### GCSE Maths

Numbers
Algebra
Geometry and Measures
Probability
Statistics

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

To say that 16 belongs to set A, we write To say that 20 is not in A, we write 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: The dots in the bracket indicates that the elements keep on going for ever. The set of even numbers, E, can be written as: 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: ## 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:  Then , 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: 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 , 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.

## Subset  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.