# Boolean algebra

Boolean algebranoun

An algebra in which variables may have one of two values and the operations defined on them are logical OR, a type of addition, and logical AND, a type of multiplication.

## boolean-algebra

Noun

(*uncountable*)

- (algebra) A De Morgan algebra which also satisfies the law of excluded middle and the law of noncontradiction.
*The set of divisors of 30, with binary operators: g.c.d. and l.c.m., unary operator: division into 30, and identity elements: 1 and 30, forms a Boolean algebra.*-
*The nodes N*_{i}of a Boolean lattice can be labeled with Boolean formulae F(N_{i}), such that if node C is the meet of nodes A and B, then F(C) = F(A)F(B); if node C is the join of nodes A and B, then F(C) = F(A)+F(B); if node C is the complement of node A, then F(C) = F(A)'; and the 'â‰¤' order relation corresponds to logical entailment. A set of 'n' Boolean formulae could be called a "basis" for a 2^{n}-element Boolean algebra iff they are all mutually disjoint (i.e., the product of any pair is 0) and their Î£ (collective sum) is equal to 1. A set S of formulae could then generate a Boolean algebra inductively as follows: (base step) let P_{0}= S âˆª {(Î£S)'}, (inductive step) if a pair of formulae F and G in P_{i}are non-disjoint (i.e., FGâ‰ 0), then let P_{i+1}= (P_{i}âˆª {FG, F'G, FG'}) \ {F, G}, otherwise P_{i}is a basis. If CARD(P_{i})=n then the Boolean algebra will have 2^{n}elements which are all "linear combinations" of the basis elements, with a coefficient of either 0 or 1 for each term of each linear combination.

- (algebra, logic, computing) Specifically, an algebra in which all elements can take only one of two values (typically 0 and 1, or "true" and "false") and are subject to operations based on AND, OR and NOT
- (mathematics) The study of such algebras; Boolean logic, classical logic.

## boolean algebra - Computer Definition

See Boolean logic.