Éléments de logique - Raisonnements I Le calcul propositionnel

V F
π . . .
. . .
P P (P)
P P
(P)
P(P)
V
F
P Q P Q P Q
P Q
P Q P Q
V V
V F
F V
F F

P Q P Q P Q
P Q
P Q P Q
V V
V F
F V
F F
P Q P Q P ⇒Q
P Q Q
P Q P ⇒Q
V V
V F
F V
F F
P⇒Q P P Q
P Q P Q P ⇔Q P
Q . . .
P Q P ⇔Q
V V
V F
F V
F F

f g
f≡g
[( P)Q]
[P Q] [( P)Q]
. . . . . .
(P)≡
[Q(P)] ≡
[(P⇒Q) (Q⇒P)] ≡
(P)⇔P
P(P)
P⇒P
P Q
(P⇒Q)≡[( Q)⇒(P)]
(Q)⇒(P)
P⇒Q
(P Q)≡; (P Q)≡

A
A(x, y, . . .)
xRx
x A(x)
A(x)A(x)
A(x)x
∀ ∃
• ∀x, A(x)x A(x)
A(x)x
• ∃x, A(x)x A(x)
A(x)x
• ∃!x, A(x)x A(x)
A(x)x
∃x, A(x)
x
∃x∈R+,3x+ 2 >1x= 0 3 ×0 + 2 = 2 >1
∀x∈R+,3x+ 2 >1x
x x >0

f[0 ,+∞[f
∀x∈[0 ,+∞[, f(x) = 0.
f
(∀x, P (x))
(∃x, P (x))
∀,∃A(x)
∀ ∃ A(x)A(x)
(un)n∈N
∃M∈R,∀n∈N, un6M
(vn)n∈N