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 ?
Let X be a nonempty set, [gamma][member of] [Gamma](X) and [member of] [Omega].
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,
[15] Let X be a nonempty set and G = (V(G), E(G)) be a graph such that V(G) = X, and let T: X [right arrow] CB(X).
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 X be a nonempty set. A mapping [sigma] : X x X [right arrow] [0, [infinity]) is called metric-like if, for all x, y, z [member of] X, the following conditions are satisfied:
Let H be a nonempty set and [less than or equal to] be an ordered relation on H.
consists of a topological space X, a nonempty set D, and a family of continuous functions [[phi].sub.A] : [[DELTA].sub.n] [right arrow] X (that is, singular n- simplices) for A [member of] <D> with the cardinality [absolute value of (A)] = n + 1.
(2) A = {[a.sub.1], [a.sub.2], ..., [a.sub.m]} is a nonempty set of m categorical attributes;
Let [GAMMA] be a nonempty set, and let L be a [sigma]-algebra over [GAMMA].
In rough set theory, the quadruplet S = (U, A, V, f) is called an information system, where U = {[x.sub.1], [x.sub.2], ..., [x.sub.n]} is a nonempty set of samples, called a universe or a sample space.
* Given a nonempty set M [??] V a vertex v of G is said to be k-controlled by M if [[delta].sub.M](v) [greater than or equal to] [[delta].sub.V](v)/2+ k.