Skip to content Skip to navigation

Connexions

You are here: Home » Content » Reference: propositional equivalences

Navigation

Lenses

What is a lens?

Definition of a lens

Lenses

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.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • Rice Digital Scholarship

    This module is included in aLens by: Digital Scholarship at Rice UniversityAs a part of collection: "Intro to Logic"

    Click the "Rice Digital Scholarship" link to see all content affiliated with them.

Recently Viewed

This feature requires Javascript to be enabled.
 

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 ¬¬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 .

Content actions

Download module as:

PDF | More downloads ...

Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

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.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks