Set-builder notation

A set-builder notation describes or defines the elements of a set instead of listing the elements. For example, the set { 1, 2, 3, 4, 5, 6, 7, 8, 9 } list the elements. 

The same set could be described as { x/x is a counting number less than 10 } in set-builder notation.

We read the set { x / x is a counting number less than 10 } as the set of all x such that x is a counting number less than 10.

More examples showing how to set-builder notation works

You can list all even numbers between 10 and 20 inside curly braces separated by a comma. Again, this is called the roster notation. 

However, could you use the roster notation to list all the prime numbers? See now when it is a good idea to use the set-builder notation.

Why do we use set-builder notation?

Some sets are big or have many elements, so it is more convenient to use set-builder notation as opposed to listing all the elements which is not practical when doing math.

For example, instead of making a list of all counting numbers smaller than 1000, it is more convenient to write { x / x is a counting number less than 100 }

It is also very useful to use a set-builder notation to describe the domain of a function.

If f(x) = 2 / (x-5), the domain of f is {x / x is not equal to 5}

More examples showing the set-builder notation

1) x > 9

Unless otherwise stated, you should always assume that a given set consists of real numbers. 

Therefore, x > 9 can be written as { x / x > 9, is a real number }

2) The set of all integers that are all multiples of five.

{x / x = 5n, n is an integer }

3) { -6, -5, -4, -3, -2, ...  }

{ x / x ≥ -6, x is an integer }

4) The set of all even numbers 

{x / x = 2n, n is an integer }

5) The set of all odd numbers 

{x / x = 2n + 1, n is an integer }