Chapter 5 Sets
5.1 Basic Set Concepts
Learning Objectives
By the end of this section, you will be able to:
- Represent sets in a variety of ways
- Represent well-defined sets and the empty set with proper set notation
- Compute the cardinal value of a set
- Differentiate between finite and infinite sets
- Differentiate between equal and equivalent sets
Sets and Ways to Represent Them
Think back to your kitchen organization. If the drawer is the set, then the forks and knives are elements in the set. Sets can be described in a number of different ways: by roster, by set-builder notation, by interval notation, by graphing on a number line, and by Venn diagrams. Sets are typically designated with capital letters. The simplest way to represent a set with only a few members is the roster (or listing) method, in which the elements in a set are listed, enclosed by curly braces and separated by commas. For example, For example, if [latex]F[/latex] represents our set of flatware, we can represent [latex]F[/latex] by using the following set notation with the roster method:
[latex]F=\{ \text{fork, spoon, knife, meat thermometer, can opener} \}[/latex]
Example 1
Write a set consisting of your three favorite snow cone flavors and label it with a capital [latex]S[/latex].
There are many possible answers depending on what your three favorite snow cone flavors are, but any answer must list three different flavors separated by commas, such as the following:
[latex]S=\{ \text{wild cherry, blue raspberry, green apple} \}[/latex]
Exercise 1
Write a set consisting of four cities in Louisiana and label it with a capital [latex]C[/latex].
Solution
possible solution: [latex]C=\{ \text{Lake Charles, New Orleans, Baton Rouge, Shreveport} \}[/latex]
All the sets we have considered so far have been well-defined sets. A well-defined set clearly communicates whether an element is a member of the set or not. The members of a well-defined set are fixed and do not change over time. Consider the following question. What are your top 10 songs of 2021? You could create a list of your top 10 favorite songs from 2021, but the list your friend creates will not necessarily contain the same 10 songs. So, the set of your top 10 songs of 2021 is not a well-defined set. On the other hand, the set of the letters in your name is a well-defined set because it does not vary (unless of course you change your name). The NFL wide receiver Chad Johnson famously changed his name to Chad Ochocinco to match his jersey number of 85.
Example 2
For each of the following collections, determine if it represents a well-defined set.
(a) The group of all past governors of Louisiana.
The group of all past governors of Louisiana is a well-defined set, because you can clearly identify if any individual was or was not a member of that group. For example, Britney Spears is not a member of this set, but Huey P. Long is a member of this set.
(b) A group of old cats.
A group of old cats is not a well-defined set bec