬‫א‬‫א‬‫א‬W p  q
 K q  p  p ‫א‬
 
‫א‬‫א‬‫א‬‫א‬‫א‬ 
 W‫א‬‫א‬‫א‬
 
  p : 3  8 , q : 3  5  8 W‫א‬‫א‬‫א‬W١٤
  3  5  8  3  8 ‫א‬W p  q
K p  q ‫א‬‫א‬ p  q 
 K p  q ‫א‬ q  p 
 
  p W‫א‬، q : 4  2  3 W‫א‬‫א‬‫א‬W١٥
  4  2  3 ‫א‬‫א‬W p  q
‫א‬‫א‬  ‫א‬‫א‬ q   p 
 K‫א‬ p  q
 
W ( ) ‫א‬‫א‬ () ‫א‬‫א‬‫א‬‫א‬‫א‬L
 p q  p  q
p
pq
T T  T
F
 T
T F  F
F T  T
F
T
 F
 T
F F  T
T
 T
 pq pq
 ‫א‬   ‫א‬ 
 W‫א‬‫א‬‫א‬
 
 
 
 
  p  q  p  q W‫א‬‫א‬‫א‬‫א‬
 
 K? q  p ?? q  p ‫א‬?
39
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 W‫א‬‫א‬‫א‬W١٦
 pq , q p , pq , q p
 
 p q  p  q q  p  p
q
T T  T
 T
F
T F  F
F
F T  T
 T
 F
F
T
T
F
F F  T
 T
T
T
 p  q q  p
T
T
T
F
F
T
T
T
 
 W‫א‬
 
 
 
 
 
 W‫א‬‫א‬
 
pqq p
q p pq
 
 W
‫א‬ ‫א‬ ‫א‬  q  p    p  q    q  p 
 pq
 
 q  p ‫א‬ p  q ‫א‬‫א‬‫א‬
 K p  q 
 
 W‫א‬K p : x 2  4  q : x  4 WW١٧
 K‫א‬ x  4 E x  2 F x 2  4 ‫א‬W p  q
 x  1     ،  ‫א‬ x 2  4   x  4   ‫א‬ W q  p
 K x  3
 x  1 ،‫א‬ x  4  x 2  4 ‫א‬W p  q
 K x  3 
40
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 
 p  q  p  q  ( p  q)
 W‫א‬‫א‬‫א‬
 
T T  T
 F
T F  F
F T  T
 T
 F
  p  q  p  q W
 W‫א‬‫א‬
F F  T
 F
  ( p  q)  ( p  q)  p  q
 

 
 
 W‫א‬‫א‬.2.8
‫א‬‫א‬‫א‬ ‫א‬
 K p  q W? q ‫א‬‫א‬ p ?
 q  p ‫א‬‫א‬‫א‬
 p q  p  q
T T  T
T F  F
F T  F
F F  T
‫א‬‫א‬K‫א‬
 W‫א‬‫א‬
 W‫א‬‫א‬‫א‬
  p  q  ( p  q)  (q  p)
 E‫א‬‫א‬‫א‬F
 
 
  : ‫א‬K٩
  ‫א‬ P1 , P2 , P3 ,....., Pn  ‫א‬‫א‬        ‫א‬
 W،‫א‬ Q ‫א‬‫א‬
  P1 , P2 , P3 ,....., Pn  Q
‫א‬E F     P1 , P2 , P3 ,....., Pn  Q  ‫א‬
، P1 , P2 , P3 ,....., Pn  ‫א‬  E Q F‫א‬
 K‫א‬‫א‬EF
 
41
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 W‫א‬‫א‬  W١٨
EF، (b) p  q , q  p EF
  (a ) p , p  q  q
 W‫א‬
  ‫א‬ p  q   p   (a
K‫א‬‫א‬‫א‬ q ‫א‬
 W‫א‬‫א‬‫א‬‫א‬
 p q  p  q
T
T
T
T
F
F
F
T
T
F
F
T
 
 
 
 
 
 
‫א‬‫א‬‫א‬
  q ‫א‬‫א‬‫א‬،‫א‬‫א‬
 p , p  q  q  W‫א‬K‫א‬‫א‬‫א‬‫א‬‫א‬
K
 
 W‫א‬ (a ‫א‬‫א‬‫א‬ (b
‫א‬‫א‬ q  p  q ‫א‬‫א‬
‫א‬‫א‬ p ‫א‬‫א‬،‫א‬‫א‬
‫א‬    ، E p  F ‫א‬ ‫א‬ ‫א‬    
 KEF p  q , q  p
 p q  p  q
T T
T
T F
F
F T
T
42
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
F F
T
 
 P1 , P2 , P3 ,....., Pn ‫א‬‫א‬‫א‬
،E ‫א‬F‫א‬ P1  P2  P3  .....  Pn ‫א‬‫א‬
 Q ‫א‬‫א‬EF P1 , P2 , P3 ,....., Pn  Q ‫א‬
 W،  P1  P2  P3  .....  Pn 
 W‫א‬‫א‬‫א‬KE‫א‬F ( P1  P2  P3  .....  Pn  Q )  T
 
  p  q , q  r  p  r ‫א‬  W١٩
 W‫א‬
‫א‬‫א‬ ( p  q )  ( q  r )  ( p  r ) ‫א‬
 W‫א‬
qr  pr
( p  q )  (q  r ) ( p  q )  (q  r )  ( p  r )
p
q
r
pq
T
T
T
T
T
T
T
T
T
T
F
T
F
F
F
T
T
F
T
F
T
T
F
T
T
F
F
F
T
F
F
T
F
T
T
T
T
T
T
T
F
T
F
T
F
T
F
T
F
F
T
T
T
T
T
T
F
F
F
T
T
T
T
T
 
‫א‬ ( p  q )  ( q  r )  ( p  r )  T 
 KEF p  q , q  r  p  r
 
 ‫؟‬‫א‬‫א‬W٢٠
 K    ‫א‬  ‫א‬  ‫א‬ ‫א‬   ‫א‬
‫א‬K‫א‬‫א‬، ‫א‬‫א‬
 K
43
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 
 W‫א‬
  P1 W‫א‬



