Difference between revisions of "Empty set"
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| − | In [[set theory]], the '''empty set''', usually denoted <math>\emptyset</math>, is the unique | + | In [[set theory]], the '''empty set''', usually denoted <math>\emptyset</math>, is the "set" without any members. For example, the "set of all blue-eyed lions," the "set of all mermaids," and the "set of all people born in 1700 who are still alive" are all empty sets. |
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| + | The empty set is unique, since any two empty sets have precisely the same members (that is, none) and is a [[subset]] of every other set. | ||
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| + | Its existence is postulated by the [[Axiom of empty set]] in the axioms of [[Zermelo–Fraenkel set theory]]. Though this seems counterintuitive, the empty set is a subset of the empty set, and the empty set is [[disjoint]] with itself. More generally, the union of any set with the empty set is the original set, while the intersection of any set with the empty set is itself the empty set. | ||
[[Category: set theory]] | [[Category: set theory]] | ||
Revision as of 05:32, January 8, 2010
In set theory, the empty set, usually denoted , is the "set" without any members. For example, the "set of all blue-eyed lions," the "set of all mermaids," and the "set of all people born in 1700 who are still alive" are all empty sets.
The empty set is unique, since any two empty sets have precisely the same members (that is, none) and is a subset of every other set.
Its existence is postulated by the Axiom of empty set in the axioms of Zermelo–Fraenkel set theory. Though this seems counterintuitive, the empty set is a subset of the empty set, and the empty set is disjoint with itself. More generally, the union of any set with the empty set is the original set, while the intersection of any set with the empty set is itself the empty set.