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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 the axiom of empty set it is known that the set <math>\emptyset = \{\}</math> exists. From the axiom of pairing it is then known that the 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 the axiom of empty set it is known that the set <math>\emptyset = \{\}</math> exists. From the axiom of pairing it is then known that the set |
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<math>\{\emptyset,\emptyset\} = \{\emptyset\}</math> exists, and thus the set <math>\{\{\emptyset\},\emptyset\}</math> exists. This latter set has two elements. |
<math>\{\emptyset,\emptyset\} = \{\emptyset\}</math> exists, and thus the set <math>\{\{\emptyset\},\emptyset\}</math> exists. This latter set has two elements.{{fact}} |
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==See also== |
==See also== |
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Revision as of 15:33, 8 September 2008
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 the same set as {a}, and is thus 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 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, and thus the set exists. This latter set has two elements.[citation needed]
