Element (mathematics)

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

Sets

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

Sets can themselves be elements. 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 and terminology

The relation "is an element of", also called set membership, is denoted by ∈. Writing

$x \in A \,$

means that "x is an element of A". Equivalent expressions are "x is a member of A", "x belongs to A", "x is in A" and "x lies in A". The expressions "A includes x" and "A contains x" are also used to mean set membership, however some authors use them to mean instead "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]

The LaTeX command for this symbol is "\in".

The negation of set membership is denoted by ∉.

Cardinality of sets

Main article: cardinality

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 either 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, N = { 1, 2, 3, 4, ... }.

Examples

Using the sets defined above:

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

References

1. ^ Eric Schechter (1997). Handbook of Analysis and Its Foundations. Academic Press. ISBN 0-12-622760-8.  p. 12
2. ^ George Boolos (February 4, 1992). 24.243 Classical Set Theory (lecture). (Speech). Massachusetts Institute of Technology, Cambridge, MA. 

• 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".