Chapter 5 Sets
5.2 Subsets
Learning Objectives
By the end of this section, you will be able to:
- Represent subsets and proper subsets symbolically
- Compute the number of subsets of a set
- Apply concepts of subsets and equivalent sets to finite and infinite sets
The rules of Major League Soccer (MLS) allow each team to have up to 30 players on their team. However, only 18 of these players can be listed on the game day roster, and of the 18 listed, 11 players must be selected to start the game. How the coaches and general managers form the team and choose the starters for each game will determine the success of the team in any given year.
The entire group of 30 players is each team’s set. The group of game day players is a subset of the team set, and the group of 11 starters is a subset of both the team set and the set of players on the game day roster.
Set [latex]A[/latex] is a subset of set [latex]B[/latex] if every member of set [latex]A[/latex] is also a member of set [latex]B[/latex]. Symbolically, this relationship is written as [latex]A \subseteq B[/latex].
Sets can be related to each other in several different ways: they may not share any members in common, they may share some members in common, or they may share all members in common. In this section, we will explore the way we can select a group of members from the whole set.
Every subset is also a subset of itself, [latex]B \subseteq B[/latex].
Recall the set of flatware in our kitchen drawer from Section 5.1, [latex]F= \{ \text{fork, spoon, knife, meat thermometer, can opener} \}[/latex]. Suppose you are preparing to eat dinner, so you pull a fork and a knife from the drawer to set the table. The set [latex]D= \{ \text{knife, fork} \}[/latex]is a subset of set [latex]F[/latex] because every member or element of set [latex]D[/latex] is also a member of set [latex]F[/latex]. More specifically, set [latex]D[/latex] is a proper subset of set [latex]F[/latex], because there are other members of set [latex]F[/latex] not in set [latex]D[/latex]. This is written as [latex]D \subset F[/latex]. The only subset of a set that is not a proper subset of the set would be the set itself.
The empty set or null set, [latex]\emptyset[/latex], is a proper subset of every set except itself.
Graphically, sets are often represented as circles. In the following graphic, set [latex]A[/latex] is represented as a circle completely enclosed inside the circle representing set [latex]B[/latex], showing that set [latex]A[/latex] is a proper subset of set [latex]B[/latex]. The element [latex]x[/latex] represents an element that is in both set [latex]A[/latex] and set [latex]B[/latex].
Example 1
Set [latex]L[/latex] is a set of reading materials available in a shop at the airport, [latex]L = \{ \text{newspaper, magazine, book} \}[/latex]. List all the subsets of set [latex]L[/latex].
Step 1: It is best to begin with the set itself, as every set is a subset of itself. In our example, the cardinality of set [latex]L[/