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 ?
A nonempty set H with a hyper operation "[omicron]" and a constant 0 is called a hyper BCK-algebra (See [16]), if it satisfies the following conditions: for any x,y,z [member of] H,
Under the assumptions that {[[xi].sub.k]} is bounded, i.e., [parallel][[xi].sub.k][parallel] [less than or equal to] N, [for all]n [greater than or equal to] 1, [[lambda].sub.n] [member of] [a, b] [subset] (0, 2/[N.sup.2]), [for all]n [greater than or equal to] 1 and the set [OMEGA] := {u [member of] Fix(T) : f ([y.sup.n], u) [less than or equal to] 0, n [greater than or equal to] 1} is nonempty, the authors proved that the sequence {[x.sup.n]} generated by the above algorithm converges to a solution of (1.1).
Let CB(X) be the collection of all nonempty closed bounded subsets of X.
(2) A nonempty subset A of an ordered AG-groupoid S is called a left (right) ideal of S if we have the following:
Let C be a nonempty closed convex subset of a real Hilbert space H and let S : C [right arrow] C be a nonexpansive mapping with Fix (S) [not equal to] 0.
It is clear that [X.sup.Par.sub.u] [subset] [X.sup.wPar.sub.u] for every positive integer m, every nonempty set X, and every function u = ([u.sub.1],..., [u.sub.m]) : X [??] [R.sup.m].
Let ([OMEGA], [SIGMA]) be a measurable space and C be a nonempty closed convex sunset of a separable Banach space E.
(6) If {[X.sub.n]} is a sequence of nonempty, bounded (in norm), weakly closed subsets of [W.sup.n,1]([[0, T].sup.N]) and [X.sub.1] [subset] [X.sub.2] [subset] ...
where [alpha], [[alpha].sub.n] [member of] (0,1) and [x.sub.1] is an any given element in a nonempty closed convex subset E [subset not equal to] X.
If x [member of] H and A is a nonempty subset of H, then [A.sub.x] = {(y, z) [member of] H x H | x [less than or equal to] y [omicron] z}.
In [9], Schirmer observed that for a given self map f : [S.sup.n] [right arrow] [S.sup.n] of an n-sphere, n [greater than or equal to] 2, any closed nonempty proper subset A of [S.sup.n] can be realized as the fixed point set of a map g [member of] [f] with Fix(g) = A.

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