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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. |
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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Revision as of 23:56, 18 August 2005
A binary set is a set with (exactly) two distinct elements, or, equivalently, a set whose cardinality is two.
Examples:
- The set {a,b} is binary.
- The set {a,a} is not binary, since it is equivalent to the set {a}, which is a singleton.
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 {{}}. From the axiom of empty set it is known that the set {} exists. From the axiom of pairing it is then known that the set {{},{{}}} exists, which contains both {} and {{}}. But the set {{},{{}}} is a binary set.