Empty Set

 

The set containing no elements, commonly denoted  or , the former of which is used in this work. These correspond to Wolfram Language and TeX characters summarized in the table below.

symbol	TeX	Wolfram Language
	varnothing	[Diameter]
	emptyset	[EmptySet]

Unfortunately, some authors use the notation 0 instead of  for the empty set (Mendelson 1997). The empty set is generally designated using {" src="https://mathworld.wolfram.com/images/equations/EmptySet/Inline6.svg" style="height:22px; width:6px" />}" src="https://mathworld.wolfram.com/images/equations/EmptySet/Inline7.svg" style="height:22px; width:6px" /> (i.e., the empty list) in the Wolfram Language.

A set that is not the empty set is called a nonempty set. The empty set is sometimes also known as the null set (Mendelson 1997).

The complement of the empty set is the universal set.

Strangely, the empty set is both open and closed for any set  and topology.

A groupoid, semigroup, quasigroup, ringoid, and semiring can be empty. Monoids, groups, and rings must have at least one element, while division algebras and fields must have at least two elements.

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REFERENCES

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 266, 1996.

Mendelson, E. Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, 1997.

مواضيع ذات صلة

Subset

Set Class

Set

Second Category

Proper Subset

Power Set

Matroid

Inductive Set

Independent Vertex Set

Independent Set

Graphoid

G_delta Set