Cours polycopié pour le module Mathématique II

N Q
R C
:=
x x2:= x.x
R(R,+)

2>3
P(x)x > 3P(2)
P(4)
P(x, y)x+y= 2 P(1,1)
P(2,1)
P(nonP )¯ P
P P
P(x)x > 3 (nonP )(x)x≤3x < 3
non(nonP ) = P.
∀ ∃
∀ ∃
∀x∈E, P (x)x E P (x)
P
”x E

P(x) : x > 3∀x > 4, P (x)
∀x > 2, P (x)
∃x∈E, P (x)x E P (x)
∃x∈E|P(x)
P(x) : x > 3∃x > 2|P(x)∃x < 2|P(x)
E
2 2
E
∃y > 2,∀x > y, P (x)
Q(z) : ∀y > z, ∃x < y |P(x).
∀x∈E, P (x)∀a∈E, P (a)
∃y > 2| ∀x≥y, P (x) ”∃x > 2| ∀x≥x, P (x)
non”∀x∈E, P (x)” = ”∃x∈E, nonP (x)”,
non”∃x∈E, P (x)” = ”∀x∈E, nonP (x)”.
∀x∈E, P (x)∃x∈Ec|P(x)
EcE
∀x > 2, x > 3∃x > 2|x≤3
∃x≤2|x≤3
∃x > a, ∀y < x, y > 4
∀ ∃
∀x∈E, ∃y∈F, P (x, y)∃y∈F, ∀x∈E, P (x, y)
∀y∈E, ∃x∈F, P (y, x)
∀x > 3,∃y > 2, y > x ∃y > 2,∀x > 3, y > x

∀x∈E, ∀y∈F, P (x, y, z)∀y∈F, ∀x∈E, P (x, y, z)
∃x∈E, ∃y∈F, ∃y∈F, ∃x∈E,
∀x∈
E, ∃y∈F, ∀z∈G, P (x, y, z)∀z∈G, ∃y∈F, ∀x∈
E, P (x, y, z)
x, y, z {1,2}
∀x, ∃y, ∀z, x =y y ≤z+ 1
∀z, ∃y, ∀x, x =y y ≤z+ 1
∀ ∃ ∃y, ∀x, P (x, y)
∀x, ∃y, P (x, y)
∨ ∧
P∨Q P Q P Q
P∧Q P Q P Q
P(x) : ”x > 2” Q(y) : ”y < 3” (P∨Q)(x)
(P∧Q)(x) 2 < x < 3
non(P∧Q) = (nonP )∨(nonQ),non(P∨Q) = (nonP )∧(nonQ).
∨ ∧
P∧Q=Q∧P, P ∨Q=Q∨P.
∨ ∧
(P∧Q)∧R=P∧(Q∧R),(P∨Q)∨R=P∨(Q∨R).
P∧Q∧R P ∨Q∨R
∨ ∧
P∧(Q∨R) = (P∧Q)∨(P∧R), P ∨(Q∧R) = (P∨Q)∧(P∨R).
P P ∧(nonP )P∨(nonP )