1.4 Set Operations with Two Sets

The Union of Two Sets

Like the union of two families in marriage, the union of two sets includes all the members of the first set and all the members of the second set. To be in the union of two sets, an element must be in the first set, the second set, or both. In this way, the union of two sets is a logical inclusive OR statement. Symbolically, $A$ union $B$ is written as: $A\cup B.$ $A$ union $B$ is written in set builder notation as: $A\cup B=\left\{x\right|x\in A\text{or}x\in B\}.$

Let us consider the sets of Helen's and Frank's children from the movie Yours, Mine, and Ours again. Helen's children is set $H=\left\{\text{Colleen, Nick, Janette, Tommy, Jean, Phillip, Gerald, Theresa, Joseph}\right\}$ and Frank's children is set $F=\left\{\text{Mike, Rusty, Greg, Rosemary, Loise, Susan, Veronica, Mary, Germaine, Joan, Joseph}\right\}$. The union of these two sets is the collection of all nineteen of their children, $\begin{array}{l}H\cup F=\{\text{Colleen, Nick, Janette, Tommy, Jean, Phillip, Gerald, Theresa, Joseph,}\\ \text{Mike, Rusty, Greg, Rosemary, Loise, Susan, Veronica, Mary, Germaine, Joan}\}\text{.}\end{array}$
Notice, Joseph is in both set $H$ and set $F$, but he is only one child, so, he is only listed once in the union.

When observing the union of sets $A$ and $B$, notice that both set $A$ and set $B$ are subsets of $A$ union $B$. Graphically, $A$ union $B$ can be represented in several different ways depending on the members that they have in common. If $A$ and $B$ are disjoint sets, then $A$ union $B$ would be represented with two disjoint circles within the universal set, as shown in the Venn diagram below.

Figure 1.21 $A\cup B$

If sets $A$ and $B$ share some, but not all, members in common, then the Venn diagram is drawn as two separate circles that overlap.

Figure 1.22

If every member of set $A$ is also a member of set $B$, then $A$ is a subset of set $B$, and $A$ union $B$ would be equal to set $B$. To draw the Venn diagram, the circle representing set $A$ should be completely enclosed in the circle containing set $B$.

Figure 1.23

Tech Check

Set Operation Practice

Sets Challenge is an application available on both Android and iPhone smartphones that allows you to practice and gain familiarity with the operations of set union, intersection, complement, and difference.

Figure 1.24 Google Play Store image of Sets Challenge game. (credit: screenshot from Google Play)

The Sets Challenge application/game uses some notation that differs from the notation covered in the text.

• The complement of set $A$ in this text is written symbolically as $A^{\prime },$ but the Sets Challenge game uses $A^C$ to represent the complement operation.
• In the text we do not cover set difference between two sets $A$ and $B$, represented in the game as $A-B.$ In the game this operation removes from set $A$ all the elements in $A\cap B.$ For example, if set $A=\{a,b,c,d\}$ and set $B=\{b,d,f,h\}$ are subsets of the universal set $U=\{a,b,c,\dots ,z\},$ then $A-B=\{a,b,c,d\}-\{b,d\}=\{a,c\},$ and $B-A=\{b,d,f,h\}-\{b,d\}=\{f,h\}.$ There is a project at the end of the chapter to research the set difference operation.

Determining the Cardinality of Two Sets

The cardinality of the union of two sets is the total number of elements in the set. Symbolically the cardinality of $A$ union $B$ is written, $n\left(A\cup B\right)$. If two sets $A$ and $B$ are disjoint, the cardinality of $A$ union $B$ is the sum of the cardinality of set $A$ and the cardinality of set $B$. If the two sets intersect, then $A$ intersection $B$ is a subset of both set $A$ and set $B$. This means that if we add the cardinality of set $A$ and set $B$, we will have added the number of elements in $A$ intersection $B$ twice, so we must then subtract it once as shown in the formula that follows.

