١١٥אא ١١٥ Jאא אאאא ١١٥אא אאאא ، א ، א א، א א א א אא א א א א W אאא،אאאאאא א א אאאא ؛אאאאאא K אאא אאאאאאא א ،א אא א א א א א א א ، א،אאאא אאאאאאא אאאאא אאא،א Kאא،א ? ٤١١٥ ?אא אא א א א ? א א Kאאא אאאאאאא ،א ،אאאא Kאאאאא ،אאאא אאאא Kא ١١٥אא אאאא א א א ٢ אWאא ٢ אא ٣ אא ٤ אאאאא ٩ אאאא 14 אאאאאא ١٧ אא 18 אאאאא 20 אאאאא 22 26 אאאאWאא 26 אאאאא 27 אאא 31 אאא 32 אאא 33 אא 34 אא 37 אאאא 40 א 43 אא 45 50 אאאאאWאא ١١٥אא אאאא 51 52 א 53 אאאאא 54 אאא 60 NANDאאא 63 67 אWאאא 69 70 אEאFאאא 71 אא 72 אא 74 אאא 77 אאאאאאא 78 א 79 אLאאאא 81 אא 82 אאאאאא 83 אא 84 אא 84 א 85 אLאא 87 89 אא ١١٥אא אאאא Kאאאא، א،אא אא W אא אאאאאאא א א _א א _ א Kא אא אאאאאאא א א א א א א א Kאאא Wאאא Kאאאא Kאאאאא Kאאאאאא Kאאא _א_אא Wאאא א א א א א א א א א אאאאאאאאא Kאאאאאא Kאאאאאאאאאא א אא א א א א א K אאאאאא Kאאאאאאא אא ٤١١٥ אא אאאא אא א 0 ٤١١٥ אא אאאא אאאאWאא Kאאא،אא Wאאא Kאאא Kאאאאא Kאאאאאאא Kאאאאא Kאאאאאאא Kאאאאאאאא 1 ٤١١٥ אא אאאא ◌ّ א אא،אאא ،אאאאאאא Kאאא א א א א ١٢٣ א Wא אאאK K١١١١٠١١אא א א א א א א אא( ?אASCII) אאא K?The American Standard Code for Information Interchangeא W (Decimal System) אאK١ 0, 1, 2, 3, 4, 5, 6, Wאא א 7, 8, 9 א ،9 و0 א Kאאאא،א אאאא אאא Kא 9 K4.5אא אאW١ 2 Kאאא א א א א Wאאאאא، 1 (10) 0 , 10 (10) 1 , 100 (10) 2 , 1000 (10) 3 , ...... Wאאאאא 0.1 (10) 1 , 0.01 (10) -2 , 0.001 (10) -3 , 0.0001 (10) -4 , ...... 2 ٤١١٥ אא אאאא א 9א 0אאאאאא W Kא١٠א 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)אאK٢ א 1 ٠ א ٢ א א אא??א??אא KE2F2אא?? Wאאאא?א?אאא 1 2 0 , 2 21 , 4 2 2 , 8 2 3 , 16 2 4 , 32 2 5 , ...... Wאא?א?אאאאא 0.5 2 1 , 0.25 2 -2 , 0.125 2 3 , 0.0625 2 -4 , ...... ١٠אאאאאא K٢אאאאא אא אא ٢٠ 0אאאאאW٣ Wאא 3 ٤١١٥ אא אאאא אא אא 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 WאאאאאK٣ WאאאאאאK١{٣ אאאאא אאא Kאא،٢א WאאאאאאאW٤ 4 ٤١١٥ אא אאאא (1) 110101 , (2) 0.1101 , (3) 101.111 Wא (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 WאאאאאאK٢{٣ ؛אאאא Wאאאא، WאאאאאאK١{٢{٣ Kא אאאא2אE١ Kא אאאא2אאE٢ K٠אE٣ WאאאאאאאW٥ (1) 12 , (2) 19 , (3) 45 5 ٤١١٥ אא אאאא Wא Wאאא2E١ 12 0א 6 2 6 0א 3 2 3 1א 1 2 1 1א 0 2 KE1210 1100 2 F1100Wאא12 Wאאא2E٢ 19 1א 9 2 9 1א 4 2 4 0א 2 2 2 0א 1 2 1 1א 0 2 KE1910 10011 2 F10011Wאא19 Wאאא2E٣ 6 ٤١١٥ אא אאאא 45 22 2 22 0א 11 2 11 1א 5 2 5 1א 2 2 2 0א 1 2 1 1א 0 2 1א KE 4510 101101 2 F101101Wאא45 WאאאאאאK٢{٢{٣ א א ، 2 א E١ K،אאא אא 2אאאאE٢ K،אאא אאאא، K0אאE٣ WאאאאאאאW٦ (1) 0.6875 , (2) 0.3125 Wא א א א א 2 E1 1אא 0.6875 2 1.375 0אא 0.375 2 0.75 1אא 0.75 2 1.5 7 Wא ٤١١٥ אא אאאא 1אא 0.5 2 1.0 K٠אא KE 0.687510 0.1011 2 F0.1011Wאא0.6875א א א א א 2 E2 0אא 0.3125 2 0.625 Wא 1אא 0.625 2 1.25 0אא 0.25 2 0.5 1אא 0.5 2 1.0 K٠אא KE 0.312510 0.01012 F0.0101Wאא0.3125א ؟אאאא91.15625אאW٧ Wא W2אא 91 1א 45 2 45 1א 22 2 22 0א 11 2 11 1א 5 2 5 1א 2 2 8 ٤١١٥ אא אאאא 2 0א 1 2 1 1א 0 2 KE 9110 1011011 2 F١٠١١٠١١Wאא٩١ W2אא 0אא 0.15625 2 0.3125 0אא 0.3125 2 0.625 1אא 0.625 2 1.25 0אא 0.25 2 0.5 1אא 0.5 2 1.0 F 0.00101 Wא א 0.15625 א KE 0.15625 10 0.00101 2 1011011.00101Wאא91.15625אא KE 91.15625 10 1011011 .00101 2 F WאאאאK٤ WאאאK١{٤ ?אאאאאאאא אאא9אאאאאא אאאאאK?אאא،א Wאא 000 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 אא אאאא WאאאאW٨ (1) 10 11 , (2) 11 11 , (3) 110 100 , (4) 111 101 Wא 1 1 0 1 1 (1) 1 0 1 10 11 101W 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 WאאאK٢{٤ אאאאאא אאאאא،א K 10 ٤١١٥ אא אאאא WאאאאאאW٩ 1101 , 1010.01 , 0.010101 , 10000 Wא 10000 1101 1010.01 0.010101 אאאאW אאאאאאא אא Wא 000 1 0 1 11 0 0 1 1 , , with a borrow of 1 from the next column WאאאאW١٠ (1) 11 10 , (2) 101 11 1 1 Wא 1 0 0 1 (1) 11 10 1W 0 1 1 0 1 1 1 ( 2) 0 1 0 101 11 10 W 11 ٤١١٥ אא אאאא WאאאK٣{٤ א א א א א א אא Wאאאאאא 00 0 0 1 0 1 0 0 11 1 WאאאאW١١ (1) 11 11 , (2) 101 111 , (3) 1001 1011 1 1 1 1 1 1 (1) Wא 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 אא אאאא 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 WאאאK٤{٤ Kאאאאא WאאאאW١٢ (1) 110 11 , (2) 110 10 Wא 10 11 110 11 000 110 11 10 W 13 ٤١١٥ אא אאאא 11 10 110 10 010 10 00 110 10 11 W WאאאאאאK٥ 0 אאאאאא n אW١ K 2 n 1 0אאאאאא8אW١٣ K655350אאאא16א K255 ؟אאא אאאא K W Jאא.1.5 1א 0 אא אא Kאאאאא،א 8א JאאאאאאאW١٤ (1) 19 , (2) 19 W Wא K10011Wאא19א٥א2אE1 K00010011W8א Jאא K10010011W Jאא-19E2 Wאא JאאאאאW١٥ 14 ٤١١٥ אא אאאא (1) 10010 , (2) 00101 Wא (1) 10010 (10) (1 21 0 2 0 ) 2 (2) 00101 (101) (1 2 2 0 21 1 2 0 ) 4 0 1 5 אאא Jאא n אW٢ ( 2 n 1 1) ( 2 n 1 1) א אאא Jאא8אW١٦ K127 -127 א K32767 -32767אאאא16א W1אא.2.5 J א א א אא א א Kא10،אאאאאאאא W8א1אאאאאאאW17 (1) 19 , (2) 19 Wא Jאא 19א 14א 1אE١ K1אאא00010011W8א 00010011،1אא-19אE٢ K11101100W1 Wאא1אאאאאW18 (1) 00010111 , (2) 11101000 15 Wא ٤١١٥ אא אאאא (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 אאאא1אא n אW٣ n 1 n 1 ( 2 1 ) ( 2 1) אאא1אא8אW19 K127 -127 א K32767 -32767אאאא16א W2אא.3.5 J א א א אא א א א10،אאאאאאאא Kאא 1 W8א2אאאאאאאW20 (1) 19 , (2) 19 Wא Jאא 19א 14א 1אE١ K2אאא00010011W8א 00010011،2אא-19אE٢ K11101101W111101100W1 16 ٤١١٥ אא אאאא WE8אFאאאא 2 0 , 21 , 2 2 , 2 3 , 2 4 , 2 5 , 2 6 , 2 7 Wאא2אאאאאW21 (1) 01010110 , (2) 10101010 Wא (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 אאאא2אא n אW٤ ( 2 n 1 1) ( 2 n 1 ) אאא2אא8אW22 K127 -128 א K32767 -32768אאאא16א א2אאאאאאאא אאאא א א1אא W 00000000 , 11111111 W Jא 17 ٤١١٥ אא אאאא 00000000 , 10000000 W(Hexadecimal System)אאK6 W אאאא 0 , 1 ,2 , 3 , 4, 5, 6 , 7 , 8 , 9 , A , B , C , D , E , F KE16F16אא KאאאאאאWא אאאאאאאאW23 Wאא 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 WאאאאאאאK7 Wאאאאאא.1.7 18 ٤١١٥ אא אאאא א א א א א אא 2א? 16 2אא א א א אאא 16א 16א 2א K?