  P2 W‫א‬‫א‬

 W‫א‬‫א‬‫א‬
  P1 W‫א‬
 P2 W‫א‬‫א‬
  P1 W‫א‬ p1  p 2 , p 2 W
W‫א‬‫א‬ p1  p 2 , p 2  p1 W‫א‬
  ( p1  p 2 )  p 2  p1  T
 P1  P2
 P1
 P2
T
T
F
F
 T
F
T
F
F
T
F
T
T
T
 F
 T
 T
F
F
 T
F
F
T
T
 T
T
 P1  p 2  ( p1  p 2 )  p 2  ( p1  p 2 )  p 2  p1
 T
 T
 
 K‫א‬ ( p1  p 2 )  p 2  p1  T ‫א‬‫א‬
 
 ‫א‬‫א‬K١٠
 Q ( p, q,...)   ‫א‬ Q ( p, q,...)  ‫א‬   P ( p, q,...)  ‫א‬  
 W، P ( p, q,...) 
  P ( p, q,...)  Q ( p, q,...)
p
q  p  q
T
T  T
T
F  T
F
T  T
F
F  F
 
 p  q  p W٢١
 W‫א‬ p  q
 
 
 
44
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 
  ، p  q    ‫א‬ ‫א‬     p   
 K p  p  q ‫א‬K‫א‬‫א‬‫א‬
 
، P ( p, q,...)  Q ( p, q,...) ‫א‬‫א‬
K‫א‬ P ( p, q,...)  Q ( p, q,...)  
 ‫א‬ P  Q  ‫א‬ ‫א‬  ‫א‬  ‫א‬  P  Q  ‫א‬ 
 W‫א‬‫א‬
 ‫א‬‫א‬K‫א‬
 W‫א‬‫א‬ Q ( p, q,...)  P ( p, q,...) 
Q ( p, q,...)  P ( p, q,...) 
EF P ( p, q,...)  Q ( p, q,...) ‫ א‬
E ‫א‬F‫א‬ P  Q ‫א‬‫ א‬
 
 
 
 
 
 
 
 
 
 
 
 
 
 
45
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 
 
 W‫א‬‫א‬‫א‬W١
K‫א‬ Ea
K‫א‬ Eb
K Ec
  45  58 Ed
K‫א‬‫א‬‫א‬‫א‬ Ee
 ‫؟‬ Ef
K‫א‬‫א‬ 2  2  5 ‫א‬ Eg
 
  p W?‫?א‬ q W?‫?א‬W‫א‬‫א‬‫א‬W٢
  a ) p , b ) p  q , c ) p  q , d ) q  p W‫א‬
 
 W‫א‬‫א‬‫א‬W٣
  p :‫א‬
  q : 
r : ‫א‬
 W  ,  ,
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 K‫א‬‫א‬ Ea
 K‫א‬‫א‬‫א‬ Eb
 K‫א‬‫א‬ Ec
       ‫א‬       Ed
K‫א‬
 
  p :‫א‬
46
 W‫א‬‫א‬‫א‬W٤
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
  q : 
 
r : ‫א‬
s : ‫א‬
 W‫א‬‫א‬‫א‬
a) p  q , p  r , p  s , q  r , q  s , r  s
b) p  q , p  r , p  s , q  r , q  s , r  s
 
  ( p  q )  ( p  q )  F W‫א‬‫א‬W٥
 
 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W٦




  a ) p  ( p  r ) , b) ( p  q )  ( p  q )  q , c ) p  ( p  q )  q
 
 W‫א‬‫א‬‫א‬‫א‬W٧
a) p  ( p  r )  ( p  q)  ( p  r )
  b) p  ( p  q )  p
c) ( p  q )  p  q
 
 W‫א‬‫א‬‫א‬‫א‬W٨
a) p  ( p  q)  T
 
b) { p  ( p  q )}  q  T
c) { p  ( p  q)}  {q  ( p  q)}  q
d ) {( p  q )  ( p  q)}  q  T
 
a) ( p  q)  ( p  q  r )
 W‫א‬W٩
  b) p  {q  ( p  q )}
c) ( p  q)  ( p  q  r )  ( p  q)
47
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 
 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W١٠
 K‫א‬‫א‬‫א‬ (a
‫א‬‫א‬،‫א‬‫א‬‫א‬ (b
 K‫א‬
 K ‫א‬‫א‬‫א‬‫א‬ (c
 
 W‫א‬‫א‬‫א‬W١١
  a) 3  8  11 ‫א‬ 1  3  4  b) 3  11  10 ‫א‬ 1  3  7 
  c) 3  8  11 ‫א‬ 1  3  7  d ) 3  8  10 ‫א‬ 1  3  4 
 
 W‫א‬‫א‬‫א‬W١٢
  p : ‫א‬ ABC ‫א‬ q : ‫א‬‫א‬ ABC ‫א‬
 p  q ‫א‬‫א‬‫א‬،‫א‬
 
:‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W١٣
 a) ‫א‬‫א‬‫א‬
  b) ‫א‬‫א‬‫א‬
 E p  q  p  q W‫א‬F
 
 W‫א‬‫א‬‫א‬W١٤
 K‫א‬‫א‬ (a)
 K ‫א‬‫א‬‫א‬‫א‬ (b)
 K‫א‬‫א‬ (c)
 
 
48
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 
 W  W١٥
(a) p  p  T
(b) [ p  ( p  q )]  q  T
(c ) ( p  q )  ( p  q )  T
(d ) ( p  q)  ( p  q)  T
(e) [( p  q )  (q  r )]  ( p  r )  T
( f ) ( p  q)  (q  p)  T
a) ( p  q)  (q  p)
 W‫א‬‫א‬  W١٦
  b) p  q  ( p  q )  ( p  q )
c) ( p  q)  r  p  (q  r )
 
  a) p  q , p  q 
 W  W١٧
  b) p  q, q  p 
  c) p  q, r  q, r  p 
 
 W‫א‬‫א‬  W١٨
 K‫א‬‫א‬،‫א‬‫א‬‫א‬
 
 W‫א‬‫א‬W١٩
 K 7 ‫א‬ 4  7 K 7  4  7 ‫א‬
 
 
 
 
 
 
49
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 
 
 
 
 
 
 
 
 
 
‫א‬‫א‬
 
 
‫א‬‫א‬‫א‬‫א‬‫א‬
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
50
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W‫א‬‫א‬
 
