Element (mathematics)

In mathematics, an element is an object contained in a class, often a set.

Here is an example, consider a set X:

<math>X=\{\,1,2,3,4\}<math>

The elements of this set are 1, 2, 3 and 4 respectively. Groups of elements i.e. <math>\{\,1,2\}<math> are subsets.

Not all elements of sets are elements singular in nature. Some sets have elements that are sets of multiple elements themselves. Consider this set Y:

<math>Y=\{\,1, 2, \{\,3,4\}\}<math>

The elements of this set are not 1, 2, 3, and 4. Here the elements of this set are 1, 2, and <math>\{\,3,4\}<math>, where the set <math>\{\,3,4\}<math> is a single element of Y by itself.

Just to illustrate as well, you can have a set whose elements are non-numerical:

<math>Q=\{\,Red,Green,Blue\}<math>

There are two special symbols in math that indicate if a particular object is an element of a set or not.

This symbol is used to indicate a particular object is an element of a set (you can think of this as saying "is an element of"): <math>\boldsymbol \in<math>

This symbol is used to indicate a particular object is not an element of a set (you can think of this as saying "is not an element of"): <math>\boldsymbol \not\in <math>

Here is an example of some statements we can make about the set X from above using the element symbols:

<math>4 \in X<math>

<math>5 \not\in X<math>

The first statement says 4 is an element of X. The second statement says 5 is not an element of X. These are both true statements based on our definition of X from above.

For information on other symbols used in set theory see the Table of mathematical symbols.

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 X is 4, while the cardinality of the sets Y and Q 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, while below we have a famous infinite set known as the set of natural numbers:

<math>N=\{\,1,2,3,4,5...\}<math>

The three dots <math>...<math> denote that the set keeps on going towards infinity.