Logique et raisonnement mathématique

X
X X
P X P
x≥3R[3; +∞[
X∀x∈X, P (x)
∃x∈X, P (x)
P
PeP
P Q P ∨Q
P Q P ∧Q
P Q
P Q
P⇒Q
(eP)∨Q
P Q
=P Q
Q P
P⇔Q
=. . .
X P A
Q B P Q

P Q P
Q P Q
∃ ∃x0∈X
∀
∀x∈X, ...
n(n+ 1)
n∈N
X=NP(n)n n 2n
X=N×NQ(k, n)k n ⇐⇒ ∃n0∈Nn=n0k
X=RP(x)x2+ 5x+ 1 = 0
P∃x∈R, x2+ 1 = 0
X=N×RP(n, x) : n≥x
X=RP(x)∀n∈N, n ≥x
∀n∈N,∃x∈R, n ≤x
∃x∈R,∀n∈N, n ≤x
∀x∈R,∀n∈N, n ≤x
∀x∈R,(∀y∈R, x ≤y2)⇒x≤0
∀x∈R,∀y∈R,(x≤y2⇒x≤0)
• ∃x∈X∀x∈X
•
•
•P P (x, y, z, . . . )
∀x∈X, P (x, y, z, . . . )∃x0∈X, P (x, y, z, . . . )
Q(y, z, . . . )X
•
•
• ∀ ∃ ∃ ∀ ∀x∈X, P (x)
∃x0∈X P (x0)∃x0∈X, P (x0)∀x∈X P (x)
• ∀x∈X, P (x)∃x∈X, P (x)x
•P⇒Q Q P P
Q P ⇐⇒ Q P Q

P⇒Q
eQ⇒eP
PeQ
P⇔Q P ⇒Q Q ⇒P
A=B A ⊂B B ⊂A
P P
P(n)
∀x∈X
x∈X
∀
x∈Rx2
x
x
x∈R
x
∃
x0∈X
x0∈X
x0X
n2k
k n = 2k n
x≥0
x≥0
x < 0
x≥0
x≥0
x≥0
x≥0
x≥0
P Q
P Q
P Q
P Q

P⇒(Q R) (P Q)) ⇒R
P⇒Q Q ⇒P
P⇒Q
P⇐⇒ Q P ⇐⇒ Q
Q,R,C
fR