Idempotence

A ∧ A ≡ A

A ∨ A ≡ A

Commutativity

A ∧ B ≡ B ∧ A

A ∨ B ≡ B ∨ A

Associativity

A ∧ (B ∧ C) ≡ (A ∧ B) ∧ C

A ∨ (B ∨ C) ≡ (A ∨ B) ∨ C

Absorption

A ∧ (A ∨ B) ≡ A

A ∨ (A ∧ B) ≡ A

Distributivity

A ∧ (B ∨ C) ≡ (A ∧ B) ∨ (A ∧ C)

A ∨ (B ∧ C) ≡ (A ∨ B) ∧ (A ∨ C)

Negation

A ∧ (¬A) ≡ false

A ∨ (¬A) ≡ true

Double Negation

¬(¬A) ≡ A

de Morgan

¬(A ∧ B) ≡ (¬A) ∨ (¬B)

¬(A ∨ B) ≡ (¬A) ∧ (¬B)

Implication

A ⇒ B ≡ (¬A) ∨ B

Bi-Implication

A ⇔ B ≡ (A ⇒ B) ∧ (B ⇒ A)

Rules of Precedence

Highest first: ¬, ∧, ∨, ⇒, ⇔