Theory

Law of negation:    ¬¬p⟺p

Law of equality:

  • p ∧ p ⟺ p
  • p ∨ p ⟺ p

Law on complements:

  • p ∧ ¬p ⟺ 0
  • p ∨ ¬p ⟺ 1

The Law of Neutrality:

  • p ∧ 1 ⟺ p
  • p ∨ 0 ⟺ p

Law of Domination:

  • p ∧ 0 ⟺ 0
  • p ∨ 1 ⟺ 1

Commutative Law:

  • p ∨ q ⟺ q ∨ p
  • p ∧ q ⟺ q ∧ p

Association Law:

  • (p ∨ q) ∨ r ⟺  p ∨ (q ∨ r)
  • (p ∧ q) ∧ r ⟺  p ∧ (q) r)

Distribution Law:

  • p ∨ (q ∧ r) ⟺  (p ∨ q) ∧ (p ∨ r)
  • p ∧ (q ∨ r) ⟺  (p q) (p ∧ r)

De Morgan’s Law:

  • ¬(p ∨ q) ⟺ ¬p ∧ ¬q
  • ¬(p ∧ q) ⟺ ¬p ∨ ¬q

Law of absorption:

  • p ∧ (p ∨ q) ⟺ p
  • p ∨  (p ∧ q) ⟺ p

Consequences:

  •  p → q ⟺ ¬p ∨ q
  • (q ∧ ¬p) ∨ (q ∧ p) ⟺ q