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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 ...
The completion is the Boolean algebra of regular open sets in the Stone space of prime ideals of A. Each element x of A corresponds to the open set of prime ideals not containing x (which is open and closed, and therefore regular). The completion is the Boolean algebra of regular cuts of A. Here a cut is a subset U of A+ (the non-zero elements ...
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 ...
In Boolean algebra, the consensus theorem or rule of consensus [1] is the identity: The consensus or resolvent of the terms and is . It is the conjunction of all the unique literals of the terms, excluding the literal that appears unnegated in one term and negated in the other. If includes a term that is negated in (or vice versa), the ...
Canonical normal form. In Boolean algebra, any Boolean function can be expressed in the canonical disjunctive normal form ( CDNF) [1] or minterm canonical form, and its dual, the canonical conjunctive normal form ( CCNF) or maxterm canonical form. Other canonical forms include the complete sum of prime implicants or Blake canonical form (and ...
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 ...
Free Boolean algebra. In mathematics, a free Boolean algebra is a Boolean algebra with a distinguished set of elements, called generators, such that: Each element of the Boolean algebra can be expressed as a finite combination of generators, using the Boolean operations, and. The generators are as independent as possible, in the sense that ...
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 ...
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