Applying Concepts of “AND” and “OR” to Set Operations

To become a licensed driver, you must pass some form of written test and a road test, along with several other requirements depending on your age. To keep this example simple, let us focus on the road test and the written test. If you pass the written test but fail the road test, you will not receive your license. If you fail the written test, you will not be allowed to take the road test and you will not receive a license to drive. To receive a driver's license, you must pass the written test AND the road test. For an “AND” statement to be true, both conditions that make up the statement must be true. Similarly, the intersection of two sets $A$ and $B$ is the set of elements that are in both set $A$ and set $B$. To be a member of $A$ intersection $B$, an element must be in set $A$ and also must be in set $B$. The intersection of two sets corresponds to a logical "AND" statement.

The union of two sets is a logical inclusive "OR" statement. Say you are at a birthday party and the host offers Leah, Lenny, Maya, and you some cake or ice cream for dessert. Leah asks for cake, Lenny accepts both cake and ice cream, Maya turns down both, and you choose only ice cream. Leah, Lenny, and you are all having dessert. The “OR” statement is true if at least one of the components is true. Maya is the only one who did not have cake or ice cream; therefore, she did not have dessert and the “OR” statement is false. To be in the union of two sets $A$ and $B$, an element must be in set $A$ or set $B$ or both set $A$ and set $B$.

Drawing Conclusions from a Venn Diagram with Two Sets

All Venn diagrams will display the relationships between the sets, such as subset, intersecting, and/or disjoint. In addition to displaying the relationship between the two sets, there are two main additional details that Venn diagrams can include: the individual members of the sets or the cardinality of each disjoint subset of the universal set.

A Venn diagram with two subsets will partition the universal set into 3 or 4 sections depending on whether they are disjoint or intersecting sets. Recall that the complement of set $A$, written $A^{\prime },$ is the set of all elements in the universal set that are not in set $A.$

Figure 1.25 Side-by-side Venn diagrams with disjoint and intersecting sets, respectively.

Example 1.33

Using a Venn Diagram to Draw Conclusions about Set Membership

Figure 1.26

1. Find $A\cup B.$
2. Find $A\cap B.$
3. Find $B^{\prime }$.
4. Find $n(B^{\prime }).$

Solution

1. $A\cup B=\{1,2,3,4,5,6,7,8\},$ because $A$ union $B$ is the collection of all elements in set $A$ or set $B$ or both.
2. Because $A$ and $B$ are disjoint sets, there are no elements that are in both $A$ and $B$. Therefore, $A$ intersection $B$ is the empty set, $A\cap B=\varnothing .$
3. The complement of set $B$ is the set of all elements in the universal set that are not in set $B$: $B^{\prime }=\{0,1,3,5,7,9\}.$
4. The cardinality, or number of elements in set $B^{\prime },\text{is}n(B^{\prime })=6.$

Your Turn 1.33

Venn diagram with two intersecting sets and members.

1.

Find $A\cap <!--\; \cap \; -->B$.

2.

Find $A\cup <!--\; \cup \; -->B$.

3.

Find $A\cap <!--\; \cap \; -->B^{\prime }$.

4.

Find $n(A\cap <!--\; \cap \; -->B^{\prime })$.

Example 1.34

Using a Venn Diagram to Draw Conclusions about Set Cardinality

Figure 1.27 Venn diagram with two intersecting sets and number of elements in each section indicated.

1. Find $n(A\text{or}B).$
2. Find $n(A\text{and}B).$
3. Find $n(A).$

Solution

1. The number of elements in $A$ or $B$ is the number of elements in $A$ union $B$: $n(A\cup B)=n(\{2,5,7\})=14.$
2. The number of elements in $A$ and $B$ is the number of elements in $A$ intersection $B$: $n(A\cap B)=5.$
3. The number of elements in set $A$ is the sum of all the numbers enclosed in the circle representing set $A$: $n(A)=n(\{7,5\})=12.$

Your Turn 1.34

Venn diagram with two disjoint sets and number of elements in each section.

1.

Find $n(A\text{ or }B)$.

2.

Find $n(A\text{ and }B)$.

3.

Find $n(A^{\mathrm{\prime <!--\; \prime \; -->}})$.