אא WאאאאאאאW24 (1) 9719 , (2) 0.78125 Wא Wאאא16E١ 9719 607 7א 16 607 37 15א 16 37 5א 2 16 2 2א 0 16 25F 716 Wאא9710 (9710 )10 (15 F 7)16 W א א א א 16 E2 12אא 0.78125 16 12.5 8אא 0.5 16 8.0 Wא 0.C 816 Wאא0.78125 (0.78125 )10 (0.C 8)16 W 19 ٤١١٥ אא אאאא Wאאאאאא.2.7 אא אאאאאאא WK162אא WאאאאאאW25 39.B816 Wא 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 57.71875Wאא 39.B816 (39.B8)16 (57.71875 )10 W Wאאאאאא.3.7 Wאאאאאא K4 אאאE١ Kא אא4אE٢ KאאאאאE٣ WאאאאאאאW26 (1) 1100101000 10111 , (2) 1111110000 1101001 20 Wא ٤١١٥ אא אאאא (1) 1100101000 10111 0110010100 010111 0110 0101 0001 0111 (6517)16 (2) 1111110000 1101001 0001111110000 1101001 0001 1111 1000 0110 1001 (1F 869)16 Wאאאאאא.4.7 Wאאאאאא K4אאאE١ KאאE٢ WאאאאאאאW27 (1) 10 A416 , ( 2) CF 8 E16 (1) (10 A4)16 0001 0000 1010 0100 1000010100100 Wא (2) (CF 8 E )16 1100 1111 1000 1110 1100111110 001110 WW28 (59.125)10 (111011 .001) 2 (3B.2)16 21 ٤١١٥ אא אאאא אW1 W10אאאא (1) 10 , (2) 100 , (3) 10000 , (4) 0,001 ؟٤אW2 WאאאאW3 (1) 11 , (2) 100 , (3) 1110 , (4) 100101 WאאאאW4 (1) 11 .01 , (2) 10.01 , (3) 0.1110 , (4) 1001.011 ؟אאאאאאW5 (1) 17 , (2) 35 , (3) 49 , (4) 68 , (5) 82 , (6) 205 WאאאאW6 (1) 10 , (2) 17 , (3) 63 , (4) 628 , (5) 125 , (6) 186 WאאאאW7 (1) 0.32 , (2) 0.246 , (3) 0.0981 , (4) 12.34 WאאאW8 (1) 10 11 , (2) 110 10 , (3) 111 111 , (4) 1010 1011 WאאאW9 22 ٤١١٥ אא אאאא (1) 11 10 , (2) 110 10 , (3) 111 101 , (4) 1110 1011 Wאאא W10 (1) 10 11 , (2) 110 10 , (3) 111 111 , (4) 1010 1011 Wאאא W11 (1) 100 10 , (2) 1001 11 , (3) 1100 100 EאאאF8אאאא אW12 Wאאאא (1) 29 , (2) 85 , (3) 100 , (4) 123 , (5) 99 , (6) 57 WאאאאאאאW13 (1) 110011 , (2) 10110 , (3) 1100010 , (4) 010100101 WאאאאW14 (1) 5516 , (2) A7516 , (3) 8 B 5 D16 , (4)1CF 316 WאאאאW15 (1) 110011 , (2) 10110 , (3) 1100010 , (4) 010100101 WאאאאW16 (1) 5516 , (2) A7516 , (3) 8 B 5 D16 , (4)1CF 316 WאאאאW17 (1) 16 , (2) 377 , (3) 2784 , (4) 1024 23 ٤١١٥ אא אאאא אא אאאא 24 ٤١١٥ אא אאאא אא אאאא 25 ٤١١٥ אא אאאא א،אאאאWאא Kאא WאאאWאאא Kאאאאא Kאאאא Kאאאאא Kאאאאא א Kאאאא 26 ٤١١٥ אא אאאא אאאא WK1 א א א א ،א א ؛ א אא אאאא ؛ KאEאאFאא Kאאאא (True : T ) אאאאא KאאK ( False : F ) WאאאאאK2 אא Wאא Kא W١ Kאא Ea 180 אאאא Eb 4 2 1 Ec 2 x 16 W x 3 Ed ؟א Ee Kא Ef אאאאאא א א K א א ، Kא ،א אאאא אאאאKא K 27 ٤١١٥ אא אאאא W٢ Ea 4 cm ABCD אEb א אEc א א ، א א א א א א K????Wאאאאא ?א Wא א ? 4 cm ? ? ABCD ?א Wא א K??א? אאאאא Kאאאאאא אאאא א א א אא p , q , r , s , .... א א א Kאא אאאK3 Wאאאאאא W "and " ??W(conjunction)אא W " or" ??W (disjunction)אא W " not " ?KKKK?W(negation) אא p q א.1.3 אא p q WאאKE "and " F??אא אאא p q אK? q p ? q p א p q אK q p Kאא p q ،א 28 ٤١١٥ אא אאאא א p q א K 1 אא p q q p א א א א q p א א א א ، Kא p q p q pq T T T T F F F T F F F F 1 א (F ) א (T ) אאא א_א p q ، q p אאא K q p _ WאאאאW٣ אאאא אE١ 4 2 7 א אE٢ 2א6א E٣ 06א E٤ ، א אא א א א א Kאאאאאאא 29 ٤١١٥ אא אאאא p q אK٢{٣ אאא p q WאאאKE " or" F??