empty set

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empty set

Mathematics
The set that has no members or elements.
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[1] Let X be a non empty set, then A = {(x, [[mu].sub.A] (x)) : x [member of] X} is called a fuzzy set on X, where [[mu].sub.A](x) [member of] [0, 1] is the degree of membership function of each x [member of] X to the set A.
Suppose ([c.sub.J](A) [intersection] [M.sub.J]) [intersection] (X - U) is a non empty subset.
We have E = 1 + [E.sub.+], where [E.sub.+] is the species of non empty finite sets and F = 1 + [F.sub.+], where [F.sub.+] is the species of F-structures on non empty finite sets (ii).
Theorem: 3.13: An intuitionistic fuzzy sets [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are an IF closed ideals of BCK-algebra [X.sub.i] for i=1,2, ..., n if and only if the non empty upper s-level cut [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and the non empty lower t-level cut [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are closed ideals of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for any s, t [member of] [0, 1].
For example, if we insert the precondition "non empty" for the pop operation, then the pop operation makes no promises when is called on an empty stack.
A non empty subset S of an [H.sub.v]-group (H, [omicron]) is called [H.sub.v]-subgroup of H if (S, [omicron]) is an [H.sub.v]-group.
In vg-[R.sub.1] 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 V [subset] U, x [not member of] vg[bar.(V)].
Definition 2.4 A hypertree is a non empty hypergraph H such that, given any vertices v and w in H,
If non empty set [D.sub.0] [subset] D is open, [partial derivative][g.sub.1] (t,[x.sub.1] (t), [x.sub.2] (t)) / [partial derivative[x.sub.j], [partial derivative][g.sub.2] (t,[x.sub.1] (t), [x.sub.2] (t)) / [partial derivative] [x.sub.j] the derivatives are continuous functions on the domain D for every j [member of] {1,2}, than the one and only one integral curves [??] [member of] D of the system passes through every point ([t.sub.0], [x.sup.0.sub.1], [x.sup.0.sub.2]) [member of] [D.sub.0].
[3] A neutrosophic topology on a non empty set X is a family t of neutrosophic subsets in X satisfying the following axioms:
[1] Let X be a non empty fixed set and I the closed interval [0,1].
* [[alpha].sub.i] and [[beta].sub.i+1] must have the same width, for every i [not equal to] 1; if i = 1, we have that [[alpha].sub.1] is always non empty and the width of [[alpha].sub.1] is equal to the width of [[beta].sub.2] plus one;

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