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Reference: propositional equivalences

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

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.

Table 1: Propositional Logic Equivalences
Double Complementation ¬¬φφ φ φ
Complement φ¬φtrue φ φ φ¬φfalse φ φ
Identity φfalseφ φ φ φtrueφ φ φ
Dominance φtruetrue φ φfalsefalse φ
Idempotency φφφ φ φ φ φφφ φ φ φ
Absorption φ(φψ)φ φ φ ψ φ φφψφ φ φ ψ φ
Redundancy φ(¬φψ)φψ φ φ ψ φ ψ φ¬φψφψ φ φ ψ φ ψ
DeMorgan's Laws ¬(φψ)¬φ¬ψ φ ψ φ ψ ¬(φψ)¬φ¬ψ φ ψ φ ψ
Associativity φ ψθ φψ θ φ ψ θ φ ψ θ φ ψθ φψ θ φ ψ θ φ ψ θ
Commutativity φψψφ φ ψ ψ φ φψψφ φ ψ ψ φ
Distributivity φ(ψθ)φψφθ φ ψ θ φ ψ φ θ φψθ(φψ)(φθ) 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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Definition of a lens

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A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

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