א אאא p q אK? q p p אאא א p q אK q p q p א p q ، q Kא אא p q א K 2 א p א א p q אא Kא q p q pq T T T T F T F T T F F F ٢א WאאאאW٤ אא אE١ 4 2 7 א אE٢ 2א6א E٣ 06א E٤ ؛אאאאאאאא Kאאאא 30 ٤١١٥ אא אאאא Kא??L א، q p ? q p ?א K? q p ?א??Kא א א ، q p K א??א L2 L1 ?אא KEאF Kאאא??אא K? q | p ? א p q p א.3.3 Kאא p אאאא،א?KKKKKKK?א K p p W א p p א، p p Wא p p p p T F F T WאאאW٥ אE١ אא E٢ א E٣ K??א?א?Wאא K?אא??אא?Wאא 31 ٤١١٥ אא אאאא Wא?א?Wאא Kא?א? ، q p אאאא אאא Kא אאאא א p , q , r א Wאא 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 WEאFאאאK٤ א א אא א P ( p, q, r ,....) KE F , , אאאEאאF p, q, r ,.... ،אאא אאא Kאא אא אאאאאאא Wאאאאא 32 ٤١١٥ אא אאאא ( p q ) WאאאW٦ Wא 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 Wא א א א א، q p א א Kאא (F ) (T ) אא א א א א א Kא א א א אKא , , אאאאא Kאאאא ( p q ) א א א א א א ،א א q p א WאאאKא p q ( p q ) T T T F F T F F T F T T WאאאK٥ אא،א (T ) אאא אאא،אא אEאF K א 33 ٤١١٥ אא אאאא אאF (F ) אא KאE (F ) a ) p ( p q ) א b) ( p q ) ( p q ) W W٧ Wא 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 ) Wא T T Wא T T p ( p q) T Eאא F Wאא (b pq ( p q) ( p q) T T F F F F T F F T F F p q pq pq 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 WאאK٦ א א E F P ( p, q,...) Q ( p, q,...) Wאא، KEאא F 34 ٤١١٥ אא אאאא p q p q WאאEאFאאW٨ WאWא p q p q T T T T F F F T F F F F p q pq T T T F F T F T T T F F p q pq F F F F T T T F T T T T W p q p q אא pq pq WאאK٧ אאאא، p, q, r אאאאא Kאאאא אאאE١ a) p q q p אאE٥ b) p q q p a) p T T b) p F p c) p F F d) p T p אאE٢ a) p (q r ) ( p q) r a ) p (q r ) ( p q ) ( p r ) אאE٦ b) p ( q r ) ( p q ) r אאE٣ a) p p T c) T F d) F T אE٧ p p b) p ( q r ) ( p q ) ( p r ) אאE٤ a) p p p b) p p F אE٨ b) p p p 35 a) ( p q) p q b) ( p q ) p q ٤١١٥ אא אאאא W٩ אאEאF (2a ) א Kאא p ( q r ) אא א ( p q ) r ( p q ) r p ( q r ) W،אא r pq ( p q) r qr 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 EאF ( p q ) p p WאאW١٠ p q p q ( p q) p T T T T F F F T F F F F T T F F Wא E p Fאא Wא،E ( p q ) p Fא ( p q) p p Kאאא ( p q ) p p א Kאאאאאאא Kאאאא 36 ٤١١٥ אא אאאא WאאאאW١١ a ) ( p q) ( p q) p b) ( p q ) ( p q ) q T Wא אא ( a E٨ a F א ( p q) ( p q) ( p q) ( p q) p ( q q ) E٣ b Fא E٦ a Fא p T E٥ d Fא p (b אא ( p q) ( p q) q q ( p p) q q F q q q א E٣ a Fא E٦ b Fא E٥ b Fא T E٦ a Fא א אאאאאאאא Wאא WאW١٢ ( p q) ( p q r ) ( p q) Wא 37 ٤١١٥ אא אאאא אא א ( p q ) [ p ( q r )] ( p q ) ( p q ) { p [( q r ) q ]} E٣ b Fא E١٠Fא ( p q) ( p q) E٣ b Fא p (q q) E٦ a Fא p T E٨ a F E٥ d F א p KאאאאK٨ Wא א.1.8 א q p (if p then q ) ? q p א?אא K p q W، K? q א p ?? q p ? p q אא Wאא p q p q T T T T F F F T T F F T p q א א p q א K אא، א א א א אאאא،א אאKא p ? q p א? אאא، q p אא W WאאאW١٣ q Wאאא، p Wא 38 ٤١١٥ אא אאאא אאאאאW p q K q p p א אאאאא Wאאא p : 3 8 , q : 3 5 8 WאאאW١٤ 3 5 8 3 8 אW p q K p q אא p q K p q א q p p Wא، q : 4 2 3 WאאאW١٥ 4 2 3 אאW p q אא אא q p Kא p q W ( ) אא () אאאאאL p q p q p pq T T T F T T F F F T T F T F T F F T T T pq pq א א Wאאא p q p q Wאאאא K? q p ?? q p א? 39 ٤١١٥ אא אאאא WאאאW١٦ pq , q p , pq , 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 Wא Wאא pqq p q p pq W א א א q p p q q p pq q p א p q אאא K p q WאK p : x 2 4 q : x 4 WW١٧ Kא x 4 E x 2 F x 2 4 אW 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 אW p q K x 3 40 ٤١١٥ אא אאאא p q p q ( p q) Wאאא T T T F T F F F T T T F p q p q W Wאא F F T F ( p q) ( p q) p q Wאא.2.8 אאא א K p q W? q אא p ? q p אאא p q p q T T T T F F F T F F F T אאKא Wאא Wאאא p q ( p q) (q p) EאאאF : אK٩ א P1 , P2 , P3 ,....., Pn אא א W،א Q אא P1 , P2 , P3 ,....., Pn Q אE F P1 , P2 , P3 ,....., Pn Q א ، P1 , P2 , P3 ,....., Pn א E Q Fא KאאEF 41 ٤١١٥ אא אאאא Wאא W١٨ EF، (b) p q , q p EF (a ) p , p q q Wא א p q p (a Kאאא q א Wאאאא p q p q T T T T F F F T T F F T אאא q אאא،אא p , p q q WאKאאאאא K Wא (a אאא (b אא q p q אא אא p אא،אא א ، E p F א א א KEF p q , q p p q p q T T T T F F F T T 42 ٤١١٥ אא אאאא F F T P1 , P2 , P3 ,....., Pn אאא ،E אFא P1 P2 P3 ..... Pn אא Q אאEF P1 , P2 , P3 ,....., Pn Q א W، P1 P2 P3 ..... Pn WאאאKEאF ( P1 P2 P3 ..... Pn Q ) T p q , q r p r א W١٩ Wא אא ( p q ) ( q r ) ( p r ) א Wא qr pr ( p q ) (q r ) ( p q ) (q r ) ( p r ) p q r pq 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 KEF p q , q r p r ؟אאW٢٠ K א א א א א אKאא، אא K 43 ٤١١٥ אא אאאא Wא P1 Wא P2 Wאא Wאאא P1 Wא P2 Wאא P1 Wא p1 p 2 , p 2 W Wאא p1 p 2 , p 2 p1 Wא ( 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 Kא ( p1 p 2 ) p 2 p1 T אא אאK١٠ 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 Wא p q 44 ٤١١٥ אא אאאא ، p q א א p K p p q אKאאא ، P ( p, q,...) Q ( p, q,...) אא Kא P ( p, q,...) Q ( p, q,...) א P Q א א א א P Q א Wאא אאKא Wאא Q ( p, q,...) P ( p, q,...) Q ( p, q,...) P ( p, q,...) EF P ( p, q,...) Q ( p, q,...) א E אFא P Q א א 45 ٤١١٥ אא אאאא WאאאW١ Kא Ea Kא Eb K Ec 45 58 Ed Kאאאא Ee ؟ Ef Kאא 2 2 5 א Eg p W??א q W??אWאאאW٢ a ) p , b ) p q , c ) p q , d ) q p Wא WאאאW٣ p :א q : r : א W , , אאאאאאא Kאא Ea Kאאא Eb Kאא Ec א Ed Kא p :א 46 WאאאW٤ ٤١١٥ אא אאאא q : r : א s : א Wאאא 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 WאאW٥ WאאאאאאW٦ a ) p ( p r ) , b) ( p q ) ( p q ) q , c ) p ( p q ) q WאאאאW٧ a) p ( p r ) ( p q) ( p r ) b) p ( p q ) p c) ( p q ) p q WאאאאW٨ 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 ) WאW٩ b) p {q ( p q )} c) ( p q) ( p q r ) ( p q) 47 ٤١١٥ אא אאאא WאאאאאאאW١٠ Kאאא (a אא،אאא (b Kא K אאאא (c WאאאW١١ 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 WאאאW١٢ p : א ABC א q : אא ABC א p q אאא،א :אאאאאאאW١٣ a) אאא b) אאא E p q p q WאF WאאאW١٤ Kאא (a) K אאאא (b) Kאא (c) 48 ٤١١٥ אא אאאא 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) Wאא 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 Wאא W١٨ Kאא،אאא WאאW١٩ K 7 א 4 7 K 7 4 7 א 