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# propositional equivalences

Summary: (Blank Abstract)

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.

 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 ab¬ab a b a b .

## Example 1

For example, using Identity and Commutativity, we have trueb¬truebfalsebbfalseb b b b b b .

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