4.3: Subsets

Figure 4.3.1: The players on a soccer team who are actively participating in a game are a subset of the greater set of team members. (Credit: “PAFC-Mezokovesd-108” by Puskás Akadémia/Flickr, Public Domain Mark 1.0)

Learning Objectives

After completing this section, you should be able to:

1. Represent subsets and proper subsets symbolically.
2. Compute the number of subsets of a set.
3. 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.

Subset

Set $A$ is a subset of set $B$ if every member of set $A$ is also a member of set $B$. Symbolically, this relationship is written as $A\subseteq B$.

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.

Checkpoint

Every set is also a subset of itself, $B\subseteq B$

Recall the set of flatware in our kitchen drawer from Section 4.1, $F=\left\{\text{fork, spoon, knife, meat thermometer, can opener}\right\}$. Suppose you are preparing to eat dinner, so you pull a fork and a knife from the drawer to set the table. The set $D=\left\{\text{knife, fork}\right\}$ is a subset of set $F$, because every member or element of set $D$ is also a member of set $F$. More specifically, set $D$ is a proper subset of set $F$, because there are other members of set $F$ not in set $D$. This is written as $D\subset F$.

Proper Subset

Set $A$ is a proper subset of set $B$ if every member of set $A$ is also a member of set $B$, but B also contains at least one element that is not in A. Symbolically, this relationship is written as $A\subset B$.

The only subset of a set that is not a proper subset of the set would be the set itself.

Checkpoint

The empty set or null set, $\varnothing$, is a proper subset of every set, except itself.

Graphically, sets are often represented as circles. In the following graphic, set $A$ is represented as a circle completely enclosed inside the circle representing set $B$, showing that set $A$ is a proper subset of set $B$. The element $x$ represents an element that is in both set $A$ and set $B$.

Checkpoint

While we can list all the subsets of a finite set, it is not possible to list all the possible subsets of an infinite set, as it would take an infinitely long time.

Example 4.3.1

Listing All the Proper Subsets of a Finite Set

Set $L$ is a set of reading materials available in a shop at the airport, $L=\left\{\text{newspaper, magazine, book}\right\}$. List all the subsets of set $L$.

Answer

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 $L$ is $n(L)=3$. There is only one subset of set $L$ that has the same number of elements of set $L:\left\{\mathrm{newspaper},\mathrm{magazine},\mathrm{book}\right\}$.

Step 2: Next, list all the proper subsets of the set containing $n(L)-1$ elements. In this case, $3-1=2$. There are three subsets that each contain two elements: $\left\{\mathrm{newspaper},\mathrm{magazine}\right\}$, $\left\{\mathrm{newspaper},\mathrm{book}\right\}$, and $\left\{\mathrm{magazine},\mathrm{book}\right\}$.

Step 3: Continue this process by listing all the proper subsets of the set containing $n(L)-2$ elements. In this case, $3-2=1$. There are three subsets that contain one element: $\left\{\mathrm{newspaper}\right\}$, $\left\{\mathrm{magazine}\right\}$, and $\left\{\mathrm{book}\right\}$.

Step 4: Finally, list the subset containing 0 elements, or the empty set: $\left\{\right\}$.

Your Turn 4.3.1

Consider the set of possible outcomes when you flip a coin, S = {heads, tails}. List all the possible subsets of set S.

Example 4.3.2

Determining Whether a Set Is a Proper Subset

Consider the set of common political parties in the United States, $P=\left\{\mathrm{Democratic,\; Green,\; Libertarian,\; Republican}\right\}$. Determine if the following sets are proper subsets of $P$.

