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Equivalence

  • If A,B are formulas are such that

    v ( A ) = v ( B )

    for all interpretations v , A is (logically) equivalent to B:

    A ≡ B

  • Example: ¬p V q ≡ p → q since both formulas are true in all interpretations except when v ( p ) = T, v ( q ) = F and are false for that particular interpretation

  • Caution : ≡ does not mean the same thing as :

    • A ↔ B is a formula (syntax)

    • A ≡ B is a relation between two formula (semantics)

    • Theorem : A ≡ B if and only if A ↔ B is true in every interpretation; i.e. A ↔ B is a tautology .


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