A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is generally also ignored (unlike a list or multiset). Members of a set are often referred to as elements and the notation is used to denote that is an element of a set . The study of sets and their properties is the object of set theory.

Older words for set include aggregate and set class. Russell also uses the unfortunate term manifold to refer to a set (Russell 1903, §68).

Historically, a single horizontal overbar was used to denote a set stripped of any structure besides order, and hence to represent the order type of the set. A double overbar indicated stripping the order from the set and hence represented the cardinal number of the set. This practice was begun by set theory founder Georg Cantor (Cantor 1895, §§1 and 7).

Symbols used to operate on sets include (which means "AND" or intersection), and (which means "OR" or union). The symbol is used to denote the set containing no elements, called the empty set.

There are a number of different notations related to the theory of sets. In the case of a finite set of elements, one often writes the collection inside curly braces, e.g.,

for the set of natural numbers less than or equal to three. Similar notation can be used for infinite sets provided that ellipses are used to signify infiniteness, e.g.,

for the collection of natural numbers greater than or equal to three, or

for the set of all even numbers.

In addition to the above notation, one can use so-called set-builder notation to express sets and elements thereof. The general format for set-builder notation is

where denotes an element and denotes a property satisfied by . (4) can also be expanded so as to indicate construction of a set which is a subset of some ambient set , e.g.,

The shorthand (4) does not assert that every property defines a set. In Zermelo-Fraenkel set theory, the axiom of subsets justifies the bounded form (5). Allowing every property to define a set leads to Russell's antinomy (Jech 2003, pp. 3-4).

It is worth noting that the ":" in (4) and (5) is sometimes replaced by a vertical line, e.g.,

Also worth noting is that the sets in (1), (2), and (3) can all be rewritten in set-builder notation as subsets of the set of integers, namely

respectively.

Other common notations related to set theory include , which is used to denote the set of maps from to where and are arbitrary sets. For example, an element of would be a map from the natural numbers to the set . Call such a function , then , , etc., are elements of , so call them , , etc. This now looks like a sequence of elements of , so sequences are really just functions from to . This notation is standard in mathematics and is frequently used in symbolic dynamics to denote sequence spaces.

Let , , and be sets. The operations on these sets using the and binary operators are commutative

associative

and distributive

More generally, we have the infinite distributive laws

where runs through any index set . The proofs follow trivially from the definitions of union and intersection.

See also

Aggregate, Analytic Set, Borel Set, C, Cayley Number, Coanalytic Set, Collection, Definable Set, Derived Set, Double-Free Set, Doublestruck, Element, Ground Set, Inclusion-Exclusion Principle, Intension, Intersection, Kinney's Set, List, Manifold, Multiset, N, Partially Ordered Set, Perfect Set, Proper Class, Q, R, Real Matrix, Set Class, Set Difference, Set Extension, Set Theory, Triple-Free Set, Union, Venn Diagram, Well Ordered Set, Z, Z^-, Z⁺ Explore this topic in the MathWorld classroom

Portions of this entry contributed by Christopher Stover

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References

Cantor, G. "Beiträge zur Begründung der transfiniten Mengenlehre." Math. Ann. 46, 481-512, 1895. https://doi.org/10.1007/BF02124929.Courant, R. and Robbins, H. "The Algebra of Sets." Supplement to Ch. 2 in What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 108-116, 1996.Jech, T. Set Theory: The Third Millennium Edition, Revised and Expanded. Berlin, Germany: Springer-Verlag, 2003. https://doi.org/10.1007/3-540-44761-X.Numberphile. "Let's Talk About Sets." May 17, 2016. https://www.youtube.com/watch?v=y12Tt3bOmKA.Russell, B. The Principles of Mathematics, Vol. I. Cambridge University Press, 1903. https://people.umass.edu/klement/pom/pom.html.

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Set

Cite this as:

Weisstein, Eric W., with contributions by Christopher Stover. "Set." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Set.html

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