Proofs (laws, and truth values)

And (∧)

only true if both values are true

Or (∨)

only false if both values are false

Imply (→)

Only false if true followed by false

IFF (↔)

Only true if both values are the same

Exclusive or (⊻)

Only true if both values are different

commutative law (comm)

a ∧ b ≡ b ∧ a
a ∨ b ≡ b ∨ a

associative law (assoc)

(a ∧ b) ∧ c ≡ a ∧ (b ∧ c)
(a ∨ b) ∨ c ≡ a ∨ (b ∨ c)

distributive law (dist)

a ∧ (b ∨ c) ≡ (a ∧ b) ∨ (a ∧ c)
a ∨ (b ∧ c) ≡ (a ∨ b) ∧ (a ∨ c)

De Morgan's Law (deM)

~(a ∧ b) ≡ ~a ∨ ~b
~(a ∨ b) ≡ ~a ∧ ~b

double negation (DN)

~(~a) ≡ ~~a ≡ a

contrapositive/transposition (trans)

a → b ≡ ~b → ~a

material equivocation (equiv)

a ↔ b ≡ (a → b) ∧ (b → a)

material implication (imp)

a → b ≡ ~a ∨ b

tautology (taut)

a ∧ a ≡ a
a ∨ a ≡ a

absorption (exp)

a ∧ (a ∨ b) ≡ a
a ∨ (a ∧ b) ≡ a