How to Write Sets in Roster Form

How to Write Sets in Roster Form
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  • 0:04 Roster Form of a Set
  • 1:13 Writing a Set in Roster Form
  • 3:13 Limitations of Roster Form
  • 5:17 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

We see lists of objects all the time in the world around us. This lesson will explain how to represent a set in roster form, or as a list of elements. We will also look at some pros and cons to roster form of a set.

Roster Form of a Set

Quick question! What is your favorite sports team? Now that you're thinking of it, have you ever taken a look at a list of your team's members, or in other words, the team roster? You see, a team roster is basically a list of the members of the team. For example, suppose a baseball team, the Derivatives, has 15 members. A list of these members would make up the team roster.

Now, let's think about this mathematically. Mathematically speaking, roster form of a set is a list of elements that are in the set. To illustrate this, consider the Derivatives' team roster again. The team is the set, and the team members are the elements of that set. Mathematically, we would represent this set in roster form as follows:

Team Roster = {Harry, Jake, Ryan, Nathan, Paul, Jesse, Casey, Scott, Jeff, Eric, Dan, Heath, Jeremy, Peyton, Alex}

Basically, to represent a set in roster form, we simply list the elements of the set, separated by commas, within braces. Now that we're familiar with what the roster form of a set is, let's dig a bit deeper.

Writing a Set in Roster Form

There are many ways that a set can be represented. We can use a verbal description, set notation, roster form, etc. In order to write a set in roster form, we simply need to follow these steps:

  1. Identify all of the elements in the set.
  2. List the elements, separated by commas, within braces.

For example, consider the set, S, described verbally:

  • S = {all integers that are strictly greater than 0 and less than or equal to 4}

To write this in roster form, we would first identify all the elements in the set. Let's see. . . the integers that are strictly greater than 0 and less than or equal to 4 would be the integers that are between 0 and 4, not including 0, but including 4, so 1, 2, 3, and 4.

Now we just write these integers, separated by commas, within braces.

  • S = {1, 2, 3, 4}

Ta-da! We've got S in roster form! That's not so hard, is it?

Let's consider another example. Set notation is a representation of a set of the form {element | properties of that element}. Consider the set T given in set notation.

  • T = {x | x is prime and 3 ≤ x ≤ 10}

To write T in roster form, we first figure out all of its elements. We see that T consists of all elements, x, such that x is prime and 3 ≤ x ≤ 10. Prime numbers are positive integers that are only divisible by 1 and themselves, so T is all the integers between 3 and 10 (3 and 10 included) that are only divisible by 1 and themselves. Those numbers would be 3, 5, and 7. Now we just list them, separated by commas, within braces.

  • T = {3, 5, 7}

Once again, that was pretty easy!

Limitations of Roster Form

As we've seen roster form is one of the most basic ways to represent a set. However, that doesn't mean that it's always the most appropriate representation to use. For instance, consider the set, R, of all real numbers. Hmmm. . . if we tried to write this in roster form, we would need to list all of the elements, but this leads to a few questions:

  1. Where do we start?
  2. Where do we end?
  3. How is it possible to list all the real numbers, when we can always come up with another one?

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