What is Set Builder Notation ?

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}

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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