Logical Equivalence

Two statement patterns are logically equivalent if they have identical final truth-table columns — that is, they take the same truth value for every assignment of truth values to their variables. Equivalence is written with the symbol $\equiv$. If two patterns are logically equivalent, one may always be replaced by the other in any argument or circuit without changing anything, which is what makes equivalence the workhorse for simplifying compound statements.

To test whether $A \equiv B$, build one truth table containing a column for $A$ and a column for $B$; if the two columns agree in every row, they are equivalent, and if they differ in even a single row, they are not.

Several standard equivalences recur constantly and are worth committing to memory.

Conditional as a disjunction. $p \rightarrow q \equiv \sim p \vee q.$ This single equivalence lets every conditional be rewritten without the arrow, and is the key to negating conditionals.

Contrapositive. $p \rightarrow q \equiv \sim q \rightarrow \sim p.$ A conditional and its contrapositive always share a truth table — "if it rains then the ground is wet" says exactly the same as "if the ground is not wet then it did not rain". (By contrast the converse $q \rightarrow p$ and the inverse $\sim p \rightarrow \sim q$ are not equivalent to the original.)

Biconditional as two conditionals. $p \leftrightarrow q \equiv (p \rightarrow q) \wedge (q \rightarrow p).$

De Morgan's laws. The negation of a conjunction is the disjunction of the negations, and the negation of a disjunction is the conjunction of the negations: $\sim(p \wedge q) \equiv \sim p \vee \sim q, \qquad \sim(p \vee q) \equiv \sim p \wedge \sim q.$ …