Empty Set

Related to Empty Set: Infinite set

empty set

[′em·tē ′set]

(mathematics)

The set with no elements.

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 It might be outdated or ideologically biased.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 ?

[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.

Neutrosophic Supra Topological Applications in Data Mining Process

If [x.sub.1], [x.sub.1] is the real zero of a[x.sup.2] + bx + c then the solution of (1) is the empty set [empty set], and the solution of (2) is {[x.sub.1]}; see remarks (a) and (c).

An algebraic approach for solving quadratic inequalities

The formal proof therefore focuses on the hypothesis "x [member of] [empty set]" and applies the superdeduction rule named "b_in_empty" and corresponding to the empty set. In B, the empty set is defined as follows: [empty set] [??] BIG--BIG.

Recovering intuition from automated formal proofs using tableaux with superdeduction

Columns 3 and 4 show how these constructions act on two important special classes: the empty set ([empty set]) that does not contain any structure and the neutral class [epsilon] consisting of a single structure of size 0.

Boltzmann oracle for combinatorial systems

Such is the power of the empty set, the set with no elements.

God is great

Peano's construction of the set of natural numbers from an empty set gives us an illustration of the concept of creatio ex nihilo.

Creatio ex nihilo, number theory, quantum vacuum, and the big bang

(9) The axioms that do not conform to simple rules of manipulation are the two explicitly existential axioms of the Empty Set and Infinity (BE 62).

The limits of the subject in Badiou's Being and Event

Students work through the physical meaning of empty set, intersection, and union of sets.

Counter Logic

An assignment is said to satisfy a diagram D if and only if it satisfies all the representing facts of D, where an assignment satisfies the representing facts if, basically, any shaded region is assigned the empty set and any region with an x-sequence in it is assigned a non-empty set.

The Logical Status of Diagrams

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: