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Logical Equivalence

Logical equivalence is when two logical statements always have the same truth value — whenever one is true the other is true, and whenever one is false the other is false.

Two propositions pp and qq are logically equivalent, written p≡qp \equiv q (or p⇔qp \Leftrightarrow q), if they have identical truth values under every possible assignment of truth values to their component variables. In other words, the biconditional p↔qp \leftrightarrow q is a tautology. Logical equivalence can be verified by constructing a truth table or by applying known equivalence laws such as De Morgan's laws, double negation, or distribution.

Key Formula

p≡qif and only ifp↔q is a tautologyp \equiv q \quad \text{if and only if} \quad p \leftrightarrow q \text{ is a tautology}
Where:
  • p,qp, q = two logical propositions (statements)
  • ≡≡ = the logical equivalence symbol
  • ↔↔ = the biconditional (if and only if) connective

Worked Example

Problem: Show that ¬(p∧q)\neg(p \land q) is logically equivalent to ¬p∨¬q\neg p \lor \neg q (De Morgan's Law) using a truth table.
Step 1: List all possible truth value combinations for pp and qq.
p∈{T,T,F,F},q∈{T,F,T,F}p \in \{T, T, F, F\}, \quad q \in \{T, F, T, F\}
Step 2: Evaluate p∧qp \land q for each row, then negate it to get ¬(p∧q)\neg(p \land q).
p∧q:{T,F,F,F}  ⟹  ¬(p∧q):{F,T,T,T}p \land q: \{T, F, F, F\} \implies \neg(p \land q): \{F, T, T, T\}
Step 3: Evaluate ¬p\neg p and ¬q\neg q, then find ¬p∨¬q\neg p \lor \neg q.
¬p:{F,F,T,T},¬q:{F,T,F,T}  ⟹  ¬p∨¬q:{F,T,T,T}\neg p: \{F, F, T, T\}, \quad \neg q: \{F, T, F, T\} \implies \neg p \lor \neg q: \{F, T, T, T\}
Step 4: Compare the two result columns. They match in every row.
¬(p∧q):{F,T,T,T}=¬p∨¬q:{F,T,T,T}\neg(p \land q): \{F, T, T, T\} = \neg p \lor \neg q: \{F, T, T, T\}
Answer: Since ¬(p∧q)\neg(p \land q) and ¬p∨¬q\neg p \lor \neg q produce the same truth value in every row, they are logically equivalent: ¬(p∧q)≡¬p∨¬q\neg(p \land q) \equiv \neg p \lor \neg q.

Why It Matters

Logical equivalence is essential for simplifying complex logical expressions in mathematics, computer science, and circuit design. When you know two expressions are equivalent, you can swap one for the other — choosing whichever form is simpler or more convenient. Proof techniques like contrapositive proofs rely directly on the logical equivalence between a conditional statement and its contrapositive.

Common Mistakes

Mistake: Confusing logical equivalence (≡\equiv) with a simple biconditional (↔\leftrightarrow).
Correction: A biconditional p↔qp \leftrightarrow q is itself a proposition that can be true or false depending on pp and qq. Logical equivalence p≡qp \equiv q is a stronger claim: it means p↔qp \leftrightarrow q is true in *every* possible case (a tautology).
Mistake: Assuming two statements are equivalent because they agree on one or two cases.
Correction: You must check all possible truth value combinations. Two statements are logically equivalent only if they agree in every single row of the truth table.

Related Terms