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A '''binary set''' is a [[set]] with (exactly) two distinct elements, or, equivalently, a set whose [[cardinality]] is [[two]]. |
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Examples: |
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* The set {''a'',''b''} is binary. |
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* The set {''a'',''a''} is not binary, since it is equivalent to the set {''a''}, which is a [[singleton (mathematics)|singleton]]. |
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In [[axiomatic set theory]], the existence of binary sets is a consequence of the [[axiom of empty set]] and the [[axiom of pairing]]. From these two axioms can be derived the existence of the singleton <nowiki>{{}}</nowiki>. From the axiom of empty set it is known that the set <nowiki>{}</nowiki> exists. From the axiom of pairing it is then known that the set <nowiki>{{},{{}}}</nowiki> exists, which contains both <nowiki>{}</nowiki> and <nowiki>{{}}</nowiki>. But the set <nowiki>{{},{{}}}</nowiki> is a binary set. |
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==See also== |
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* [[ordered pair]] |
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* [[binary relation]] |
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[[Category:Set theory]] |
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