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Boolean algebras are named after [[George Boole]], an English mathematician at [[University College Cork]]. The algebraic system of logic Boole formulated is distinct from that described in this article in some small but important respects.
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== Formal definition ==
A '''Boolean algebra''' is a [[set]] ''A'', supplied with two [[binary operation]]s <math>\land</math> (logical AND), <math>\lor</math> (logical OR), a [[unary operation]] <math>\lnot</math> (logical NOT) and two elements 0 (logical FALSE) and 1 (logical TRUE), such that, for all elements ''a'', ''b'' and ''c'' of set ''A'', the following [[axioms]] hold:
:{| cellpadding=5
|<math> a \lor (b \lor c) = (a \lor b) \lor c </math>
|<math> a \land (b \land c) = (a \land b) \land c </math>
| [[associativity]]
|-
|<math> a \lor b = b \lor a </math>
|<math> a \land b = b \land a </math>
| [[commutativity]]
|-
|<math> a \lor (a \land b) = a </math>
|<math> a \land (a \lor b) = a </math>
| absorption
|-
|<math> a \lor (b \land c) = (a \lor b) \land (a \lor c) </math>
|<math> a \land (b \lor c) = (a \land b) \lor (a \land c) </math>
| [[distributivity]]
|-
|<math> a \lor \lnot a = 1 </math>
|<math> a \land \lnot a = 0 </math>
| [[complemented lattice|complements]]
|}
The first three pairs of axioms above: associativity, commutativity and absorption, mean that (''A'', <math>\land</math>, <math>\lor</math>) is a [[lattice (order)|lattice]]. Thus a Boolean algebra can also be equivalently defined as a [[distributive lattice|distributive]] [[complemented lattice]].
From these [[axioms]], one can show that the smallest element 0, the largest element 1, and the complement ¬''a'' of any element ''a'' are uniquely determined. For all ''a'' and ''b'' in ''A'', the following [[identity (mathematics)|identities]] also follow:
:{| cellpadding=5
| <math> a \lor a = a</math>
|<math> a \land a = a </math>
| [[Idempotent|idempotency]]
|-
|<math> a \lor 0 = a </math>
|<math> a \land 1 = a </math>
| rowspan=2 | [[bounded poset|boundedness]]
|-
|<math> a \lor 1 = 1 </math>
|<math> a \land 0 = 0 </math>
|-
|<math> \lnot 0 = 1 </math>
|<math> \lnot 1 = 0 </math>
| 0 and 1 are complements
|-
|<math> \lnot (a \lor b) = \lnot a \land \lnot b</math>
|<math> \lnot (a \land b) = \lnot a \lor \lnot b</math>
| [[de Morgan's laws]]
|-
| <math> \lnot \lnot a = a </math>
|
| [[involution]]
|}
== Examples ==
| 