Logique, ensembles, applications, dénombrement

∃x∈R,∀y∈R, x +y > 0∀x∈R,∀y∈R, x +y > 0
∃a∈R,∀b > 0,|a|< b ∀b > 0,∃a∈R,|a|< b
f g R R
f
x f(x)>2
x y x < y f(x)< f(y)
f
f
f
f g
A B E
A∩B=A∪B A =B
A⊂B, A ∪B=B, Ac∪B=E.
A B E A B
A∆B E
A∆B={x∈A∪B\x /∈A∩B}= (A∪B)\(A∩B).
A∆B= (A∩Bc)∪(B∩Ac)
A∆A A∆∅A∆E A∆Ac
C E
(A∆B)∩C= (A∩C)∆(B∩C).
(A∆B)∪C⊃(A∪C)∆(B∪C).
f:Z→[−1; +∞[f(x) = x2x≤0f(x)=2x−1x > 0
f
E, F G u :E→F v :F→G
u v v ◦u
u v v ◦u
v◦u u
v◦u v
v◦u u v

E F f :E→F
A B E
A⊂B f(A)⊂f(B)
f(A∩B)⊂f(A)∩f(B)
f(A∪B) = f(A)∪f(B)
f A B E f(A∩B) =
f(A)∩f(B)
R
R=⊥
P(E)E ARB A =B A =Bc
Z∼
n∼m⇐⇒ n+m .
∼
0 1
CR
zRz0⇐⇒ |z|=|z0|.
R
z∈C
n p 1≤p≤n
n
p=n n−p
n+ 1
p+ 1=n p+ 1+n p
nn−1
p−1=pn p
p, q n n ≤p n ≤q
p+q
n=
n
X
j=0 p
j q n−j.
(1 + x)p+qx∈R
p
X
k=0 p
k2n
X
k=0
kn
k2
10
10
n n ≥1k1≤k≤n
5 32
5 26