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}.
