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

When the set is written as { 1, 2, 3, 4, 5, 6, 7, 8, 9 } , we call it the roster method.

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 

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 }

Recent Articles

  1. High School Math Teacher

    Jan 20, 20 01:57 PM

    Top-notch information for those who want to become a high school math teacher. All important stuff that you need to know before making a move can be found right here on this website.

    Read More

New math lessons

Your email is safe with us. We will only use it to inform you about new math lessons.

                                 Follow me on Pinterest

Real Life Math Skills

Learn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball.