Boolean algebra

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

The algebra, developed by Boole, of events, unions, intersections and complementary events in a sample space S. For any events A, B, C the following algebraic laws hold:

The algebra of logic. It is sometimes called symbolic logic. Boolean algebra is a method of expressing logic in a mathematical context. It is primarily concerned with binary operations and based on the cumulative, associative, and distributive laws of binary operations. Boolean algebra provides the theoretical concepts for computer design. It is not used to solve managerial problems directly.

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A Boolean algebra is a system consisting of a set S and two operations, ∩ and ∪ (cap and cup), subject to the following axioms. For all sets a,b,c, that are members of S:

1 a ∩ (b ∩ c) = (a ∩ b) ∩ c.

Also a ∪ (b ∪ c) = (a ∪ b) ∪ c

(associativity)

2 a ∩ b = b ∩ a.

Also a ∪ b = b ∪ a (commutativity)

3 a ∩ (b ∪ c) = (a ∩ b) ∪ (a ∩ c).

Also a ∪ (b ∩ c) = (a ∪ b) ∩ (a ∪ c)

(distributivity)

4 There belong to S two elements, 0 and 1, with the properties

a ∪ 0 = a; a ∩ 1 = a (identity)

5 For each set a in S there exists a set a′ with the properties that a ∪ aprime; = 1, a ∩ aprime; = 0 (complementation).

The propositional calculus can be represented as a Boolean algebra, with ∩ representing &, ∪ representing ∨, and 1 = T, 0 = F. The Boolean operators are then the truth functors, such as &, ∨, and ¬. A Boolean search is a search for things meeting a condition defined with these operators.

Boolean algebra may mean:

• Boolean algebra (logic), the algebra of truth values and operations on them
• Boolean lattice, a class of mathematical structures that can be described in terms of an ordering or in terms of operations on a set

For introductory articles, see:

• Boolean algebra (introduction), a general overview
• Boolean logic, including applications such as computer science
• Boolean algebras canonically defined: Boolean algebra as the laws of the algebra of finitary operations on {0,1}

See also

• Boolean (disambiguation)

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