Set builder notation is another way of writing out a set. As you already know, normally your write out a set as a list of all the elements in a set, seperated by commas, with curly brackets around them, such as {1,2,3}. There is, however, another way of writing a set, one which is sometimes shorter and sometimes longer, and you will most likely have to know for tests.

## How do you use set builder notation?

set builder notation starts off just like normal set notation- everything is surrounded in curly brackets {}. However, instead of wrting out the *members* of a set, you write the *rule* that defines them. For example, if you had the set {Monday, Tuesday, Wensday, Thursday, Friday, Saturday, Sunday}, the rule would be all the days in the week. The rule is placed inside the curly brackets to the right of a vertical line |, to the left of which is placed “x”. The rule is always written out in terms of x. Sound confusing? It’s really not. Take a look below for a few examples that will help clarify.

{X | X is a day in the week}

{X | X is a month in the year}

{X | X is a natural number}

As you can see, it really isnt quite as complicated as it sounds. It always starts with a curly bracket, then the letter x, a vertical bar (If you’re typing, its shift+\, right under the backspace key on most keyboards), the rule in terms of x (defined by what x is) and then a closing curly brace. Written out in list notation (standard notation), the examples above would be as follows:

{Monday, Tuesday, Wensday, Thursday, Friday, Saturday, Sunday}

{January, February, March, April, May, June, July, August, September, October, November, December}

{1,2,3,4,5…}

Number three has the “dot dot dot” because natural numbers go on to infinity. You only need to include enough numbers to get the pattern for list form.

## What to Remember

Set Builder notation is made up of five parts:

- Open Curly Bracket {
- x
- Veritical bar |
- The rule in terms of x
- Closing Curly Bracket }

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