Boolean algebra

:For a basic intro to sets, Boolean operations, Venn diagrams, truth tables, and Boolean applications, see Boolean logic.

Formal definition

A Boolean algebra is a set A, supplied with two binary operations land (logical AND), lor (logical OR), a unary operation lnot / ~ (logical NOT) and two elements 0 (logical FALSE) and 1 (logical TRUE), such that, for all elements a, b and c of set A, the following axioms hold:

Related Topics:
Set - Binary operation - Unary operation - Axioms

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The first three pairs of axioms above: associativity, commutativity and absorption, mean that (A, land, lor) is a lattice. Thus a Boolean algebra can also be equivalently defined as a distributive complemented lattice.

Related Topics:
Lattice - Distributive - Complemented lattice

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From these axioms, one can show that the smallest element 0, the largest element 1, and the complement ¬a of any element a are uniquely determined. For all a and b in A, the following identities also follow:

Related Topics:
Axioms - Identities

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