ensembles 1 définitions d`ensembles de nombres : N ⊂ Z ⊂ D ⊂ Q

N⊂Z⊂D⊂Q⊂R⊂C
N N ={0,1,2,3....}
Z Z ={..., −11,−10...0,1,2,3....}
D
D={...0,1...0,569... −0,25...2569,5...}
Q
Q={..., −11
1... 10 3... −58 −256 ... 2,7 580 ...}
R R ={..., −11,−10...0,569... −0,25...0,1,2,3...π...√10}
C{..., −11,−10...0,569... −0,25...0,1,2,3...π...(1 + i√2)...e3i....√10} ∈ C
(E, •)
E•
a, b, c A
a•b∈A⇔
(a•b)•c=a•(b•c)⇔
∃e a •e=e•a=a⇔
a•b=b•a⇔
•
• ∀(a, b)∈E2, a •b∈E
∀(a, b, c)∈E3, a •(b•c)=(a•b)•c
∀a∈E, a •=•a=a
∀a∈E, ∃b∈E tel que a •b= 0
Z

(A, +,×)
A+×(A, +)
a, b, c A
(a+b) + c=a+ (b+c)
a+ 0 = 0 + a=a
a+ (−a)=(−a) + a= 0
a+b=b+a
⇔(A, +)
(a×b)×c=a×(b×c)⇔
a×1=1×a=a⇔
a×(b+c) = a×b+a×c⇔
(b+c)×a=b×a+c×a⇔
∀a∈A, ∀b∈A, a ×b=b×a
(A, +,×)⇔(A, +,×)∀a∈A∗,∃b∈A tq a ×b=b×a= 1
(A, +,×
A+×(A, +)
(A\{∅},×)
a, b, c A
(a+b) + c=a+ (b+c)
a+ 0 = 0 + a=a
a+ (−a) = (−a) + a= 0
a+b=b+a
⇔(A, +)
(a×b)×c=a×(b×c)
a×1=1×a=a
a×(a−1) = (a−1)×a= 1
a×b=b×a
⇔(A\{∅},×)
a×(b+c) = a×b+a×c⇔
(b+c)×a=b×a+c×a⇔