Element (mathematics)

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


Writing means that the elements of the set are the numbers 1, 2, 3 and 4. Sets of elements of, for example, are subsets of.
Sets can themselves be elements. For example, consider the set. The elements of are not 1, 2, 3, and 4. Rather, there are only three elements of, namely the numbers 1 and 2, and the set.
The elements of a set can be anything. For example,, is the set whose elements are the colors, and.

Notation and terminology

The relation "is an element of", also called set membership, is denoted by the symbol "∈". Writing
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". Logician George Boolos strongly urged that "contains" be used for membership only and "includes" for the subset relation only.
For the relation ∈, the converse relationT may be written
The negation of set membership is denoted by the symbol "∉". Writing
The symbol ∈ was first used by Giuseppe Peano 1889 in his work Arithmetices principia, nova methodo exposita. Here he wrote on page X:
Signum ∈ significat est. Ita a ∈ b legitur a est quoddam b; …

which means
The symbol ∈ means is. So a ∈ b is read as a is a b; …

The symbol itself is a stylized lowercase Greek letter epsilon, the first letter of the word, which means "is".

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 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 positive integers,.


Using the sets defined above, namely A =, B = and C = :