The following laws of logic are valid in propositional logic and can be proved with truth tables. They are also valid in any boolean algebra. See logical operator for the meaning of the symbols. This table uses the symbols '∧' for AND, '∨' for OR, '¬' for NOT, and '≡' to denote equivalence. Logical TRUE and FALSE values are indicated by 'T' and 'F'.
Summary of the Laws of Logic
| Idempotent |
p ∨ p ≡ p
p ∧ p ≡ p |
| Associative |
(p ∧ q ) ∧ r ≡ p ∧ ( q ∧ r )
(p ∨ q ) ∨ r ≡ p ∨ ( q ∨ r ) |
| Commutative |
p ∧ q ≡ q ∧ p
p ∨ q ≡ q ∨ p |
| Distributive |
p ∨ ( q ∧ r ) ≡ ( p ∨ q ) ∧ ( p ∨ r )
p ∧ ( q ∨ r ) ≡ ( p ∧ q ) ∨ ( p ∧ r ) |
| Identity |
p ∧ T ≡ p
p ∨ F ≡ p |
| Annihilation |
p ∨ T ≡ T
p ∧ F ≡ F |
| Complement |
p ∨ ¬ p ≡ T
p ∧ ¬ p ≡ F
¬ T ≡ F
¬ F ≡ T |
| Involution |
¬ ¬ p ≡ p |
| DeMorgan's |
¬ ( p ∨ q ) ≡ ( ¬ p ∧ ¬ q )
¬ ( p ∧ q ) ≡ ( ¬ p ∨ ¬ q ) |
| Absorption |
p ∧ ( p ∨ q ) ≡ p
p ∨ ( p ∧ q ) ≡ p |