 W‫א‬‫א‬‫א‬W‫א‬‫א‬‫א‬
K‫א‬‫א‬‫א‬‫א‬‫ א‬
K‫א‬‫א‬‫א‬ 
K‫א‬‫א‬‫א‬‫ א‬
K‫א‬‫א‬‫א‬‫א‬‫א‬‫ א‬
 K NAND ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
51
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 
 ‫א‬‫א‬‫א‬‫א‬‫א‬
 K١
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ K  
 W‫א‬‫א‬ ‫א‬
  p ‫א‬ a K١
  p ‫א‬ a K٢
 E ab  a  b 
 ،  ‫א‬F  ‫א‬  K٣
   ‫א‬  K٤
  (T ) W‫א‬ 1 K٥
  (F ) W‫א‬ 0 K٦
 
‫א‬    ‫א‬ ‫א‬‫א‬    ‫א‬ ‫א‬ ‫א‬ 
 W‫א‬

‫א‬‫א‬‫א‬

52
 ‫א‬
 ٤١١٥
 ‫א‬‫א‬
 
 
 
1
pqq p
abba
2
3
pqq p
( p  q)  r  p  (q  r )
( p  q)  r  p  (q  r )
p  (q  r )  ( p  q)  ( p  r )
p  (q  r )  ( p  q)  ( p  r )
ab ba
( a  b )  c  a  (b  c )
4
5
6
7
8
9
10
11
 
‫א‬‫א‬‫א‬‫א‬
12
( a  b )  c  a  (b  c )
a  (b  c )  ( a  b )  ( a  c )
a  (b  c )  ( a  b )  ( a  c )
( p  q)  p  q
a b  ab
( p  q)  p  q
p p p , p p p
pT T , pT  p
ab  a b
a aa , aaa
a  1  1 , a 1  a
p pT , p p  F
a  a 1 , a  a  0
a0a , a00
pF  p , pF  F
 
 
 
 W‫א‬‫א‬‫א‬W١
  a) Z  (a  bc)(ab  c)
a)
5
Z abc
 
 0  b  ac  a  0  c  bc
9 , 11
 0  ac  0  bc
 ac  bc
12
‫א‬
 b)
 
 a ab  ac  abbc  bcc
Z  c( a  b)
 W‫א‬
‫ א‬ 
Z  (a  bc)(ab  c)
 a (ab  c)  bc(ab  c)
b) Z  a  bc
 a (b c)
 a (b  c )
 
7
8
 a (b  c)
12
5
 
 W‫א‬K٢
53
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
‫א‬  
  K   ‫א‬ ‫א‬‫א‬    ‫א‬ 
 W‫א‬
 ?‫א‬‫א‬‫א‬‫א‬‫א‬?
 W
  Z : ‫א‬، a : ‫א‬، b : ‫א‬‫א‬  : OR
 WEF‫א‬
 Z  a  b
 W‫א‬‫א‬
a
b
Z ab
0
0
0
1
0
1
1
0
1
1
1
1
 
 
 
 
 
 
 
  K‫א‬  ‫א‬  ‫א‬   ‫א‬  
‫א‬ ‫א‬ K   ‫א‬       ‫א‬ 
 W
 
  ab  ac  a (b  c ) W  W٢
 W‫א‬
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 WFT01‫א‬‫א‬
 
a
b
c
ab
ac
54
ab  ac
bc
a(b  c)
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
1
1
0
0
0
1
1
1
0
0
1
0
1
0
0
0
0
0
1
0
0
1
1
0
1
0
0
1
1
1
1
1
0
1
1
1
0
1
1
1
1
1
1
1
 
،E a(b  c) F‫א‬‫א‬E ab  ac F6‫א‬‫א‬
 W‫א‬
  ab  ac  a (b  c ) 
 
 W‫א‬‫א‬‫א‬‫א‬‫א‬K٣
E‫א‬‫א‬F‫א‬‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬K‫א‬‫א‬‫א‬‫א‬‫א‬
 K‫א‬‫א‬ L‫א‬
 
 1 0‫א‬‫א‬  L
K‫א‬‫א‬   1  0 ‫א‬   ‫א‬  ‫א‬ ‫א‬  
 F    ‫א‬ ‫א‬‫א‬     
 L E‫א‬‫א‬‫א‬‫א‬‫א‬
‫א‬‫א‬ ،‫א‬‫א‬‫א‬
 K‫א‬‫א‬‫א‬
 
 W‫א‬‫א‬‫א‬K٤
55
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
K‫א‬ ،‫א‬ 
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 K‫א‬‫א‬‫א‬‫א‬
 
 WEF AND ‫א‬‫א‬K١{٤
‫א‬‫א‬E 1 F AND ‫א‬
‫א‬FKE 1 F‫א‬‫א‬
 F ‫א‬  K Z  ab  W ‫א‬   K AND ‫א‬ 
  E 1a
b  1 AND  a  1 F B AND A ‫א‬‫א‬‫א‬E 1 ‫א‬
 K AND ‫א‬‫א‬ 1b ‫א‬KE
 
 K b  a EF‫א‬
 
AND ‫א‬ Z ‫א‬‫א‬
  Z  abc W‫א‬ a , b , c ‫א‬ K b  0  a  0 ‫א‬ Z  0 ‫א‬
a
a
Z=ab
AND
b
1a ‫א‬
 
 
b
1b ‫א‬
a
b
Z  ab
0
0
0
1
0
0
1
0
0
1
1
1
 
 WEF OR ‫א‬‫א‬K٢{٤
‫א‬ ‫א‬  ‫א‬  ‫א‬ E 1 F     OR ‫א‬‫א‬ 
K OR ‫א‬
 E 2a ‫א‬F‫א‬KE 1 F
  ‫א‬ E Z  1  F ‫א‬  K Z  a  b  W ‫א‬  
 K AND ‫א‬‫א‬  2b ‫א‬ KE b  1 OR  a  1  F  ‫א‬ 
‫א‬ 
 K b  a EF‫א‬ OR ‫א‬ Z ‫א‬‫א‬
  K b  1 a  0 ‫א‬ Z  1 
 
56
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
a


 
b
a
Z=a+b
OR
b
2a ‫א‬
2b ‫א‬
 
a
b
Z ab
0
0
1
0
1
0
0
1
1
1
1
1
 
 WE‫א‬F NOT ‫א‬‫א‬K٣{٤
،‫א‬‫א‬‫א‬
 W
  Z  1  a  0  Z  0  a  1
 
 W،‫א‬‫א‬،‫א‬ Z ‫א‬‫א‬
  Z  a 
 W‫א‬ NOT ‫א‬‫א‬
 
a
Z a
NOT
a
Z a
0
1
1
0
 
 
 
 

