Laws of logic
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.
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 |
