Empty Set

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empty set

[′em·tē ′set]
(mathematics)
The set with no elements.

Empty Set

 

(or null set), the set that contains no elements. The concept of the empty set, like the concept of zero, arises from the need to have the result of any operation on sets also be a set. The source of the concept of the empty set is the very method of defining a set by a characteristic property of its elements, since it may not be known beforehand whether elements possessing the property do in fact exist. Thus, it still is not known whether the equation xn + yn = zn, where n is an integer greater than 2, can be solved for x, y, and z if x, y, and z are natural numbers. In other words, it still is not known whether the set of those n > 2 for which the equation is solvable is empty or nonempty.

References in periodicals archive ?
[10] Let X be a non empty set, then A = {(x, [[mu].sub.A](x), [[gamma].sub.A](x)) : x [member of] X} is called an intuitionistic set on X, where 0 [less than or equal to] [[mu].sub.A](x) + [[gamma].sub.A](x) [less than or equal to] 1 for all x [member of] X, [[mu].sub.A](x),[[gamma].sub.A](x) [member of] [0, 1] are the degree of membership and non membership functions of each x [member of] X to the set A respectively.
Finally, we need the following special function [pi] defined on the set, [P.sup.+], of non empty partitions and the "Mobius-like" arithmetical function, [upsilon], defined on the set, [N.sup.+], of positive integers.
In a vg-regular space X, if X is a vg-[T.sub.0]-limit point of X, then for any non empty vg-open set U, there exists a non empty vg-open set V such that vg[bar.(V)] [subset] U, x [not member of] vg[bar.(V)].
Let X be a non empty set and A [member of][[zeta].sup.X] Then [A.sup.~] = <[[mu].sub.A](x) , [[gamma].sub.A](x) > for every x [member of] X.
* [[beta].sub.i] and [[alpha].sub.i+1] must have the same height, for every i [not equal to] 1; if i = 1, we have that [[beta].sub.1] is always non empty and the height of [[beta].sub.1] is equal to the width of [[alpha].sub.2] plus one;
[4] A collection G of non empty subsets of a space X is called a grill on X
When the proportion of non empty urns reaches r, one ball is removed in every non empty urn.
An interval neutrosophic set [50] [??] of a non empty set H is expreesed by truth-membership function [t.sub.[??]](h) the indeterminacy membership function [i.sub.[??]](h) and falsity membership function [f.sub.[??]](h).
Definition 2.4.[5] An intuitionistic fuzzy topology (IFT) in Coker's sense on a non empty set X is a family [tau] of IFSs in X satisfying the following axioms.
Definition 2.1 [5] Let X be a non empty set, then [tau].sub.1], [[tau].sub.2], ..., [[tau].sub.2] be N-arbitrary topologies defined on X and the collection [N.sub.[tau]] = {S [subset or equal to] X : S = ([[universal].sup.N.sub.i=1] [A.sub.i]) [union] ([[??].sup.N.sub.i=1] [B.sub.i]), [A.sub.i], [B.sub.i] [member of] [[tau].sub.i]} is, called a N-topology on X if the following axioms are satisfied:
A non empty set X together with two topologies [[tau].sub.1] and [[tau].sub.2] is called a Bitopological space[2].
A neutrosophic crisp topology (NCT) on a non empty set X is a family of [GAMMA] of neutrosophic crisp subsets in X satisfying the following axioms:

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