NAND ‫א‬‫א‬‫א‬K NOT
 W NAND ‫א‬‫א‬K٤{٤
AND ‫א‬‫א‬ NAND ‫א‬‫א‬
 W، AND ‫א‬‫א‬ NAND ‫א‬‫א‬‫א‬K AND ‫א‬‫א‬‫א‬
Z  ab
 
 W‫א‬ NAND ‫א‬‫א‬
 
a
a
NAND
b
Z  ab
57
b
ab
Z  ab
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
0
0
0
1
0
0
1
1
1
0
0
1
1
1
1
0
 
 
 
 
 
 
K OR ‫א‬‫א‬ NOR ‫א‬‫א‬‫א‬K NOT
 W NOR ‫א‬‫א‬K٥{٤
OR ‫א‬‫א‬ NOR ‫א‬‫א‬
W، OR ‫א‬‫א‬ NOR ‫א‬‫א‬‫א‬
 Z  a  b
 
 W‫א‬ NOR ‫א‬‫א‬
 
a
NOR
Z  ab
b
a
b
ab
0
0
1
0
1
0
0
1
1
1
1
1
Z ab
 
 
1
 
0
 
0
 
0
 
 
 W‫א‬‫א‬‫א‬ Z ‫א‬W٣
a
b

AND
NOT
OR
Z
NOT
 
c
OR
NOT
 
58
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 W‫א‬
  ab W AND ‫א‬‫א‬‫א‬ a and b W AND ‫א‬‫א‬‫א‬
  a WE‫א‬F NOT ‫א‬‫א‬‫א‬ ab WE‫א‬F NOT ‫א‬‫א‬‫א‬
  a  c W OR ‫א‬‫א‬‫א‬ a and c W OR ‫א‬‫א‬‫א‬
  a  c and ab W OR ‫א‬‫א‬‫א‬ a  c W NOT ‫א‬‫א‬‫א‬
  Z  ab  a  b W OR ‫א‬‫א‬ Z ‫א‬‫א‬
 
 
 W
‫א‬‫א‬‫א‬‫א‬
 K‫א‬‫א‬‫א‬
 
 
‫א‬ ‫א‬ ‫א‬‫א‬ ‫א‬  ‫א‬ ‫א‬ ‫א‬ Z  ‫א‬ ‫א‬  W٤ 
 K‫א‬‫א‬
a

b
OR
NOT
Z
 
 
 
AND
 
 
 
 W‫א‬
  a W NOT ‫א‬‫א‬‫א‬
  a and b W AND ‫א‬‫א‬‫א‬
  ab W AND ‫א‬‫א‬‫א‬
  ab and a W OR ‫א‬‫א‬‫א‬
Z  a  ab W OR ‫א‬‫א‬‫א‬‫א‬
 W‫א‬‫א‬
59
 ٤١١٥

 ‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬
  Z  a  ab  (a  a)(a  b) E 6 F

Z  a  b E1011F

 
 W‫א‬ OR
O ‫א‬‫א‬

‫א‬
‫א‬
 
 
a
OR
 
 
Z
b
 
 
 W‫א‬‫א‬‫א‬‫א‬W٥
 
 
 
 W‫א‬
W
  abc W
‫א‬ a annd b andd c WE‫א‬
‫א‬F AND ‫א‬‫א‬‫א‬
  abc W
‫א‬  a andd b and c WE‫א‬F AND ‫א‬‫א‬‫א‬
  ab W
‫א‬  a and b WE‫א‬F AND ‫א‬‫א‬‫א‬
  abc annd abc and ab W OR ‫א‬‫א‬‫א‬‫א‬
 
  Z  abc  abc  ab W Z 
‫א‬
 
60
 ٤١١٥

 ‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬
 
‫א‬
‫א‬
،
 ‫א‬‫א‬
‫א‬ W٦
 K‫א‬
 
a
b
0
0
1
 
0
1
1
1
0
1
1
0
1
 
Z
 
 
 
 W‫א‬
W
 W
 1 ‫א‬

Z  ab
OR
ab
OR
R
ab
 ab  ab  ab
   a b  b ( a  a )

 ab  b1
 ab  b
 
 W‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬
 
 
 
 
 
 
61
 ٤١١٥

 ‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬
  NA
AND ‫א‬‫א‬‫א‬‫א‬‫א‬K٥
‫א‬ 
‫א‬
‫א‬‫א‬‫א‬‫א‬ ‫ א‬‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬K NAND ‫א‬‫א‬

‫א‬
‫א‬


‫א‬‫א‬ ‫א‬‫א‬‫א‬
‫א‬‫א‬  
 K‫א‬‫א‬
‫א‬‫א‬
 W NAND ‫א‬‫א‬‫א‬
‫א‬‫א‬

NO 
  NOT
 
 
 AN
ND 
 
  
 O
OR 
 
a
a

Z  aa  a
NAND
 
a
b

NA
AND
N
NAND
ab

NAND
a
NAND
 
 
b

Z  ab
b  ab
Z  a b  a  b  a  b
N
NAND
b
 
 
 
 
62
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬ NAND ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
‫א‬‫א‬     K ‫א‬
NAND ‫א‬ ‫א‬  ‫א‬
 K‫א‬‫א‬‫א‬‫א‬ NAND
 
 
 K NAND ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W٧
 
 
 W‫א‬
  bc W‫א‬ b and c W AND ‫א‬‫א‬
  bc  d W‫א‬ bc and d W OR ‫א‬‫א‬‫א‬
  bc  d , e and a W AND ‫א‬‫א‬
  Z  ae(bc  d )  abce  ade W‫א‬‫א‬
‫א‬ OR ‫א‬‫א‬ NAND ‫א‬‫א‬‫א‬‫א‬
‫א‬ Z  abce  ade  abce  ade W‫א‬
 W
 
  Z  abce  ade
 
 W NAND ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 
63
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
64
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W١
 
a ) ab(a  b) b) (a  bc)(a  bc) c) (a  ac)(b  bc)(c  ca) d ) (a  b)(a  b)
e) (a  b)(a  b 2  b)
f ) a 4  a 3  a 2  a g ) (ab  c)
h) ( y  yz )  ( y  yz )
 
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W٢
 K‫א‬‫א‬
 
a) a  ab  a
b ) a ( a  b)  a
c) a(a  b)  ab
d ) a  bc  (a  b)(a  c)
e) a  b  a b
f ) ab  a  b

