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Binary set
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.
See also: ordered pair, binary relation.
03-10-2013 05:06:04
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The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details