# Empty Set

(redirected from Non-empty subset)
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Related to Non-empty subset: Proper subset

## empty set

[′em·tē ′set]
(mathematics)
The set with no elements.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## 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 ?
A map [??] : H x H [right arrow] P*(H) is called a hyperoperation or a join operation on the set H, where H is a non-empty set and P*(H) = P(H)\{0} denotes the set of all non-empty subsets of H.
A non-empty subset S of the partially ordered Banach space E is called partially bounded if every chain C in S is bounded.
Then {[F.sub.n] ; n [member of] N} is a decreasing sequence of [tau]-closed non-empty subsets of the countably [tau]-compact space [[bar.O.sub.x].sup.[tau]]; therefore, [[intersection].sup.[infinity].sub.n=1] [F.sub.n] [not equal to] [empty set]; and this implies a fortiori, the existence of [xi] [member of] [[intersection].sup.[infinity].sub.n=1] [bar.co]([z.sub.i] ; i [greater than or equal to] n).
We will order all non-empty subsets A [subset or equal to] V such that [absolute value of A] [less than or equal to] C with a ranking function [bar.[rho]](A) defined as,
A topological space (X,[[tau].sub.[alpha]]) is [alpha]-[tau]-connected if and only if one non-empty subset which is both [alpha]-open and [alpha]-closed in X is X itself.
Let [A.sub.[omega]] (I) be a non-empty subset of [X.sub.[omega]] (I).
Let S be a non-empty subset of a valued field (Kv).
[3] A non-empty subset I of a BT-algebra X is said to be an ideal of X if
A plan at its initial node is a non-empty subset of {[z.sub.g1], [z.sub.g2], [Mathematical Expression Omitted], [Mathematical Expression Omitted]).
(iii) a non-empty subset I of X is called an ideal if
A non-empty subset A of S is called a subsemigroup of S if A2 A .
Remark: Suppose that T = (V, E) is an X-tree and that U [[subset].bar] V is a T-core as defined in (Dress et al., 2011b, Section 5), i.e., a non-empty subset of V for which the induced subgraph [T.sub.U] := (U, [E.sub.U] := {e e E : e [[subset].bar] U}) of T with vertex set U is connected (and, hence, a tree) and the degree deg[T.sub.U]([upsilon]) of any vertex [upsilon] in [T.sub.U] is either 1 or coincides with the degree [deg.sub.T]([upsilon]) of [upsilon] in T.

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