Logique et raisonnement mathématique

3>1
√2>2
x x3−1=(x−1)(x2+x+ 1)

n>1 2n>n2
x, y, n, . . .
P(x)ex>1x
x < 0P(x)x>0P(x)
ln(x)>0x>1x∈R
P(P)P
PP
P P (P) ( (P)) =
P
P Q P Q
P Q
P Q P Q P Q
P Q P Q
π>2e>1
n n n
P(P)

P Q P Q
P Q
P Q P Q
P Q P Q
π π > 2
exp x∈Rexp0(x) = exp(x)
P Q (P) (Q)
P Q (P) (Q)
P Q (P⇒Q)P Q
(P⇒Q)P Q
(P⇒Q)P Q
P Q P Q
(P⇒Q)P Q P
Q Q P
(P⇒Q) (Q⇒P)P
Q
⇒
P P ⇒Q Q

(P=⇒Q P (Q)
(P=⇒Q P Q
=⇒
(P⇒Q(Q⇒P)
P Q (P⇒Q(Q⇒P)
P⇐⇒ Q
P⇐⇒ Q
(P⇐⇒ Q)
P Q
P Q
P Q
Q P
(x6y)⇐⇒ (x < y x =y)
(x2= 9) ⇐⇒ (x= 3 x=−3)
(x2>9x>0) ⇐⇒ (x > 3)
(P=⇒Q) (Q) =⇒(P)
(Q) =⇒(P) (P=⇒Q)
∃
∀

∃!
/
∀x∈R, x2>0
f∃x∈R/ f(x)=0
x3+x+ 1 = 0 ∃!x∈R/ x3+x+ 1 = 0
E P (x)x x ∈E
(∀x∈E, P (x)) ( ∃x∈E / (P(x)))
(∃x∈E, P (x)) ( ∀x∈E / (P(x)))
f g R
f=g
(f=g)⇐⇒ (∀x∈R, f(x) = g(x))
f g x
(un)
((un) ) ⇐⇒ ∀n∈N, un6un+1
(un)
((un) ) ⇐⇒ ∀n∈N, un+1 6un
(un)
((un) ) ⇐⇒ ∃M∈R/∀n∈N, un6M
(un)
((un) ) ⇐⇒ ∃m∈R/∀n∈N, m 6un
(un)
((un) ) ⇐⇒ ∃m∈R,∃M∈R/ m 6un6M
∀x∈R,∃y∈R/ x 6y∃y∈R/∀x∈R, x 6y