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Reference: propositional equivalences

Module by: Ian Barland, John Greiner, Phokion Kolaitis, Moshe Vardi, Matthias Felleisen. E-mail the authors

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The following lists some propositional formula equivalences. Remember that we use the symbol ≡≡ as a relation between two WFFs, not as a connective inside a WFF. In these, φφ, ψψ, and θθ are meta-variables standing for any WFF.

Table 1: Propositional Logic Equivalences
Double Complementation ¬¬φ≡φ φ φ
Complement φ∨¬φ≡true φ φ φ∧¬φ≡false φ φ
Identity φ∨false≡φ φ φ φ∧true≡φ φ φ
Dominance φ∨true≡true φ φ∧false≡false φ
Idempotency φ∨φ≡φ φ φ φ φ∧φ≡φ φ φ φ
Absorption φ∧(φ∨ψ)≡φ φ φ ψ φ φ∨φ∧ψ≡φ φ φ ψ φ
Redundancy φ∧(¬φ∨ψ)≡φ∧ψ φ φ ψ φ ψ φ∨¬φ∧ψ≡φ∨ψ φ φ ψ φ ψ
DeMorgan's Laws ¬φ∧ψ≡¬φ∨¬ψ φ ψ φ ψ ¬φ∨ψ≡¬φ∧¬ψ φ ψ φ ψ
Associativity φ∧ ψ∧θ ≡ φ∧ψ ∧θ φ ψ θ φ ψ θ φ∨ ψ∨θ ≡ φ∨ψ ∨θ φ ψ θ φ ψ θ
Commutativity φ∧ψ≡ψ∧φ φ ψ ψ φ φ∨ψ≡ψ∨φ φ ψ ψ φ
Distributivity φ∧(ψ∨θ)≡φ∧ψ∨φ∧θ φ ψ θ φ ψ φ θ φ∨ψ∧θ≡(φ∨ψ)∧(φ∨θ) φ ψ θ φ ψ φ θ

Equivalences for implication are omitted above for brevity and for tradition. They can be derived, using the definition a⇒b≡¬a∨b a b a b .

Example 1

For example, using Identity and Commutativity, we have true⇒b≡¬true∨b≡false∨b≡b∨false≡b b b b b b .

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