Posted on Leave a comment

Logical Equivalences

Given any statement variables p, q, and r, a tautology t and a contradiction c, the following logical equivalences hold.

Commutative laws p \land q \equiv q \land p p \lor q \equiv q \lor p
Associative laws (p \land q) \land r \equiv p \land (q \land r) (p \lor q) \lor r \equiv p \lor (q \lor r)
Distributive laws p \land (q \lor r) \equiv (p \land q) \lor (p \land r) p \lor (q \land r) \equiv (p \lor q) \land (p \lor r)
Idempotent laws p \land p \equiv p p \lor p \equiv p
Absorption laws p \lor (p \land q) \equiv p p \land (p \lor q) \equiv p
De Morgan’s laws \neg (p \land q) \equiv \neg p \lor \neg q \neg (p \lor q) \equiv \neg p \land \neg q
Identity laws p \land t \equiv p p \lor c \equiv p
Universal Bound laws p \lor t \equiv p p \land c \equiv c
Negation laws p \lor \neg p \equiv t p \land \neg p \equiv c
Double Negative law \neg (\neg p) \equiv p
Negations of t and c \neg t \equiv c \neg c \equiv t