How to Write Sets Using Set Builder Notation

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  • 0:02 Set Notation
  • 2:48 Set Notation with Numbers
  • 5:56 Lesson Summary
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Lesson Transcript
Instructor: Maria Airth

Maria has a Doctorate of Education and over 15 years of experience teaching psychology and math related courses at the university level.

Set builder notation allows the user to indicate specific items that are to be included within a set. In this lesson, we will learn how to write sets using set builder notation.

Set Notation

Today we are going to learn the proper notation for mathematical sets. First, you really need to know what a set is. A set is just a gathering of defined items.

You can have a set of cars, or a set of collectible dolls, or a set of sheep. Let's stick with the sheep for a while.

Let's pretend you are a farmer with a field of sheep and cows. Some of those sheep are white with black faces, some black with white faces, some are completely white and some are completely black. You've heard a rumor that black wool is selling by the bag-fulls at the market, so you want to gather together all the sheep that give black wool so you can shear them.

You want only the sheep that are either all black or have just a white face, since you don't get wool from the face of the sheep. You tell your farmhand that you want him to fence all the sheep that have black wool to shear. Technically, you are saying you want only sheep and that they must have black wool on their bodies.

To state this in mathematical set terms, you would say, 'All sheep such that the wool on the body of the sheep is black.' Using math symbols, the same sentence is {sheep | sheep = black wool}.

Note that there are two parts to the notation with a separator in between. The first thing inside the brackets tells us what item we are looking for. The straight line literally means 'such that.' In English, that means 'with this criteria to define it;' and the last thing in the brackets gives the specific criteria for selection into the set. So, this set would mean that you want only sheep with black wool gathered together within the fence.

But, what if you decide that you need to gather the cows together because they need to be milked? Well, to give that message in set notation, you could write {animals | the animal is a cow}. This set notation literally translates to: 'Gather together all the animals that are cows.'

Set Notation with Numbers

You probably aren't a farmer and probably won't be working with sheep when you are using sets, so let's look at sets using numbers. Remember, a mathematical set is just gathering things together.

If you wanted to indicate that your set of numbers included the integers from one to five, you would simply have to write {1, 2, 3, 4, 5} without using the 'such that' language.

But, if you wanted to define a larger set, say, all values that are bigger than four, you would have to write {x | x > 4}. Remember that x is just a variable, a letter representing any number. It is defined, meaning the criteria for a number being included, in the second part of the bracket notation.

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