Element (mathematics)

In mathematics, the elements (∈) or members of a set (or more generally a class) are all those objects which when collected together make up the set (or class).

Set theory and elements

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

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

The elements of a set can be anything. For example, <math>C=\{ \mbox{red, green, blue} \}</math>, 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 <math>\in</math>, and writing

<math>x \in A</math>

means that <math>x</math> is an element of <math>A</math>. Equivalently one can say or write "<math>x</math> is a member of <math>A</math>", "<math>x</math> belongs to <math>A</math>", "<math>x</math> is in <math>A</math>", or "<math>A</math> contains <math>x</math>". The negation of set membership is denoted by <math>\notin</math>.

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 <math>A</math> is 4, while the cardinality of the sets <math>B</math> and <math>C</math> 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, <math>\mathbb{N} = \{ 1, 2, 3, 4 \ldots \}</math>.

Examples

Using the sets defined above as

• <math>2 \in A</math>
• <math>\{3, 4 \} \in B</math>
• <math>\{3, 4 \}</math> is a member of <math>B</math>
• <math>\mbox{Yellow} \notin C</math>
• The cardinality of <math>D = \{ 2, 4, 6, 8, 10, 12 \}</math> is finite and equal to 6.
• The cardinality of <math>P = \{ 2, 3, 5, 7, 11, 13 \ldots \}</math> (the prime numbers) is infinite.

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