49 ٤١١٥ אא אאאא אא אאאאא 50 ٤١١٥ אא אאאא KאאאאאאWאא WאאאWאאא Kאאאא א Kאאא Kאאא א Kאאאאא א K NAND אאאאאא 51 ٤١١٥ אא אאאא אאאאא K١ אאאאאא אאאאאא K Wאא א p א a K١ p א a K٢ E ab a b ، אF א K٣ א K٤ (T ) Wא 1 K٥ (F ) Wא 0 K٦ א א אא א א א Wא אאא 52 א ٤١١٥ אא 1 pqq p abba 2 3 pqq 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 ) ab ba ( a b ) c a (b c ) 4 5 6 7 8 9 10 11 אאאא 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 ab ( p q) p q p p p , p p p pT T , pT p ab a b a aa , aaa a 1 1 , a 1 a p pT , p p F a a 1 , a a 0 a0a , a00 pF p , pF F WאאאW١ a) Z (a bc)(ab c) a) 5 Z abc 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) Wא 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 WאK٢ 53 ٤١١٥ אא אאאא א K א אא א Wא ?אאאאא? W Z : א، a : א، b : אא : OR WEFא Z a b Wאא a b Z ab 0 0 0 1 0 1 1 0 1 1 1 1 Kא א א א א א K א א W ab ac a (b c ) W W٢ Wא אאאאאא WFT01אא a b c ab ac 54 ab ac bc a(b c) ٤١١٥ אא אאאא 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) FאאE ab ac F6אא Wא ab ac a (b c ) WאאאאאK٣ EאאFאאא אאאאKאאאאא Kאא Lא 1 0אא L Kאא 1 0 א א א א F א אא L Eאאאאא אא ،אאא Kאאא WאאאK٤ 55 ٤١١٥ אא אאאא Kא ،א אאאאאאאאא Kאאאא WEF AND אאK١{٤ אאE 1 F AND א אFKE 1 Fאא F א K Z ab W א K AND א E 1a b 1 AND a 1 F B AND A אאאE 1 א K AND אא 1b אKE K b a EFא AND א Z אא Z abc Wא 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 WEF OR אאK٢{٤ א א א א E 1 F OR אא K OR א E 2a אFאKE 1 F א E Z 1 F א K Z a b W א K AND אא 2b א KE b 1 OR a 1 F א א K b a EFא OR א Z אא K b 1 a 0 א Z 1 56 ٤١١٥ אא אאאא a b a Z=a+b OR b 2a א 2b א a b Z ab 0 0 1 0 1 0 0 1 1 1 1 1 WEאF NOT אאK٣{٤ ،אאא W Z 1 a 0 Z 0 a 1 W،אא،א Z אא Z a Wא NOT אא a Z a NOT a Z a 0 1 1 0 NAND אאאK NOT W NAND אאK٤{٤ AND אא NAND אא W، AND אא NAND אאאK AND אאא Z ab Wא NAND אא a a NAND b Z ab 57 b ab Z ab ٤١١٥ אא אאאא 0 0 0 1 0 0 1 1 1 0 0 1 1 1 1 0 K OR אא NOR אאאK NOT W NOR אאK٥{٤ OR אא NOR אא W، OR אא NOR אאא Z a b Wא NOR אא a NOR Z ab b a b ab 0 0 1 0 1 0 0 1 1 1 1 1 Z ab 1 0 0 0 Wאאא Z אW٣ a b AND NOT OR Z NOT c OR NOT 58 ٤١١٥ אא אאאא Wא ab W AND אאא a and b W AND אאא a WEאF NOT אאא ab WEאF NOT אאא a c W OR אאא a and c W OR אאא a c and ab W OR אאא a c W NOT אאא Z ab a b W OR אא Z אא W אאאא Kאאא א א אא א א א א Z א א W٤ Kאא a b OR NOT Z AND Wא a W NOT אאא a and b W AND אאא ab W AND אאא ab and a W OR אאא Z a ab W OR אאאא Wאא 59 ٤١١٥ אא אאאא Z a ab (a a)(a b) E 6 F Z a b E1011F Wא OR O אא א א a OR Z b WאאאאW٥ Wא W abc W א a annd b andd c WEא אF AND אאא abc W א a andd b and c WEאF AND אאא ab W א a and b WEאF AND אאא abc annd abc and ab W OR אאאא Z abc abc ab W Z א 60 ٤١١٥ אא אאאא א א ، אא א W٦ Kא a b 0 0 1 0 1 1 1 0 1 1 0 1 Z Wא W W 1 א Z ab OR ab OR R ab ab ab ab a b b ( a a ) ab b1 ab b Wאאאא אאא 61 ٤١١٥ אא אאאא NA AND אאאאאK٥ א א אאאא אאא אאאאK NAND אא א א אא אאא אא Kאא אא W NAND אאא