Exercices de logique et théorie des ensembles

ε≥0q∈Q∗0≤q≤ε
x∈Rx2<0
P, Q, R
Pet Qnon Pet non Q
P⇒Qnon Q⇒non P
P⇒Q P et Q
a, b, c
a≤ −2a≥3
a≤b|a|> c
3≤a≤7 10 ≤b≤17
a a2a/2
2x7−4x4+ 2x+ 3 = 0
x5−2x4−8x+ 16 = 0
nPn
k=0 2k= 2n+1 −1
nPn
k=0 k2=n(n+ 1)(2n+ 1)/6
n n ≥10 2n≥n3n≥10
2≥(1 + 1/n)3
A={x∈Q:x≤0}
B={a
b:a, b ∈Z, b 6= 0 b}
C={z∈C:z2=−4}
D={z∈R:z2=−4}
E=Q
F=∅

E={1,2,3,4,5,6,7}E
A={1,2,3,4}
B={4,5,6,7}
C={1,3,5,7}
D={2,3,4,5,6}
(A∩B)∪(C∩D) (A∪C)∩(B∪D) (Ac∩D)c∩(B∪C)c
A, B E
(A∩B)c⊂Ac∩Bc
(A∩B)c⊃Ac∩Bc
(A∪B)c⊂Ac∪Bc
(A∪B)c⊃Ac∪Bc
R2
([−1,1] ×[−1,1])c
[−1,1]c×[−1,1]c
{(a, b)∈Z2:a2+b2≤4}
A, B, C, D E
(A∩B)×(C∩D)⊂(A×C)∩(B×D)
(A∩B)×(C∩D)⊃(A×C)∩(B×D)
(A∪B)×(C∪D)⊂(A×C)∪(B×D)
(A∪B)×(C∪D)⊃(A×C)∪(B×D)
f:R→R;x7→ x3−x+ 2 g:R→R;x7→ x2−4f◦g g ◦f
fg :R→R;x7→ f(x)g(x)
f:R→Rf(−x) = f(x)x∈Rf(−x) = −f(x)
x∈Rs:R→R;x7→ −x
R R
g:R→Rg◦s=g
g:R→Rg◦s=s◦g

f(]0,1[) f([1,2[) f([−3,−2]) f−1(]0,1[) f−1([1,2[) f−1([−3,−1]) f−1([−1
2,3])
f:R→R
f:x7→ x2
f:x7→ xx
f:x7→ 1
1+x
f:x7→ sin x
f:E→F A, B ⊂E C, D ⊂F
f(A∩B) = f(A)∩f(B)
f(A∪B) = f(A)∪f(B)
f−1(C∩D) = f−1(C)∩f−1(D)
f−1(C∪D) = f−1(C)∪f−1(D)
f−1(f(A)) = A
f(f−1(A)) = A
n Sn{1, . . . , n}
Sn
f: ]0,1[ →]0,1[×]0,1[ g: ]0,1[×]0,1[ →]0,1[
A FAA{0,1}FA
P(A)
P(N)R
E i :E→E
NE