# Set Builder Form – Definition and Examples

Here you will learn what is set builder form and how to represent sets in set builder form with examples.

Let’s begin –

## Set Builder Form

Definition : In this form, a set is described by a characterizing property P(x) of its elements x. In such a case the set is described by {x : P(x) holds} or, {x | P(x) holds}, which is read as ‘the set of all x such that P(x) holds’. The symbol ‘|’ or ‘:’ is read as ‘such that’.

In other words, in order to describe a set, a variable x (say) (to denote each element of the set) is written inside the braces and then after putting a colon the common property P(x) possessed by each element of the set is written within the braces.

Example 1 : The set E of all even natural numbers can be written as

E = {x : x is a natural number and x = 2n for n $$\in$$ N}

or, E = {x : x $$\in$$ N, x = 2n, n $$\in$$ N}

or, E = {x $$\in$$ N : x = 2n, n $$\in$$ N}

Example 2 : The set A = {1, 2, 3, 4, 5, 6, 7, 8} can be written as A = {x $$\in$$ N, x $$\le$$ 8}.

Example 3 : The set of all real numbers greater than -1 and less than 1 can be described as {x $$\in$$ R : -1 < x < 1}.

Example 4 : The set A = {0, 1, 4, 9, 16, ….} can be written as A = {$$x^2$$ : x $$\in$$ Z}.