Feuille de TD no 1. Logique, ensembles, applications

πππππππππππππππππππππππππππππππππππππππππππππππππππππππππππ
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πππππππππππππππππππππππππππππππππππππππππππππππππππππππππππ
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   →
f:R→R
f
f
f
f
f2π
f(x)=2
f2 3
   →
   →
IRf:I→RI
∃a∈R:∀t∈I, f(t) = a
∀y∈R,∃x∈I:f(x) = y

∀M∈R,∃r∈I:f(r)≥M
∃(m, M)∈R2:∀x∈I, m ≤f(x)≤M
∀x, y ∈I, x < y ⇒f(x)< f(y)
∀x1, x2∈I, x16=x2⇒f(x1)6=f(x2)
f:R→R
f
∀x≥0,∃y≤0, f(x) = f(y)
∀x∈R,∃y∈R:x=y f(x)6=f(y)
∀x∈R,∃y∈R:f(x) = y
∃x∈R,∃y∈R:f(x) = f(y)
∀x∈R:f(x)=2f(x)
∀t∈R,∀s∈R, f(t) = s
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R
E1={5,8,11,14,17, . . .}E2={x2, x ∈[[1,5]]}E3= [−3
2,3 2]∩Z
E4={y2, y ∈[−5,−1]}E5= [[−1,1]] E6= [1,+∞[∩[0,+∞[∩]−1,25]
E7= [1,25] E8={3x+ 2, x ∈N∗}E9={m∈[1,25] : ∃k∈N, m =k2}
E10 ={−1,0,1}E11 ={n∈N∗:∃k∈N∗, n = 3k+ 2}
E12 ={3n+ 2, n ∈N∗}E13 ={m∈Z:m≤1m≥ −1}
E14 ={t2, t ∈[1,5]}E15 ={sin(kπ
2), k ∈Z}E16 ={1,4,9,16,25}
   R
x∈R
x > 4x < 7x6= 6
x > 0x < 3x= 0

x < 3x∈Nx= 2
x∈R+x=−3x < 0
∃u∈[3,+∞[: x=u2
   R2
R2
A={(x, y)∈R2:x+y≥1}
B={(x, y)∈R2:xy < 0}
C={(x, y)∈R2:y≤min(x, 2−x)y≥ −1}
D={(x, y)∈R2:y≥x2y2+ (x−1)2≤1}
A={y∈R:∃t∈[3,+∞[: y=t2}
B={y∈R:∀x≤9, y > x}
C={y∈R:∀t∈[3,+∞[, y 6=t2}
D={y∈R:∃x≤9 : y≥x}
P(E)E={1,2,3,4}
P(E)
A, B, C E
(A∪B)\A=B∩A.
A∪B=A∪C, A ∩B=A∩C.
B=C

   R2
R2
F1={(x, y)∈R2:y≤0}
F2={(x, y)∈R2:xy ≥1x≥0}.
M1M2M1M2R2
F1F2
∀ε > 0,∃M1∈F1,∃M2∈F2:M1M2< ε
∃M1∈F1,∃M2∈F2:∀ε > 0, M1M2< ε
∃ε > 0 : ∀M1∈F1,∀M2∈F2, M1M2< ε
∀M1∈F1,∀M2∈F2,∃ε > 0 : M1M2< ε
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u
u(x) = ln(x−1) + √4−x
x4−16 +cos x sin x−1.
u:R→R
x7→ cos(2x)
v:R→R
x7→ ex
w:R→R
x7→ x3
y:R→R
x7→ x3−x
f:Z→Z
x7→ 2x
g:R→R
x7→ 2x
h:Z→R
x7→ 2x
F:R→Z?
x7→ 2x

fR R E
Rf(E)
E= [π/4,5π/6], f(x) = cos x
E= [0,√17], f(x) = bxc
E= [−1,2], f(x) = x2
f:E→F g :F→G
g◦f f
g◦f g
u:E→F v :F→E u ◦v
v◦u u v
h:R→]0,+∞[
x7→ ln(1 + ex)
h h−1
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) = f(A)∩f(B)