This is a draft cheat sheet. It is a work in progress and is not finished yet.
Symbols
¬ |
NOT |
∨ |
OR |
∧ |
AND |
→ |
IMPLIES |
↔ |
EQUAL |
Logical equivalences
¬¬ p |
≡ |
p |
p ∧ q |
≡ |
q ∧ p |
p ∨ q |
≡ |
q ∨ p |
p ∧ (q ∧ r) |
≡ |
(p ∧ q) ∧ r |
p ∨ (q ∨ r) |
≡ |
(p ∨ q) ∨ r |
p ∧ p |
≡ |
p |
p ∨ p |
≡ |
p |
p ∧ (q ∨ r) |
≡ |
(p ∧ q) ∨ (p ∧ r) |
p ∨ (q ∧ r) |
≡ |
(p ∨ q) ∧ (p ∨ r) |
¬ (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 |
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Truth table
p |
q |
¬ p |
p ∨ q |
p ∧ q |
p → q |
p ↔ q |
T |
T |
F |
T |
T |
T |
T |
T |
F |
F |
T |
F |
F |
F |
F |
T |
T |
T |
F |
T |
F |
F |
F |
T |
F |
F |
T |
T |
Tautology
always true |
p ∧ (a tautology) ≡ p |
p ∨ (a tautology) is a tautology |
¬ (a tautology) is a contradiction |
Contradiction
always false |
p ∧ (a contradiction) is a contradiction |
p ∨ (a tautology) ≡ p |
¬ (a contradiction) is a tautology |
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