TD 1. Opérations ensemblistes

X A, B, C, D X
A4B A B
A4B= (A∪B)r(A∩B)=(ArB)∪(BrA)
X A 4∅A4X A 4A
(ArB)∩(CrD) = (A∩C)r(B∪D)
(A∪B)=(A4B)4(A∩B)
(A4B)4C=A4(B4C)
X Y f :X→Y
f g :Y→X
g◦f= idX
f h :Y→X
f◦h= idY
? X Y f :X→Y
A⊆X A ⊆f−1(f(A))
f
A X A =f−1(f(A)) f
B⊆Y f(f−1(B)) ⊆B
f
B Y f(f−1(B)) = B f
? X Y f :X→Y(Ai)i∈I
X(Bj)j∈JY I J
A X B Y
f[
i∈I
Ai=[
i∈I
f(Ai)f\
i∈I
Ai⊆\
i∈I
f(Ai)f
f−1[
j∈J
Bj=[
j∈J
f−1(Bj)f−1\
j∈J
Bj=\
j∈J
f−1(Bj)
c(f−1(B)) = f−1(cB)c(f(A)) f(cA)

X Y f :X→Y
Φ : P(X)→ P(Y)A X f(A)Y
fΦ
fΦ
Ψ : P(Y)→ P(X)B Y f−1(B)
X
fΨ
fΨ
? A, B, C X
X
X
A+B
A−B
A B
|A−B|
A+B−A B
|| A−B|− C|
|A−| B−C||
sup( A,B)
inf( A,B)
XP(X)
X{0,1}XX{0,1}
X n >1A1, . . . , AnX
Sn
i=1 Ai=
n
X
k=1
(−1)k+1 X
I⊆{1,...,n}
(I)=k
Ti∈IAi.
n= 1 n= 2 n= 3
A, B, C, D f :A→
B g :C→D
P(A)P(B)
A∩C=B∩D=∅A∪C B ∪D
A×C B ×D
ACBD
A, B, C AB×C(AB)C
X Y P(X×Y)P(X)×P(Y)

A⊆N NrA
?(an)NR
lim inf ] − ∞, an]⊆]− ∞,lim inf an]
]− ∞,lim inf an[⊆lim inf ] − ∞, an[
(An)n≥0X
X
∅, X, ∞
[
n=0
An,∞
\
n=0
An,lim inf An,lim sup An.
(An)n≥0(Bn)n≥0X C
X
lim
n→∞ Ac
n=lim
n→∞ Anclim n→∞(An∩C) = lim n→∞ An∩C
lim
n→∞(An∪Bn) = lim n→∞ An∪lim n→∞ Bn(∃n0≥0∀n≥n0An⊂Bn)⇒lim n→∞ An⊂lim n→∞ Bn
?
(An)n∈NX ϕ :N→N
lim sup Aϕ(n)⊆lim sup An
lim inf An⊆lim inf Aϕ(n)
(An)B⊆X(Aϕ(n))n∈N
B
(Aϕ(n)) (An)
(A2n)n(A2n+1)nB
(An)B

?
(−1
n,1)n∈N∗(−1 n,1)n∈N∗.
R]0,1] ]0,1[
(Bn)n≥1R
B2n−1=−2−1
n,1B2n=−1,2 + 1 n2.
[−1,2] [−2,1]
(an)n∈N(bn)n∈N
lim
n[an, bn]=[−1,1[.
limn[an,6] a
(An)n∈NN
AN
X6=YAX∩Y
AR

?
N{0,1}{0,1}N
?
f: [a, b]→R
J(n) = {x∈]a, b[ ; |f(x+) −f(x−)|>
1/n}R
? X Card X2= Card X
Card XX= Card P(X)
?
N R
R
P(N)P(P(N)) P(P(P(N))) Q×Q∪n∈NNn
QQQNNQR×R R ×Z
{0,1}N{0,1}RRRNRRN
R
R2
C C
R2Q2