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Element (mathematics)

ru:Элемент множества In mathematics, an element (also called a member) is an object contained in a set (or more generally a class).

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.

The relation "is an element of", also called set membership, is denoted by "∈", and writing "x ∈ A", 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", or A contains x. The negation of set membership, is denoted by "∉".

Examples (using the sets defined above):

  • 2 ∈ A
  • {3, 4} ∈ B
  • {3, 4} is a member of B
  • 3 ∉ B

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.