 
‫א‬ ‫א‬‫א‬،‫א‬‫א‬‫א‬ W٣
‫א‬‫א‬، NAND ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬،
 K‫א‬‫א‬
a

OR
b
a)
c
d
 
a
OR
AND
OR
 

OR
b)

b
AND
OR
AND
 
c

 
65
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 Z  a  b  ab ‫א‬ NAND ‫א‬‫א‬‫א‬W٤
K NAND  OR ، AND ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 K‫א‬‫א‬‫א‬‫א‬
 
‫א‬ Z  a  b  c ‫א‬ NAND ‫א‬‫א‬ W٥
 K‫א‬‫א‬‫א‬
 
 W‫א‬‫א‬‫א‬‫א‬W٦
 
NAND
a

NAND
NAND
b

NAND
NAND
 
 
 W‫א‬‫א‬‫א‬ C  ab  S  ab  ab WW٧
 
a

NAND
NAND

NAND
S
NAND
b


 
NAND
C
 
 
 
66
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
،‫א‬‫א‬‫א‬‫א‬‫א‬EFW٨
 K NAND ‫א‬‫א‬‫א‬‫א‬
 
a
b
0
0
0
1
1
0
1
0
1
1
1
0
 
Z
 
 
 
 
 
 
 K‫א‬‫א‬ ‫א‬‫א‬‫א‬W٩
a
b
c
Z
0
0
0
0
0
1
0
1
0
1
0
0
0
1
1
0
1
0
0
1
1
0
1
0
1
1
0
0
1
1
1
1
 
 
 
 
 
 
 
 
 
 
‫א‬‫א‬‫א‬‫א‬‫א‬W10
 K NAND ‫א‬‫א‬‫א‬‫א‬
 
 
 
 
 
67
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
a
b
c
Z
0
0
0
0
0
1
0
1
0
1
0
0
0
1
1
0
1
0
0
1
1
0
1
1
1
1
0
1
1
1
1
1
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
68
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 
 
 
‫א‬‫א‬‫א‬
‫א‬
69

 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 
‫א‬‫א‬ ‫א‬‫א‬‫א‬ W‫א‬‫א‬
 K‫א‬‫א‬
 
 W‫א‬‫א‬‫א‬W‫א‬‫א‬
K ‫א‬‫א‬‫א‬‫ א‬
K‫א‬‫א‬‫א‬‫א‬ 
K‫א‬ 
K‫א‬‫א‬ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
70
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 ‫א‬
WK1
‫א‬‫א‬‫א‬‫א‬
،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 K‫א‬
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ 
‫א‬‫א‬ K‫א‬‫א‬‫א‬‫א‬
K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
‫א‬ ‫א‬ ‫א‬   ‫א‬ ‫א‬  ‫א‬  ‫א‬
 K‫א‬
 
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
،‫א‬ ‫א‬ ‫א‬  
 ‫א‬ ‫א‬ ‫א‬   ‫א‬ ‫א‬ 
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 K‫א‬‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬K‫א‬‫א‬‫א‬‫א‬
‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬
 K ‫א‬‫א‬
 
EDescriptive StatisticsF‫ א‬‫א‬W‫א‬‫א‬
 WEInferential StatisticF‫ א‬‫א‬
 
71
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 Descriptive Statistics ‫א‬‫א‬E1
‫א‬‫א‬ ‫א‬‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬‫א‬
‫א‬‫א‬K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 K‫א‬‫א‬‫א‬‫א‬
 Inferential StatisticsE‫א‬F‫א‬‫א‬E2
،‫א‬‫א‬‫א‬‫א‬‫א‬
‫א‬‫א‬،‫א‬‫א‬K‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬‫א‬
 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 
 W  (Statistics)‫א‬
‫א‬‫א‬‫א‬‫א‬‫א‬
 K‫א‬‫א‬‫א‬‫א‬
 
 W‫א‬E‫א‬F‫א‬‫א‬‫א‬K2
‫א‬‫א‬‫א‬
 K
 
 W(Qualitative data)‫א‬E‫א‬F‫א‬‫א‬K١{٢
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 K‫א‬‫א‬
 W(Quantitative data)‫א‬E‫א‬F‫א‬‫א‬K٢{٢
‫א‬‫א‬EF‫א‬‫א‬‫א‬‫א‬‫א‬
 K‫א‬‫א‬‫א‬‫א‬
 
72
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 W
‫א‬‫א‬‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬‫א‬،
 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 
W‫א‬‫א‬K3
 (Population)‫א‬‫א‬K١{٣
‫א‬  ‫א‬‫א‬ ‫א‬  ‫א‬   ‫א‬ 
‫א‬‫א‬ ،
  K‫א‬  ‫א‬  ‫א‬ ‫א‬‫א‬   
‫א‬‫א‬‫א‬‫א‬‫א‬
 ‫א‬ ‫א‬‫א‬ ،‫א‬  ‫א‬   ‫א‬ ‫א‬  
‫א‬ ‫א‬‫א‬      ‫א‬   ‫א‬‫א‬ ‫א‬
 K‫א‬
‫א‬ ‫א‬‫א‬K‫א‬
‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬  ،‫א‬
،‫א‬‫א‬‫א‬ ‫א‬ ،
        ‫א‬       
 K‫א‬‫א‬
K،  ‫א‬

  ‫א‬‫א‬ ‫א‬‫א‬ ‫א‬‫א‬‫א‬
  ‫א‬   ‫א‬ 
      
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬K‫א‬
 K ‫א‬‫א‬،‫א‬
 
‫א‬ ‫א‬ ‫א‬  ‫א‬     ‫א‬  
‫א‬   ‫א‬  K
   ‫א‬ ‫א‬   
،    ،‫א‬ ‫א‬   ‫א‬   ‫א‬ ‫א‬
73
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
     ‫א‬  ،     
‫א‬  ‫א‬ ‫א‬ ‫א‬  ‫א‬    ،
‫א‬      ‫א‬ ‫א‬  ‫א‬ ‫؛‬‫א‬ ‫א‬ ‫א‬
 K‫א‬‫א‬‫א‬
 
 W(Sample)‫א‬‫א‬K٢{٣
 ‫א‬    K‫א‬‫א‬  ‫א‬ ‫א‬    ‫א‬ 
      ،‫א‬ ‫א‬ ‫א‬  ‫א‬‫א‬ ‫א‬  ،‫א‬
 K
 ‫א‬
 
