2009 Feuille d9exercices n¦3 : logique Exercice

ε q 06q6ε
x y z z < y
z < x + 1
a b c
a6−2a>3
a6b|a|> c
36a67 10 6b617
∀x∈A, ∀p∈P, x p
∃x∈A, ∀p∈P, x p
∀x∈A, ∃p∈P, x p
∀p∈P, ∃x∈A, x p
∃p∈P, ∀x∈A, x p
∃p∈P, ∃x∈A, x p
a a2a
2
2x7−4x4+ 2x+ 3 = 0
x5−2x4−8x+ 16 = 0
fR R f
f(x) = 0
f(x)6= 0
∃x∈R, f (x) = 0
∃x∈R, f (x)6= 0
∀x∈R, f (x) = 0
∀x∈R, f (x)6= 0

∀x∈R,∀ε > 0,(|x|< ε ⇒x= 0)
∀x∈R,(∀ε > 0,|x|< ε)⇒x= 0
√2√2
√2√2
u u√2
A⇒C B ⇒C
x y x < y
y x x < y
x y x =y2
x z x =z2
z y y =z2
x x >1⇒x2>1
x x x
x P (x) = ax2+bx +c
0c= 0
a b a > b a2> b2
a b a > b a2> b2
∀a∈R,∀x∈R, x2+ 2ax + 4 >0⇒∀x∈R, x2+ 2 >a
∀a∈R,∀x∈R,x2+ 2ax + 4 >0⇒x2+ 2 >a