Cours - Université Jean Monnet

EP(E)E
A, B ∈P(E)A B A ⊂B
A⊂B B ⊂A A =B
x∈E{x} ∈ P(E)
x, y ∈E{x, y} ∈ P(E)
E F
E×F:= {(x, y), x ∈E y ∈F}
E1×E2× ··· × En:= {(x1, . . . , xn), xi∈Eii}
A∪B
A∩B
ArB:= {x∈A, x 6∈ B}
A=Ac:= ErA
A∆B:= (ArB)∪(BrA)
A∪B=A∩B A ∩B=A∪B

(Ai)i∈IE
[
i∈I
Ai:= {x∈E, x ∈Aii∈I}
\
i∈I
Ai:= {x∈E, x ∈Aii∈I}
[
i∈∅
Ai=∅\
i∈∅
Ai=E
E
(Ai)i∈IE
Ai6=∅i∈I
Ai∩Aj=∅i, j i 6=j
E=∪i∈IAi
E F
E F f = (E, F, Γ) Γ ⊂E×F
f x ∈E y ∈F
(x, y)∈Γy f(x)
f:E→F x ∈E y ∈F
(x, y)∈Γ
f:E→F Dfx∈E y ∈E
(x, y)∈Γf
f= (R,R,Γ) Γ := {(x, y)∈R2, xy = 1}f:R→R
R∗f:R∗→Rx7→ f(x)=1/x
f= (R,R,Γ) Γ := {(x, y)∈R2, x =y2}
f= (R,R+,Γ) Γ := {(x, y)∈R×R+, x =y2}x∈R
y>0y2=x y √x
fR+
E A ∈P(E)A
χA:E→ {0,1}
x7→ (1x∈A
0x6∈ A.

f:E→F A ∈P(E)
A∈P(E)
f(A) := {y∈F, ∃x∈A, y =f(x)} ⊂ F
A f
B∈P(F)
f−1(B) := {x∈E, f(x)∈B} ⊂ E
B f
x∈E y =f(x)x y
f:E→F
f x1, x2∈E
f(x1) = f(x2)=⇒x1=x2.
f y ∈F x ∈E y =f(x)
f f y ∈F
x∈E y =f(x)
E F
f:E→F
E F
f:E→F
E F
f:E→F
E m ∈N∗E{1,2,3, . . . , m}
Card(E) = m.
E E
Card(∅) = 0.
E
{1,2,3, . . . , m}

E F
E F Card(E)6Card(F)
E F Card(E) = Card(F)
E F E F Card(E)<
Card(F)
E F F
E E F Card(E)6Card(F) Card(F)6
Card(E) Card(E) = Card(F)
E X E f E
X Y0=ErX Yn+1 := f(Yn)n∈NY:= ∪n∈NYn
g:E→X
g(x) = (f(x)x∈Y
x x ∈ErY
g E X
E F u E F
v F E
X E F
E F
ut
EP(E)E
E
Card(E)<Card(P(E)).
x7→ {x}
x∈E{x} ∈ P(E) Card(E)6Card(P(E))
f:E→P(E)
A:= {x∈E:x6∈ f(x)}.
A f x0∈E
A=f(x0)x0∈A x06∈ A x06∈ A x0∈Aut

E E NCard(E) = Card(N)
E E Card(E)6
Card(N)
f:n7→ (n, 0) N N ×N
g: (n, m)7→ 2n(2m+ 1) N×N N
N N ×NCard(N) = Card(N×N)
E J (Aj)j∈J
E j ∈J Aj
[
j∈J
Aj
J
j∈J Aj
J J =N
j∈J
Aj=[
k∈N{aj,k},
[
j∈N
Aj=[
j∈N[
k∈N{aj,k}=[
(j,k)∈N×N{aj,k},
N×Nut
E
Card(P(E)) >Card(E)P(N)