W‫א‬‫א‬K4
‫א‬‫א‬  ‫א‬ ‫א‬ ‫א‬ ‫א‬‫א‬  ‫א‬  ‫א‬  
‫א‬‫א‬‫א‬‫א‬‫א‬K‫א‬
‫א‬‫א‬‫א‬‫א‬K‫א‬‫א‬‫א‬
  ‫א‬ ‫א‬      ‫א‬ ‫א‬ ‫א‬‫א‬
‫א‬‫א‬ 
 K
 
      EF  ‫א‬    ‫א‬ 
،‫א‬‫א‬ ‫א‬‫א‬‫א‬ ‫א‬ ‫א‬‫א‬‫א‬  ‫א‬  ‫א‬‫א‬ ‫א‬‫א‬
 ، ‫א‬ ‫א‬  ‫א‬    ‫א‬  
 K‫א‬‫א‬‫א‬
 
‫א‬‫א‬‫א‬‫א‬
 ‫א‬     ‫א‬ ‫א‬ ‫א‬  ‫א‬
‫א‬ ‫א‬     ‫א‬‫א‬  K   ‫א‬
‫א‬     ‫א‬    K‫א‬   ‫א‬ ‫א‬
74
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
‫א‬ ‫א‬‫א‬‫א‬K‫א‬
K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 K‫א‬
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 K‫א‬
   K 120  ‫א‬ ‫א‬‫א‬   ‫א‬ ‫א‬ W 1 
 
 K‫א‬‫א‬‫א‬‫א‬‫א‬
 
3
6
1
6
1
1
7
5
1
1
2
3
2
3
7
2
6
3
7
4
7
1
2
6
4
4
5
3
7
2
6
1
7
3
7
4
2
2
1
6
3
6
3
5
4
4
3
3
7
5
5
1
5
7
3
5
7
3
1
1
5
6
4
5
3
2
6
7
6
3
3
5
2
6
3
2
2
2
2
5
1
1
3
7
6
3
7
5
6
2
4
3
5
1
1
4
3
7
1
6
3
7
6
3
6
1
4
5
5
1
3
6
3
6
3
5
2
7
5
7
 W‫א‬
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
‫א‬‫א‬‫א‬  ‫א‬  ‫א‬  ‫א‬ ‫א‬     
 K‫א‬‫א‬‫א‬‫א‬‫א‬
 
‫א‬‫א‬  ‫א‬ ،‫א‬ 10   4 ‫א‬  ‫א‬    
 K10‫א‬‫א‬‫א‬4‫א‬‫א‬‫א‬
 
 W‫א‬‫א‬‫א‬‫א‬
 
75
 
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
‫ﻋدد اﻟﺳﻳﺎرات‬
‫اﻟﺗﻛرار‬
1
18
2
15
3
25
4
10
5
17
6
18
7
17
‫اﻟﻣﺟﻣوع‬
120
 
 W
  ‫א‬‫א‬
‫א‬‫א‬،‫א‬
‫א‬K‫א‬‫א‬
‫א‬،‫א‬‫א‬ ‫א‬  
 K‫א‬‫א‬‫א‬‫א‬
 
 W‫א‬(Classes)‫א‬‫א‬K١{٤
،‫א‬‫א‬‫א‬‫א‬‫א‬E١F
 W‫א‬‫א‬‫א‬ R ‫א‬
  R  Max  Min
 W
 ‫א‬: Max 
 ‫א‬ : Min
 
‫א‬‫א‬‫א‬‫א‬W m ‫א‬E٢F
 W‫א‬‫א‬‫א‬Yule،‫א‬
m  4 n  2 .5
 W
 ‫א‬ : m 
76
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 ‫א‬‫א‬‫א‬‫א‬ : n
 W‫א‬ L ‫א‬E٣F
 L  R
m
 
 W
K‫א‬ ‫א‬ L ‫א‬ m ‫א‬
   ‫א‬ ‫א‬‫א‬ ‫א‬    ‫א‬ ‫א‬  
  ،‫א‬‫א‬ ‫א‬  (F )  ‫א‬‫א‬   ‫א‬‫א‬ ‫א‬
K‫א‬‫א‬‫א‬ ( Fi ) ‫א‬‫א‬
‫א‬‫א‬‫א‬  Fi ‫א‬‫א‬‫א‬
 K n ‫א‬‫א‬
 
‫א‬‫א‬  ‫א‬‫א‬‫א‬‫א‬‫א‬ W ٢
  ‫א‬  ‫א‬   ‫א‬ K   ‫א‬
 K‫א‬
26.9
77.8
14.9
33
46.3
71.5
46
7.9
64.5
87.8
33.7
27
71.8
33.8
59.1
28.8
80.2
81.3
43.7
41.6
7.1
64
58.3
37.9
72.2
79.9
54.3
89.7
61.3
21.5
70.1
67
15.5
2.2
28.6
31.7
4.9
26.8
99.2
0.3
10.5
28.3
99.3
14.1
47
98.1
20.3
44.2
21.7
66.3
3.2
6.2
67
39.6
39.2
62.3
21.9
72.2
67.5
64
93.7
51
40
20.2
73.9
18.2
25.4
24.5
67.3
16.4
25.4
35.9
26.5
83.1
88
37.8
19.1
80
45.4
73.8
73.9
63.8
54.2
23.2
53
52.3
63.6
46.2
24.9
55.6
 
 W‫א‬
77
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 W R ‫א‬E١F
  R  Max  Min  99.3  0.3  99
  m  5 ،‫א‬E2F
 W‫א‬E٣F
R 99

 19.8  20
m 5
‫א‬  ‫א‬‫א‬‫א‬‫א‬‫א‬
 L 
‫א‬‫א‬ ‫א‬‫א‬‫א‬
 W‫א‬‫א‬‫א‬K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
(‫ﺣدود اﻟﻔﺋﺎت )آﻟف ﷼‬
F ‫اﻟﺗﻛرار‬
0 – 20
14
20 – 40
26
40 – 60
17
60 – 80
22
80 – 100
11
‫اﻟﻣﺟﻣوع‬
90
 
 
 
 K65‫א‬‫א‬‫א‬‫א‬W٣
‫ﻓﺋﺎت اﻷﺟور اﻟﻳوﻣﻳﺔ‬
‫ﻋدد اﻟﻌﻣﺎﻝ‬
70 – 100
8
100 – 120
10
120 – 140
16
140 – 160
14
160 – 180
10
180 – 200
5
200 - 220
2
‫اﻟﻣﺟﻣوع‬
65
 
