Posted on

# Logical Equivalences

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

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