אא NO NOT AN ND O OR a a Z aa 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 אא אאאא אאאא NAND אאאאאא אא K א NAND א א א Kאאאא NAND K NAND אאאאאאאW٧ Wא bc Wא b and c W AND אא bc d Wא bc and d W OR אאא bc d , e and a W AND אא Z ae(bc d ) abce ade Wאא א OR אא NAND אאאא א Z abce ade abce ade Wא W Z abce ade W NAND אאאאאא 63 ٤١١٥ אא אאאא 64 ٤١١٥ אא אאאא WאאאאאאW١ 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 ) אאאאאאאאW٢ Kאא 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 אא،אאא W٣ אא، NAND אאאאאאא، Kאא a OR b a) c d a OR AND OR OR b) b AND OR AND c 65 ٤١١٥ אא אאאא Z a b ab א NAND אאאW٤ K NAND OR ، AND אאאאאא Kאאאא א Z a b c א NAND אא W٥ Kאאא WאאאאW٦ NAND a NAND NAND b NAND NAND Wאאא C ab S ab ab WW٧ a NAND NAND NAND S NAND b NAND C 66 ٤١١٥ אא אאאא ،אאאאאEFW٨ K NAND אאאא a b 0 0 0 1 1 0 1 0 1 1 1 0 Z Kאא אאאW٩ 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 אאאאאW10 K NAND אאאא 67 ٤١١٥ אא אאאא 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 אא אאאא אאא א 69 ٤١١٥ אא אאאא אא אאא Wאא Kאא WאאאWאא K אאא א Kאאאא Kא Kאא 70 ٤١١٥ אא אאאא א WK1 אאאא ،אאאאאאאאאא אאאאאאאאאאא Kא אאאאאאאא אא Kאאאא Kאאאאאא א אאאאאאא א א א א א א א Kא אאאאאאא ،א א א א א א א א אאאאאא Kאאא אאאאאא אאא،אאא אאאאKאאאא אאא אאא K אא EDescriptive StatisticsF אאWאא WEInferential StatisticF אא 71 ٤١١٥ אא אאאא Descriptive Statistics אאE1 אא אאא אאאאא אאKאאאאאא Kאאאא Inferential StatisticsEאFאאE2 ،אאאאא אא،אאKאא אאאאא Kאאאאאאאאאאא W (Statistics)א אאאאא Kאאאא WאEאFאאאK2 אאא K W(Qualitative data)אEאFאאK١{٢ אאאאאאאאא Kאא W(Quantitative data)אEאFאאK٢{٢ אאEFאאאאא Kאאאא 72 ٤١١٥ אא אאאא W אאאא אאאאא، Kאאאאאאא WאאK3 (Population)אאK١{٣ א אא א א א אא ، Kא א א אא אאאאא א אא ،א א א א א אא א אא א Kא א אאKא אאאא אא ،א ،אאא א ، א Kאא K، א אא אא אאא א א אאאאאאKא K אא،א א א א א א א א K א א ، ،א א א א א 73 ٤١١٥ אא אאאא א ، א א א א א ، א א א א ؛א א א Kאאא W(Sample)אאK٢{٣ א Kאא א א א ،א א א אא א ،א K א WאאK4 אא א א א אא א א אאאאאKא אאאאKאאא א א א א אא אא K EF א א ،אא אאא א אאא א אא אא ، א א א א Kאאא אאאא א א א א א א א אא K א א א Kא א א 74 ٤١١٥ אא אאאא א אאאKא Kאאאאאאא Kא אאאאאאאא Kא K 120 א אא א א W 1 Kאאאאא 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 Wא אאאאאאא אאא א א א א Kאאאאא אא א ،א 10 4 א א K10אאא4אאא Wאאאא 75 ٤١١٥ אא אאאא ﻋدد اﻟﺳﻳﺎرات اﻟﺗﻛرار 1 18 2 15 3 25 4 10 5 17 6 18 7 17 اﻟﻣﺟﻣوع 120 W אא אא،א אKאא א،אא א Kאאאא Wא(Classes)אאK١{٤ ،אאאאאE١F Wאאא R א R Max Min W א: Max א : Min אאאאW m אE٢F WאאאYule،א m 4 n 2 .5 W א : m 76 ٤١١٥ אא אאאא אאאא : n Wא L אE٣F L R m W Kא א L א m א א אא א א א ،אא א (F ) אא אא א Kאאא ( Fi ) אא אאא Fi אאא K n אא אא אאאאא W ٢ א א א K א Kא 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 Wא 77 ٤١١٥ אא אאאא W R אE١F R Max Min 99.3 0.3 99 m 5 ،אE2F WאE٣F R 99 19.8 20 m 5 א אאאאא L אא אאא WאאאKאאאאאא (ﺣدود اﻟﻔﺋﺎت )آﻟف ﷼ F اﻟﺗﻛرار 0 – 20 14 20 – 40 26 40 – 60 17 60 – 80 22 80 – 100 11 اﻟﻣﺟﻣوع 90 K65אאאאW٣ 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 Wאא 78 ٤١١٥ אא אאאא Kאאא אE١ Kאאא אE٢ K219200אאאא E٣ Kאאא E٤ K160אאא E٥ Wא 120אאא אE١ 179אאא אE٢ ٢219200אאאא E٣ 120 – 139אאא E٤ אאאא160אאא E٥ 8 + 10 +16 + 14 = 48Wאאאא W(Relative frequency)אאאK٢{٤ אאאאאאאא K100אאאאאאא F% Fi 100 Fi א א א אא Kאא א א א א אאאאאאא אא Kאאא א א א ٢ א אא W ٤ Wאאא 79 ٤١١٥ אא אאאא Wא (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)אK٣{٤ W،אאאאאא U Li xi i 2 W i : xi i אא : U i i אא : Li Wא٢W٥ (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 אא אאאא WאLאאאאאK٤{٤ אאאאאאא Kאאאאאאאא א א א אא אא א Kא אא א אאא א אא א ،א אאאאאאאא א אאא א אא א Kא אאאאא א K א אא אאא א אא אאאאאאא ?אא، () ??א،א Kאאאא () ? W אאאאאאאאא א א א אא א ، א א Kאאאא ، ٢אאאאאא W ٦ W 81 ٤١١٥ אא אאאא Wאאאא ﺣدود اﻟﻔﺋﺎت اﻟﺻﺎﻋدة اﻟﺗﻛرار اﻟﺻﺎﻋد 0 0 20 14 40 40 60 57 80 79 100 90 Wאאאא ﺣدود اﻟﻔﺋﺎت اﻟﻬﺎﺑطﺔ اﻟﺗﻛرار اﻟﻬﺎﺑط 0 90 20 76 40 50 60 33 80 11 100 0 WאאאאאW٦ K40אאא Ea K60אאאא Eb K 8040אאא Ec Wא ، 50 40אאא Ea Kאא،אאאא 82 ٤١١٥ אא אאאא Kאאא،5760אאאא Eb א398040אאא Ec אאE40F 40אאאאא א،אאE79F 80אא אאE50F 40אאא KאאE11F80א WאאK5 א א Kאאאאאאא א א Kא Wאאא אא א אא א אא א אאאא א א א אא א Kאאאאאא Kאאאאאא 83 ٤١١٥ אא אאאא W(Histogram)אאאK١{٥ אאאאאא א א א Kאא KאאאKאא אאא،אאא 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)אאאK٢{٥ א אא א א א Kאאאאאא אאא،אאא٢W٨ W 84 ٤١١٥ אא אאאא Frequency 30 27 25 23 20 15 16 14 Frequency 10 9 5 0 ٢٠ ٤٠ ٦٠ ٨٠ ١٠٠ More WאאK6 א א א א א א א א אKאאאאאא Kאאאאא אאאאאאא א Kא א Kאאא א א א א א אאאאKא Kאאאאא KEאFא אאאאאאא ،אאאאאKא א א א 85 ٤١١٥ אא אאאא א א א K א א א א א א Kאאא،אאאאא W(Mode)אאK١{٦ אKא אאאא א،אאאא Kא K5א3,5,5,5,5,9,9,7אאאאWא Kא4,2,5,7,6,1,3אאא אא א א אא K W(Median)אK٢{٦ אאאאKאאא אא א אאאאאאאא Kאא א x1 , x 2 , x3 ,....., x n אא 1 K n א x n x n 2 nא x n 1 2 2 2 2 Kא ~ x א ~ 10,8,9,6,7WאאW٩ 86 ٤١١٥ אא אאאא Wא 6,7,8,9,10 א W، n 5 x n1 x51 x 6 x3 8 2 2 2 ~ 8 א 9,10,6,2,1,7אW١٠ Wא 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)אLאאK٣{٦ אא א א א א א n K xi i 1 n Wא x1 , x 2 , x3 ,....., x n Wא K x 87 ٤١١٥ אא אאאא 8,10,13,9,7,11,10אאאאW١٠ Wא Wא n xi i 1 n 8 10 13 9 7 11 10 68 9.71 7 7 9.71א 88 ٤١١٥ אא אאאא WאW١ 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 Wא Kאאא Ea Kאאא Eb K75אא Ec Wאאאא٧٠W٢ 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 ٩٠אא Ea ؟א Kאאאא אEb Kאאאא אEc 89 ٤١١٥ אא אאאא WאאאאאאW٣ 7,9,6,8,9,10,6,11,8,7,9,10,9,7,6,10,8,11,9,8 Wא Kאאא Ea Kא אEb Kא Ec WאאאאאאאW٤ 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 Kאאאא (a Kאאאא (b Kאאא( אc 90 ٤١١٥ אא אאאא אא א אא ،אא K١٩٨٣ J١٤٠٣،،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،אא K،K ٢٠٠٣،אאא، Introductory Statistics. London : Addison-Wesley Publishing Company, 1982 Hassett, Weiss 91