 
 
 
 
 
 
 
 W‫א‬‫א‬
78
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
  K‫א‬‫א‬‫א‬‫ א‬E١
 K‫א‬‫א‬‫א‬‫ א‬E٢
 K219200‫א‬‫א‬‫א‬‫א‬ E٣
 K‫א‬‫א‬‫א‬ E٤
 K160‫א‬‫א‬‫א‬ E٥
 W‫א‬
 120‫א‬‫א‬‫א‬‫ א‬E١
  179‫א‬‫א‬‫א‬‫ א‬E٢
 ٢219200‫א‬‫א‬‫א‬‫א‬ E٣
 120 – 139‫א‬‫א‬‫א‬ E٤
‫א‬‫א‬‫א‬‫א‬160‫א‬‫א‬‫א‬ E٥
 8 + 10 +16 + 14 = 48W‫א‬‫א‬‫א‬‫א‬
 
 W(Relative frequency)‫א‬‫א‬‫א‬K٢{٤
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 K100‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
  F% 
Fi
 100
 Fi
      ‫א‬  ‫א‬    ‫א‬ ‫א‬‫א‬  
 K‫א‬‫א‬
‫א‬     ‫א‬    ‫א‬ ‫א‬  
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬
 K‫א‬‫א‬‫א‬
‫א‬ ‫א‬   ‫א‬ ٢   ‫א‬ ‫א‬‫א‬   W ٤ 
 W‫א‬‫א‬‫א‬
79
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 W‫א‬
(‫ﺣدود اﻟﻔﺋﺎت )آﻟف ﷼‬
F
F%
0 – 20
14
15.56
20 – 40
26
28.89
40 – 60
17
18.89
60 – 80
22
24.44
80 – 100
11
12.22
‫اﻟﻣﺟﻣوع‬
90
100
 
 W(Class midpoint)‫א‬K٣{٤
 W،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
U  Li
  xi  i
2
 W
  i  : xi
  i ‫א‬‫א‬ : U i
  i ‫א‬‫א‬ : Li
W‫א‬٢W٥
 
(‫ﺣدود اﻟﻔﺋﺎت )آﻟف ﷼‬
xi
‫ﻣرﻛز اﻟﻔﺋﺔ‬
F ‫اﻟﺗﻛرار‬
0 – 20
١٠
14
20 – 40
٣٠
26
40 – 60
٥٠
17
60 – 80
٧٠
22
80 – 100
٩٠
11
‫اﻟﻣﺟﻣوع‬
90
 
80
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 W‫א‬L‫א‬‫א‬‫א‬‫א‬‫א‬K٤{٤
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 
   ‫א‬ ‫א‬  ‫א‬ ‫א‬‫א‬ ‫א‬‫א‬ ‫א‬ 
K‫א‬ ‫א‬‫א‬ ‫א‬  ‫א‬‫א‬‫א‬  ‫א‬ ‫א‬‫א‬ ‫א‬ 
،‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 ‫א‬  ‫א‬‫א‬‫א‬    ‫א‬  ‫א‬‫א‬ ‫א‬ 
 K‫א‬
 
‫א‬‫א‬‫א‬‫א‬‫א‬
‫א‬    K ‫א‬ ‫א‬‫א‬ ‫א‬‫א‬‫א‬     ‫א‬
‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
?‫א‬‫א‬، () ??‫א‬،‫א‬
 K‫א‬‫א‬‫א‬‫א‬ () ?
 
 W
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
‫א‬  ‫א‬ ‫א‬ ‫א‬‫א‬   ‫א‬   ، ‫א‬ ‫א‬
 K‫א‬‫א‬‫א‬‫א‬
 
 
، ٢‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ W ٦
 W
 
 
81
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 W‫א‬‫א‬‫א‬‫א‬
‫ﺣدود اﻟﻔﺋﺎت اﻟﺻﺎﻋدة‬
‫اﻟﺗﻛرار اﻟﺻﺎﻋد‬
0
0
 20
14
 40
40
 60
57
 80
79
 100
90
 
 W‫א‬‫א‬‫א‬‫א‬
‫ﺣدود اﻟﻔﺋﺎت اﻟﻬﺎﺑطﺔ‬
‫اﻟﺗﻛرار اﻟﻬﺎﺑط‬
0
90
 20
76
 40
50
 60
33
 80
11
 100
0
 
 W‫א‬‫א‬‫א‬‫א‬‫א‬W٦
 K40‫א‬‫א‬‫א‬ Ea
 K60‫א‬‫א‬‫א‬‫א‬ Eb
 K 8040‫א‬‫א‬‫א‬ Ec
 W‫א‬
، 50 40‫א‬‫א‬‫א‬ Ea
K‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬
82
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
K‫א‬‫א‬‫א‬،5760‫א‬‫א‬‫א‬‫א‬ Eb
‫א‬398040‫א‬‫א‬‫א‬ Ec
‫א‬‫א‬E40F 40‫א‬‫א‬‫א‬‫א‬‫א‬
‫א‬،‫א‬‫א‬E79F 80‫א‬‫א‬
‫א‬‫א‬E50F 40‫א‬‫א‬‫א‬
 K‫א‬‫א‬E11F80‫א‬
 
W‫א‬‫א‬K5
‫א‬  ‫א‬
K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
       ‫א‬  ‫א‬   
 K‫א‬
 
 W‫א‬‫א‬‫א‬
 ‫א‬‫א‬‫ א‬
 ‫א‬‫א‬‫ א‬
 ‫א‬‫א‬‫ א‬
 ‫א‬‫א‬‫א‬‫א‬‫ א‬
 ‫א‬‫ א‬
 ‫א‬‫א‬‫ א‬
 
K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 
 
83
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 W(Histogram)‫א‬‫א‬‫א‬K١{٥
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 ‫א‬ ‫א‬   ‫א‬       K‫א‬‫א‬
 K‫א‬‫א‬‫א‬K‫א‬‫א‬
‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬ ٢ W ٧
 W
 
30
27
25
23
20
16
15
14
9
10
5
0
٢٠
 
٤٠
٢٠
٦٠
٤٠
٨٠
٦٠
٨٠
 
١٠٠
١٠٠
More
More
 
 W(Polygon)‫א‬‫א‬‫א‬K٢{٥
 ‫א‬      ‫א‬‫א‬ ‫א‬  ‫א‬ ‫א‬ 
 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 
‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬٢W٨
W
84
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
Frequency
30
27
25
23
20
15
16
14
Frequency
10
9
5
0
 
