A Boolean algebra is a 6-tuple consisting of a set A, equipped with two binary operations, a unary operation, and two distinguished elements. The first binary operation ∩: A × A → A is referred to as “conjunction”, “meet”, or “and”. The second binary operation ∪: A × A → A is referred to as “disjunction”, “join”, or “or”. The unary operation ¬: A → A is referred to as “compliment” or “not”. The distinguished elements are often denoted as 0 and 1. However, I will denote these elements as e∩ and e∪.
The 6-tuple (A, ∩, ∪, ¬, e∩, e∪) is called a “Boolean algebra” if it satisfies certain stated axioms. The axioms differ depending upon the needs of a particular author. The five following axiom pairs are a particularly common formulation of Boolean algebra. For all a, b, and c in A:
| The Standard Axioms of Boolean Algebra |
| |
| a ∪ (a ∩ b) = a | a ∩ (a ∪ b) = a | absorption |
| a ∪ (b ∪ c) = (a ∪ b) ∪ c | a ∩ (b ∩ c) = (a ∩ b) ∩ c | associativity |
| a ∪ ¬a = e∪ | a ∩ ¬a = e∩ | complements |
| a ∪ (b ∩ c) = (a ∪ b) ∩ (a ∪ c) | a ∩ (b ∪ c) = (a ∩ b) ∪ (a ∩ c) | distributivity |
| a ∪ b = b ∪ a | a ∩ b = b ∩ a | commutativity |
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