Ensembles et applications

A B
A∪B=B⇔A⊆B
A∩B=B⇔B⊆A
A B C
A∪(B∩C)=(A∪B)∩(A∪C)
A∩(B∪C)=(A∩B)∪(A∩C)
n∈NEn={kn |k∈N, k ≥2}
E=[
n∈N,n≥2
En, X =N\E.
P X ={0,1} ∪ P
Z→Z
n7→ 3n+ 2.
R→R
x7→ x3+ 1.
]0,+∞[→R
x7→ 1/x.
{ } → { }
x7→ x.
{ } → { }
x7→ x.

cos : R→R
cos : R→[−1,1]
cos : [0, π]→[−1,1]
sin : R→R
f:A→B g :B→C
g◦f:A→C
f g ⇒g◦f
f g ⇒g◦f
g◦f⇒f
g◦f⇒g
g◦f⇒g
g◦f⇒f
A B
A B
A B
A B A B
A B Card A
Card B A B
A B Card A= Card B
A B Card A≤Card B
A B Card A≥
Card B
f:A→BCard A≥Card B f
g:A→BCard A≤Card B g
N2N
N Z
n

n∈N2n> n
n≥1n
n≥2 1 ·2+2·3 + . . . + (n−1) ·n=1
3n(n−1)(n+ 1)
n
n
X
k=1
k=n(n+ 1)
2
n
X
k=1
k2=n(n+ 1)(2n+ 1)
6
n
X
k=1
k3=n2(n+ 1)2
4
a > −1n∈N(1 + a)n≥1 + na
n n
2n
2n≤n!
1+3+
5 + ··· + (2n−1)
n∈Nn≥2n
n= 2
n(n+ 1)
A1, A2, . . . , An+1 A1Ann
A1
A2AnA2An+1 n
An+1
A2AnA1An+1
A2An(n+ 1)
n n ∈N5n1
4

x6= 1 1 + x+x2+··· +xn−1=1−xn
1−x
n∈Nn3+ 5n6
(x+y)n=
n
X
k=0 n
kxkyn−k.
n
k=n! k!(n−k)!
n+ 1
k+ 1=n k+ 1+n k
n∈Nn unvn
(1 + √2)n=un+vn√2un+1 vn+1 unvn
n∈Nu2
n−2v2 n= (−1)n
n
n−1n−1
Rnn
R1R2R3R4
RnRn−1
Rnn

n∈Nk∈Nk≤n
n
k=n! k!(n−k)!
n+1
k+1=n k+1+n kn
k
k n
Q
a, b, c, d ∈Qa≥b c ≥d a +c≥b+d
a, b, c, d ∈Qa≥b≥0c≥d≥0ac ≥bd
a, b ∈Qa≥b≥0a2≥b2n∈Nan≥bn
x /∈Qy∈Q⇒x+y /∈Q
x /∈Qy∈Q\ {0} ⇒ xy /∈Q
3
√2/∈Q
√3/∈Q
√3 + √7/∈Q
(√3 + √7)(√3−√7)
a√a /∈Q
a∈N√a /∈Q
a, b ∈N√a+√b√a−√b