٢٠
٤٠
٦٠
٨٠
١٠٠
More
 
 
 W‫א‬‫א‬K6
 ‫א‬ ‫א‬ ‫א‬ ‫א‬   ‫א‬   ‫א‬ ‫א‬ 
 ‫א‬
‫א‬K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 K‫א‬‫א‬‫א‬‫א‬‫א‬
 
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
‫א‬     K‫א‬   ‫א‬     
 K‫א‬‫א‬‫א‬
‫א‬   ‫א‬ ‫א‬  ‫א‬ ‫א‬    
‫א‬‫א‬‫א‬‫א‬K‫א‬
 K‫א‬‫א‬‫א‬‫א‬‫א‬ KE‫א‬F‫א‬
 
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
،‫א‬‫א‬‫א‬‫א‬‫א‬K‫א‬
  ‫א‬   ‫א‬       ‫א‬ 
85
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 ‫א‬ ‫א‬  ‫א‬  K ‫א‬ ‫א‬ ‫א‬ ‫א‬ ‫א‬ ‫א‬
 K‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬
 
 W(Mode)‫א‬‫א‬K١{٦
‫א‬K‫א‬  ‫א‬‫א‬‫א‬‫א‬
 ‫א‬،‫א‬‫א‬‫א‬‫א‬
 K‫א‬
 K5‫א‬3,5,5,5,5,9,9,7‫א‬‫א‬‫א‬‫א‬W‫א‬
 K‫א‬4,2,5,7,6,1,3‫א‬‫א‬‫א‬
 
     ‫א‬‫א‬ ‫א‬ ‫א‬    ‫א‬‫א‬ 
 K
 
 
 W(Median)‫א‬K٢{٦
‫א‬‫א‬‫א‬‫א‬K‫א‬‫א‬‫א‬
    ‫א‬‫א‬  ‫א‬           
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬
 K‫א‬‫א‬
‫א‬     x1 , x 2 , x3 ,....., x n ‫א‬‫א‬

1
 K n ‫א‬  x n  x n  2  n‫א‬ x n  1 
2 2
2 
2
 
 K‫א‬ ~
x ‫א‬ ~ 
 
 10,8,9,6,7W‫א‬‫א‬W٩
86
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 W‫א‬
  6,7,8,9,10  ‫א‬
 W، n  5 
  x n1  x51  x 6  x3  8
2
2
 
2
 
  ~  8 ‫א‬
 
 
 9,10,6,2,1,7‫א‬W١٠
 W‫א‬
 1,2,6,7,9,10  ‫א‬ 
 W، n  6 
 
 1
 1
1
1
1
13
   x 6  x 6 2    x 6  x 8    x3  x4   6  7   13   6.5
2
2
2 2
2 2
2
2
2 
2 
 
  ~
x  6.5 ‫א‬
 
 
 W(Mean)‫א‬L‫א‬‫א‬K٣{٦
‫א‬‫א‬   ‫א‬ ‫א‬ ‫א‬    ‫א‬ ‫א‬
n
 K
 xi
i 1
n
W‫א‬ x1 , x 2 , x3 ,....., x n W‫א‬
 
 K x   
 
87
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
8,10,13,9,7,11,10‫א‬‫א‬‫א‬‫א‬W١٠
 W‫א‬
 W‫א‬
n
 xi
  i 1
n

8  10  13  9  7  11  10 68

 9.71
7
7
 
    9.71‫א‬
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
88
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 
 
W‫א‬W١
44
98
40
60
66
71
82
64
72
68
55
69
77
78
88
60
65
68
69
79
69
89
64
71
66
61
75
87
54
81
70
62
74
68
60
73
82
71
62
50
 
 W‫א‬
K‫א‬‫א‬‫א‬ Ea
K‫א‬‫א‬‫א‬ Eb
K75‫א‬‫א‬ Ec
 
 W‫א‬‫א‬‫א‬‫א‬٧٠W٢
 
‫فئات األجور‬
50‐59
60‐69
70‐79
80‐89
90‐99
100‐109
110‐119
‫عدد العمال‬
8
10
16
15
10
8
3
 W
K ٩٠‫א‬‫א‬ Ea
‫؟‬‫א‬
K‫א‬‫א‬‫א‬‫א‬‫ א‬Eb
K‫א‬‫א‬‫א‬‫א‬‫ א‬Ec
89
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 
 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W٣
 7,9,6,8,9,10,6,11,8,7,9,10,9,7,6,10,8,11,9,8
 
 W‫א‬
K‫א‬‫א‬‫א‬ Ea
K‫א‬‫ א‬Eb
K‫א‬ Ec
 
 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W٤
18
14
15
14
1
20
17
8
16
3
18
15
19
14
20
17
16
15
7
12
24
28
9
34
21
5
13
25
14
11
9
21
22
25
9
3
16
23
7
19
 
 W
K‫א‬‫א‬‫א‬‫א‬ (a
K‫א‬‫א‬‫א‬‫א‬ (b
 K‫א‬‫א‬‫א‬‫( א‬c
 
 

90
 ٤١١٥
 ‫א‬‫א‬
‫א‬‫א‬‫א‬‫א‬
 ‫א‬‫א‬
 
 ‫א‬
 ‫א‬‫א‬
،‫א‬‫א‬
 
 K١٩٨٣ J١٤٠٣،،‫א‬
Design and Analysis of
Alfred Aho, John Hopcroft and Computer Algorithms,
 Jeffrey Ullman
Addison Wesley, Reading,
England, 1974.
Mathematics for scientific and
technical students, Addison
Gwyn Davies and Gordon Hick
Wesley Longman, Harlow,
England, 1998.
Computer and Intractability,
Micheal Garey and David
A Guide to the Theory of NP Johnson
Completeness, Freeman, San
Francisco, 1979.
Elements of Discrete
 C. L. Lui
Mathematics, McGraw-Hill,
New York, 1977.
Theory of Linear and Integer
Programming, John Wiley &
 Alexander Schrijver
Sons, Chichester, England,
1986.
Basic Mathematics, John
Wiley & Sons, Chichester,
Peter Tebbutt
England, 1998.
‫א‬،‫א‬‫א‬
K،K
٢٠٠٣،‫א‬‫א‬‫א‬،
Introductory Statistics.
London : Addison-Wesley
Publishing Company, 1982
 Hassett, Weiss
91
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