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|}
== Examples ==
* The simplest Boolean algebra has only two elements, 0 and 1, and is defined by the rules:
{|
|-
| width="80" |
|
{| border="1" cellpadding="4" cellspacing="0"
|-
! ∧ || 0 || 1
|-
! 0
| 0 || 0
|-
! 1
| 0 || 1
|}
| width="40" |
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{| border="1" cellpadding="4" cellspacing="0"
|-
! ∨ || 0 || 1
|-
! 0
| 0 || 1
|-
! 1
| 1 || 1
|}
|}
:* It has applications in [[logic]], interpreting 0 as ''false'', 1 as ''true'', ∧ as ''and'', ∨ as ''or'', and ¬ as ''not''. Expressions involving variables and the Boolean operations represent statement forms, and two such expressions can be shown to be equal using the above axioms if and only if the corresponding statement forms are [[logical equivalence|logically equivalent]].
:* The two-element Boolean algebra is also used for circuit design in [[electrical engineering]]; here 0 and 1 represent the two different states of one [[bit]] in a [[digital circuit]], typically high and low [[voltage]]. Circuits are described by expressions containing variables, and two such expressions are equal for all values of the variables if and only if the corresponding circuits have the same input-output behavior. Furthermore, every possible input-output behavior can be modeled by a suitable Boolean expression.
:* The [[two-element Boolean algebra]] is also important in the general theory of Boolean algebras, because an equation involving several variables is generally true in all Boolean algebras if and only if it is true in the two-element Boolean algebra (which can always be checked by a trivial [[brute force search|brute force]] algorithm). This can for example be used to show that the following laws (''Consensus theorems'') are generally valid in all Boolean algebras:
:** (''a'' ∨ ''b'') ∧ (¬''a'' ∨ ''c'') ∧ (''b'' ∨ ''c'') ≡ (''a'' ∨ ''b'') ∧ (¬''a'' ∨ ''c'')
:** (''a'' ∧ ''b'') ∨ (¬''a'' ∧ ''c'') ∨ (''b'' ∧ ''c'') ≡ (''a'' ∧ ''b'') ∨ (¬''a'' ∧ ''c'')
* Starting with the [[propositional calculus]] with κ sentence symbols, form the [[Lindenbaum-Tarski algebra|Lindenbaum algebra]] (that is, the set of sentences in the propositional calculus modulo tautology). This construction yields a Boolean algebra. It is in fact the [[free Boolean algebra]] on κ generators. A truth assignment in propositional calculus is then a Boolean algebra homomorphism from this algebra to {0,1}.
* The [[power set]] (set of all subsets) of any given set ''S'' forms a Boolean algebra with the two operations ∨ := ∪ (union) and ∧ := ∩ (intersection). The smallest element 0 is the [[empty set]] and the largest element 1 is the set ''S'' itself.
* The set of all subsets of ''S'' that are either finite or [[cofinite]] is a Boolean algebra.
* For any [[natural number]] ''n'', the set of all positive [[divisor]]s of ''n'' forms a [[distributive lattice]] if we write ''a'' ≤ ''b'' for ''a'' | ''b''. This lattice is a Boolean algebra if and only if ''n'' is [[square-free integer|square-free]]. The smallest element 0 of this Boolean algebra is the natural number 1; the largest element 1 of this Boolean algebra is the natural number ''n''.
* Other examples of Boolean algebras arise from [[topology|topological spaces]]: if ''X'' is a topological space, then the collection of all subsets of ''X'' which are both open and closed forms a Boolean algebra with the operations ∨ := ∪ (union) and ∧ := ∩ (intersection).
* If ''R'' is an arbitrary [[mathematical ring|ring]] and we define the set of ''central idempotents'' by <br> ''A'' = { ''e'' ∈ ''R'' : ''e''<sup>2</sup> = ''e'', ''ex'' = ''xe'', ∀''x'' ∈ ''R'' } <br> then the set ''A'' becomes a Boolean algebra with the operations ''e'' ∨ ''f'' := ''e'' + ''f'' − ''ef'' and ''e'' ∧ ''f'' := ''ef''.
== Order theoretic properties ==
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