Logical Equivalence

De Morgan, the conditional rewrite, and the contrapositive.

The idea

Two propositions are logically equivalent, written $p \equiv q$, when they have the same truth value in every row of the truth table. Equivalent propositions are interchangeable: replacing one by the other inside an argument changes nothing the argument proves. To check a proposed equivalence, build both truth tables and compare them row by row.

Three equivalences carry most of the rewriting.

A conditional trades for a disjunction: $p \to q \;\equiv\; \lnot p \lor q.$

De Morgan's laws push a negation across a connective, flipping it and negating each part: $\lnot(p \land q) \equiv \lnot p \lor \lnot q, \qquad \lnot(p \lor q) \equiv \lnot p \land \lnot q.$

The contrapositive of $p \to q$ is $\lnot q \to \lnot p$: negate both parts and reverse the arrow. Each fails in exactly one situation — $p$ true and $q$ false — so the two are equivalent. Reversing without negating gives the converse $q \to p$, which is not equivalent to the original.

Ways to work on it

• Walkthrough. Rewrite conditionals, negate compound statements, and separate contrapositive from converse.
• Practice. Apply one standard equivalence.
• Hardest. Negate a conditional statement.

Not sure where to start? Take the ten-question placement test.