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In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values true and false, usually denoted 1 and 0, whereas in elementary algebra the values of the variables are numbers. Second, Boolean algebra uses logical operators such ...
Boole's expansion theorem. Boole's expansion theorem, often referred to as the Shannon expansion or decomposition, is the identity: , where is any Boolean function, is a variable, is the complement of , and and are with the argument set equal to and to respectively. The terms and are sometimes called the positive and negative Shannon cofactors ...
Boolean algebra (structure) In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can ...
Boolean algebra is a mathematically rich branch of abstract algebra. Stanford Encyclopaedia of Philosophy defines Boolean algebra as 'the algebra of two-valued logic with only sentential connectives, or equivalently of algebras of sets under union and complementation.'. [1] Just as group theory deals with groups, and linear algebra with vector ...
Complete Boolean algebra. In mathematics, a complete Boolean algebra is a Boolean algebra in which every subset has a supremum (least upper bound ). Complete Boolean algebras are used to construct Boolean-valued models of set theory in the theory of forcing. Every Boolean algebra A has an essentially unique completion, which is a complete ...
In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set (usually {true, false}, {0,1} or {-1,1}). [1] [2] Alternative names are switching function, used especially in older computer science literature, [3] [4] and truth function (or logical function), used in logic.
In logic, mathematics and theoretical computer science, a Boolean domain is usually written as {0, 1}, or . The algebraic structure that naturally builds on a Boolean domain is the Boolean algebra with two elements. The initial object in the category of bounded lattices is a Boolean domain.
Boolean expression. In computer science, a Boolean expression is an expression used in programming languages that produces a Boolean value when evaluated. A Boolean value is either true or false. A Boolean expression may be composed of a combination of the Boolean constants True/Yes or False/No, Boolean-typed variables, Boolean-valued operators ...
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