Set Notation

A set is a well defined collection of objects. The objects in the set are called elements or members of the set. Sets are usually shown using capital letters (A, B, X etc), but the elements of the set are shown using small letters (a, b, x etc).

Let A be the set of the first 10 square numbers. So, A=\left\{ 1,4, 9, 16, 25, 36, 49, 64, 81, 100 \right\}

See how the elements are listed with curly brackets {}, separating the elements with commas.

To say that 16 belongs to set A, we write 16\in A

To say that 20 is not in A, we write 20\notin A

A set can be infinite, which means that it contains an infinite number of elements. The set N of the natural numbers can be written as:N=\left\{1, 2, 3, 4, ...\right\}

The dots in the bracket indicates that the elements keep on going for ever. The set of even numbers, E, can be written as:

E=\left\{2, 4, 6, 8, 10, 12, ...\right\}

We also describe the elements of a set using the set builder notation. For example, if set B is defined as the set of prime numbers less than 20, then:

B=\left\{2, 3, 5, 7, 11, 13, 17, 19\right\}

The Empty Set

The empty set, also known as the null set, contains no elements. We write it as {} or ∅. An example of the empty set is the set of all rectangles with five sides. It is empty because no rectangles have five sides.

Equal Sets

If two sets have the same elements in them, we say that they are equal sets. For example:

A=\left\{1, 3, 5, 7, 9\right\}

B=\left\{x:x \: is \: the \: odd \: numbers \: less \: than \: 10\right\}

Then A=\left\{1, 3, 5, 7, 9\right\}, B=\left\{1, 3, 5, 7, 9\right\}

Here, A and B are equal, so we write A=B.

The Universal Set

The universal set is the set that contains all the elements, which is often shown using the symbol U or ξ. For example, a universal set can be given as:

U=\left\{1, 2, 3, 4, ...10\right\}

Here, the elements 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 are everything that U may contain.

If we are further given that A=\left\{x:-5 \leq x \geq 5\right\}, then the elements of A are 1, 2, 3, 4

Notice that -5, -4, -3, -2, -2, -1, 0, 5, 6, 7, 8, 9, 10 are not elements of A. This is because the universal set only contains the elements 1, 2, 3, 4, …10.


A=\left\{1, 2, 3, 4, 5\right\}


The set B is a subset of the set A, because the elements in B are 2 and 4 and these elements are also in A. Actually, B is a proper subset of A, since B is entirely inside A.

We can represent this as a diagram:

We write B ⊂ A, which means that every element of B is in A, but A has more elements – B is contained in A.

We can also write it as A ⊃ B, which means that A has B’s elements and more – A contains B.