1. $M=\left\{\mathrm{Democratic,\; Republican}\right\}$
2. $G=\left\{\mathrm{Green}\right\}$
3. $V=\left\{\mathrm{Republican,\; Libertarian,\; Green,\; Democratic}\right\}$

Answer

1. $M$ is a proper subset of $P$, written symbolically as $M\subset P$ because every member of $M$ is a member of set $P$, but $P$ also contains at least one element that is not in $M$.
2. $G$ is a single member proper subset of $P$, written symbolically as $G\subset P,$ because Green is a member of set $P$, but $P$ also contains other members (such as Democratic) that are not in $G$.
3. $V$ is subset of $P$ because every member of $V$ is also a member of $P$, but it is not a proper subset of $P$ because there are no members of $V$ that are not also in set $P$. We can represent the relationship symbolically as $V\subseteq P,$ or more precisely, set $V$ is equal to set $P$, $V=P.$

Your Turn 4.3.2

Consider the set of generation I legendary Pokémon, L = {Articuno, Zapdos, Moltres, Mewtwo}. Give an example of a proper subset containing:

1. one member

2. three members

3. no members

Example 4.3.3

Expressing the Relationship between Sets Symbolically

Consider the subsets of a standard deck of cards: $S=\left\{\mathrm{spades,\; hearts,\; diamonds,\; clubs}\right\}$; $R=\left\{\mathrm{hearts,\; diamonds}\right\}$; $B=\left\{\mathrm{spades,\; clubs}\right\}$; and $C=\left\{\mathrm{clubs}\right\}$.

Express the relationship between the following sets symbolically.

1. Set $S$ and set $B$.
2. Set $C$ and set $B$.
3. Set $R$ and $R$.

Answer

1. $B\subset S$. $B$ is a proper subset of set $S$.
2. $C\subset B$. $C$ is a proper subset of set $B$.
3. $R\subseteq R\mathrm{or}R=R$. $R$ is subset of itself, but not a proper subset of itself because $R$ is equal to itself.

Your Turn 4.3.3

Express the relationship between the set of natural numbers, ℕ = {1, 2, 3, ...}and the set of even numbers, E = {2, 4, 6, ...}.

Number of Subsets

So far, we have figured out how many subsets exist in a finite set by listing them. Recall that in Example 4.3.1, when we listed all the subsets of the three-element set L = {newspaper, magazine, book}, we discovered that there were eight subsets. In Your Turn 4.3.1, we discovered that there are four subsets of the two-element subset, S = {heads, tails}. A one-element set has two subsets, the empty set and itself. The only subset of the empty set is the empty set itself. But how can we easily figure out the number of subsets in a very large finite set? It turns out that the number of subsets can be found by raising 2 to the number of elements in the set.

FORMULA

The number of subsets of a finite set $A$ is equal to 2 raised to the power of $n\left(A\right)$, where $n\left(A\right)$ is the number of elements in set $A$: $\mathrm{Number\; of\; Subsets\; of\; Set}A=2^{n(A)}$.

Checkpoint

Note that $2^0=1$, so this formula works for the empty set, also.

Example 4.3.4

Computing the Number of Subsets of a Set

Find the number of subsets of each of the following sets.

1. The set of top five scorers of all time in the NBA: $S=\left\{\mathrm{LeBron\; James,\; Kareem\; Abdul-Jabbar,\; Karl\; Malone,\; Kobe\; Bryant,\; Michael\; Jordan}\right\}\text{.}$
2. The set of the top four bestselling albums of all time: $A=\left\{\mathrm{Thriller,\; Hotel\; California,\; The\; Beatles\; White\; Album,\; Led\; Zepplin\; IV}\right\}$.
3. $R=\left\{\mathrm{Snap,\; Crackle,\; Pop}\right\}$.

Answer

1. $n(S)=5$. So, the total number of subsets of $S\text{is}{2}^5=32$.
2. $n(A)=4$. Therefore, the total number of subsets of $A\text{is}{2}^4=16$.
3. $n(R)=3$. So, the total number of subsets of $R\text{is}{2}^3=8$.

Your Turn 4.3.4

Compute the total number of subsets in the set of the top nine tennis grand slam singles winners, T = {Margaret Court, Serena Williams, Steffi Graff, Serena Williams, Rafael Nadal, Novak Djokovic}.

Who Knew?

Employment Opportunities

You can make a career out of working with sets. Applications of equivalent sets include relational database design and analysis.

Relational databases that store data are tables of related information. Each row of a table has the same number of columns as every other row in the table; in this way, relational databases are examples of set equivalences for finite sets. In a relational database, a primary key is set up to identify all related information. There is a one-to-one relationship between the primary key and any other information associated with it.

Database design and analysis is a high demand career with a median entry-level salary of about $85,000 per year, according to salary.com.