Logique, ensembles, méthodes de démonstration

V F
P Q P ≡Q
•
•
P P, P ¬P P
P P P P
V F
F V
P P
(P)≡P
P Q
•P Q
P∧Q P Q
•P Q
P∨Q
P Q
P Q P Q P Q
V V V V
V F F V
F V F V
F F F F

P Q P ⇒Q
[P⇒Q]≡[P Q]
P⇒Q P Q
P Q P ⇒Q
V V V
V F F
F V V
F F V
P⇒Q P Q
P Q P Q
P Q Q P
Q P P Q
P Q
P⇔Q
[P⇔Q]≡[P⇒Q Q ⇒P]
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

{♠,♣,♡,♢},{1; 4},[0; 1] ,N,R,C
{M∈P:MA =MB},{z∈C:zn= 1}
n m
[[ n, m ]]= {k∈Z|n6k6m}= [n, m]∩Z
x E x ∈E x E x E
x /∈E
•x{x}
•∅{}
P x E
P(x)
•∀x∈E, P (x)
x E P (x)
∀
•∃x∈E, P (x)
x E P (x)
∃
•∃!x∈E, P (x)
x E P (x)
∃!

∀x>0,∃y>0, y =√x∃y>0,∀x>0, y =√x
(∀x∈E, P (x)) ≡ ∃x∈E, P (x)
(∃x∈E, P (x)) ≡ ∀x∈E, P (x)
(∀x>0,∃y>0, y =√x)⇐⇒
∃x∈∅, P (x)∀x∈∅, P (x)
E F
E F E F E ⊂F
F E F ⊃E E F
F
E F E ̸⊂ F
•E∅⊂E
•E⊂F F ⊂E⇐⇒ E=F
A B A ̸=B

EP(E)
E
P({1; 2}) = {∅;{1};{2};{1; 2}}
A B E
•A∩B A B A B
•A∪B A B
A B
A B E
•A∩B=B∩A A ∪B=B∪A
•A∩(B∪C) = (A∩B)∪(A∩C)A∪(B∩C) = (A∪B)∩(A∪C)
A B E A B A ∩B=∅