In mathematics, an element or member of a set is any one of the distinct objects that make up that set.

Set theory and elements

Writing A = {1,2,3,4}, means that the elements of the set A are the numbers 1, 2, 3 and 4. Groups of elements of A, for example {1,2}, are subsets of A.

Elements can themselves be sets. For example consider the set B = {1,2,{3,4}}. The elements of B are not 1, 2, 3, and 4. Rather, there are only three elements of B, namely the numbers 1 and 2, and the set {3,4}.

The elements of a set can be anything. For example, C = {red, green, blue}, is the set whose elements are the colors red, green and blue.

Notation

The relation "is an element of", also called set membership, is denoted by ∈, and writing

means that x is an element of A. Equivalently one can say or write "x is a member of A", "x belongs to A", "x is in A", "x lies in A", "A includes x", or "A contains x". The negation of set membership is denoted by ∉.

Unfortunately, the terms "A includes x" and "A contains x" are ambiguous, because some authors also use them to mean "x is a subset of A".¹ Logician George Boolos strongly urged that "contains" be used for membership only and "includes" for the subset relation only.²

Cardinality of sets

The number of elements in a particular set is a property known as cardinality, informally this is the size of a set. In the above examples the cardinality of the set A is 4, while the cardinality of the sets B and C is 3. An infinite set is a set with an infinite number of elements, while a finite set is a set with a finite number of elements. The above examples are examples of finite sets. An example of an infinite set is the set of natural numbers, .

Examples

Using the sets defined above as

• 2 ∈ A
• {3,4} ∈ B
• {3,4} is a member of B
• Yellow ∉ C
• The cardinality of D = {2,4,6,8,10,12} is finite and equal to 6.
• The cardinality of P = {2,3,5,7,11,13...} (the prime numbers) is infinite.

Notes

References

• Paul R. Halmos 1960, Naive Set Theory, Springer-Verlag, NY, ISBN 0-387-90092-6. "Naive" means that it is not fully axiomatized, not that it is silly or easy (Halmos's treatment is neither).

• Patrick Suppes 1960, 1972, Axiomatic Set Theory, Dover Publications, Inc. NY, ISBN 0-486-61630-4. Both the notion of set (a collection of members), membership or element-hood, the axiom of extension, the axiom of separation, and the union axiom (Suppes calls it the sum axiom) are needed for a more thorough understanding of "set element".