Set Operations with Two Sets

Learning Objectives

After completing this section, you should be able to:

1. Determine the intersection of two sets.
2. Determine the union of two sets.
3. Determine the cardinality of the union of two sets.
4. Apply the concepts of AND and OR to set operations.
5. Draw conclusions from Venn diagrams with two sets.

The Intersection of Two Sets

The members that the two sets share in common are included in the intersection of two sets. To be in the intersection of two sets, an element must be in both the first set and the second set. In this way, the intersection of two sets is a logical AND statement. Symbolically, \(A\) intersection \(B\) is written as: \(A\cap B\). \(A\) intersection \(B\) is written in set builder notation as: \(A\cap B=\{x|x\in A\ \text{and}\ x\in B\}\).

Let us look at Helen's and Frank's children from the movie Yours, Mine, and Ours. Helen's children consist of the set \(H=\{\text{Colleen, Nick, Janette, Tommy, Jean, Phillip, Gerald, Theresa, Joseph}\}\) and Frank's children are included in the set \(F=\{\text{Mike, Rusty, Greg, Rosemary, Loise, Susan, Veronica, Mary, Germaine, Joan, Joseph}\}\). \(H\) intersection \(F\) is the set of children they had together. \(H\cap F=\{\text{Joseph}\}\), because Joseph is in both set \(H\) and set \(F\).

Finding the Intersection of Set

Try it.

Set \(A=\{1,3,5,7,9\}\) and \(B=\{2,3,5,7\}.\) Find \(A\) intersection \(B.\)

Solution

The intersection of sets \(A\) and \(B\) include the elements that set \(A\) and \(B\) have in common: 3, 5, and 7. \(A\cap B=\{3,5,7\}.\)

Notice that if sets \(A\) and \(B\) are disjoint sets, then they do not share any elements in common, and \(A\) intersection \(B\) is the empty set, as shown in the Venn diagram below.

Determining the Intersection of Disjoint Sets

Try it.

Set \(A=\{0,2,4,6,8\}\) and set \(B=\{1,3,5,7,9\}.\) Find \(A\cap B.\)

Solution

Because sets \(A\) and \(B\) are disjoint, they do not share any elements in common. So, the intersection of set \(A\) and set \(B\) is the empty set. \(A\cap B=\emptyset .\)

Notice that if set \(A\) is a subset of set \(B\), then \(A\) intersection \(B\) is equal to set \(A\), as shown in the Venn diagram below.

Finding the Intersection of a Set and a Subset

Try it.

Set \(A=\{1,3,5,\ldots \}\) and set \(B=ℕ=\{1,2,3,\ldots \}\) Find \(A\cap B.\)

Solution

Because set \(A\) is a subset of set \(B\), \(A\) intersection \(B\) is equal to set \(A\). \(A\cap B=A=\{1,3,5,\ldots \},\) the set of odd natural numbers.

Condensed: the full section is in OpenStax Contemporary Mathematics.

Key Concepts

• The intersection of two sets, \(A\cap B\) is the set of all elements that they have in common. Any member of \(A\) intersection \(B\) must be is both set \(A\) and set \(B\).
• The union of two sets, \(A\cup B\), is the collection of all members that are in either in set \(A\), set \(B\) or both sets \(A\) and \(B\) combined.
• Two sets that share at least one element in common, so that they are not disjoint are represented in a Venn Diagram using two circles that overlap.
  • The region of the overlap is the set \(A\) intersection \(B\), \(A\cap B.\)
  • The regions that include everything in the circle representing set \(A\) or the circle representing set \(B\) or their overlap is the set \(A\) union \(B\), \(A\cup B.\)
• Apply knowledge of set union and intersection to determine cardinality and membership using Venn Diagrams, the roster method and set builder notation.

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1. Set \(A=\{1,3,5,7,9\}\) and \(B=\{2,3,5,7\}.\) Find \(A\) intersection \(B.\)

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   Antwort aufdecken

   The intersection of sets \(A\) and \(B\) include the elements that set \(A\) and \(B\) have in common: 3, 5, and 7. \(A\cap B=\{3,5,7\}.\)

2. Set \(A=\{0,2,4,6,8\}\) and set \(B=\{1,3,5,7,9\}.\) Find \(A\cap B.\)

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   Antwort aufdecken

   Because sets \(A\) and \(B\) are disjoint, they do not share any elements in common. So, the intersection of set \(A\) and set \(B\) is the empty set. \(A\cap B=\emptyset .\)

3. Set \(A=\{1,3,5,\ldots \}\) and set \(B=ℕ=\{1,2,3,\ldots \}\) Find \(A\cap B.\)

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   Antwort aufdecken

   Because set \(A\) is a subset of set \(B\), \(A\) intersection \(B\) is equal to set \(A\). \(A\cap B=A=\{1,3,5,\ldots \},\) the set of odd natural numbers.

4. Set \(A=\{1,3,5,7,9\}\) and set \(B=\{2,3,5,7\}\). Find \(A\) union \(B\).

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   Antwort aufdecken

   \(A\) union \(B\) is the set formed by including all the unique elements in set \(A\), set \(B\), or both sets \(A\) and \(B\): \(A\cup B=\{1,3,5,7,9,2\}.\) The first five elements of the union are the five unique elements in set \(A\). Even though 3, 5, and 7 are also members of set \(B\), these elements are only listed one time. Lastly, set \(B\) includes the unique element 2, so 2 is also included as part of the union of sets \(A\) and \(B\).

5. Set \(A=\{0,2,4,6,8\}\) and set \(B=\{1,3,5,7,9\}.\) Find \(A\cup B.\)

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   Antwort aufdecken

   Because sets \(A\) and \(B\) are disjoint, the union is simply the set containing all the elements in both set \(A\) and set \(B\). \(A\cup B=\{0,1,2,3,4,5,6,7,8,9\}.\)

6. Set \(A=\{1,3,5,\ldots \}\) and set \(B=ℕ=\{1,2,3,\ldots \}.\) Find \(A\cup B.\)

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   Antwort aufdecken

   Because set \(A\) is a subset of set \(B\), \(A\) union \(B\) is equal to set \(B\). \(A\cup B=ℕ=\{1,2,3,\ldots \}=B.\)

7. The number of elements in set \(A\) is 10, the number of elements in set \(B\) is 20, and the number of elements in \(A\) intersection \(B\) is 4. Find the number of elements in \(A\) union \(B\).

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   Antwort aufdecken

   Using the formula for determining the cardinality of the union of two sets, we can say \(n(A\cup B)=n(A)+n(B)-n(A\cap B)=10+20-4=26.\)

8. If \(A\) and \(B\) are disjoint sets and the cardinality of set \(A\) is 37 and the cardinality of set \(B\) is 43, find the cardinality of \(A\) union \(B\).

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   Antwort aufdecken

   To find the cardinality of \(A\) union \(B\), apply the formula, \(n(A\cup B)=n(A)+n(B)-n(A\cap B).\) Because sets \(A\) and \(B\) are disjoint, \(A\cap B\) is the empty set, therefore \(n(A\cap B)=n(\emptyset )=0\) and \(n(A\cup B)=37+43-0=80.\)

9. \(A=\{0,3,6,9,12\},B=\{0,4,8,12,16\},\) and \(C=\{1,2,3,5,8,13\}.\)

   Find the set consisting of elements in:

   1. \(A\ \text{and}\ B.\)
   2. \(A\ \text{or}\ B.\)
   3. \(A\ \text{or}\ C.\)
   4. \((B\ \text{and}\ C)\ \text{or}\ A.\)
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   1. \(A\ \text{and}\ B=A\cap B=\{0,12\},\) because only the elements 0 and 12 are members of both set \(A\) and set \(B\).
   2. \(A\ \text{or}\ B=A\cup B=\{0,3,4,6,8,9,12,16\},\) because the set \(A\) or \(B\) is the collection of all elements in set \(A\) or set \(B\), or both.
   3. \(A\ \text{or}\ C=A\cup C=\{0,1,2,3,5,6,8,9,12,13\},\) because the set \(A\) or \(C\) is the collection of all elements in set \(A\) or set \(C\), or both.
   4. \((B\ \text{and}\ C)\ \text{or}\ A=(B\cap C)\cup A.\) Parentheses are evaluated first: \((B\ \text{and}\ C)=B\cap C=\{8\},\) because the only member that both set \(B\) and set \(C\) share in common is 8. So, now we need to find \(\{8\}\ \text{or}\ \{0,3,6,9,12\},\) Because the word translates to the union operation, the problem becomes \(\{8\}\cup \{0,3,6,9,12\},\) which is equal to \(\{0,3,6,8,9,12\}.\)

10. Don Woods is serving cake and ice cream at his Juneteenth celebration. The party has a total of 54 guests in attendance. Suppose 30 guests requested cake, 20 guests asked for ice cream, and 12 guests did not have either cake or ice cream.

    1. How many guests had cake or ice cream?
    2. How many guests had cake and ice cream?
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    Antwort aufdecken

    1. The total number of people at the party is 54, and 12 people did not have cake or ice cream. Recall that the total number of elements in the universal set is always equal to the number of elements in a subset plus the number of elements in the complement of the set, \(n(U)=n(A)+n({A}^{'}).\) That means \(54=n(\text{cake or ice cream})\ +\ n(\text{not}\ (\text{cake or ice cream}))\), or equivalently,\(n(\text{cake}\ \cup \ \text{ice cream})=\ \text{54}-n({(\text{cake}\ \cup \ \text{ice cream})}^{'})=54-12=42.\) A total of 42 people at the party had cake or ice cream.
    2. To determine the number of people who had both cake and ice cream, we need to find the intersection of the set of people who had cake and the set of people who had ice cream. From Question 1, the number of people who had cake or ice cream is 42. This is the union of the two sets. The formula for the union of two sets is \(n(A\cup B)=n(A)+n(B)-n(A\cap B).\) Use the information given in the problem and substitute the known values into the formula to solve for the number of people in the intersection: \(42=30+20-n(A\cap B).\) Adding 30 and 20, the equation simplifies to \(42=50-n(\text{cake and ice cream)}\text{.}\) Which means \(n(\text{cake and ice cream)}=50-42=8.\)

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

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    Antwort aufdecken

    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, \(A\cap B=\emptyset .\)
    3. The complement of set \(B\) is the set of all elements in the universal set that are not in set \(B\): \({B}^{'}=\{0,1,3,5,7,9\}.\)
    4. The cardinality, or number of elements in set \({B}^{'},\ \text{is}\ n({B}^{'})=6.\)

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

    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 \(A\): \(n(A)=n(\{7,5\})=12.\)

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x \in Aelement of, subset
x belongs to A; every element of A is in B.A \cup Bunion, intersection, difference
In either; in both; in A but not B.\neqnot equal
The two sides are different.\inftyinfinity
Not a number: "grows without bound" in limits and intervals.\mathbb{N}number sets
Naturals, integers, rationals, reals, complex numbers.\emptysetempty set, cardinality
The set with no elements; the number of elements of A.\forallfor all, there exists
Quantifiers: every x; at least one x.\negnot, and, or, implies, iff
Logical connectives.\blacksquare\ \text{or}\ \squareend of proof (halmos)
Marks the point where the statement has been established.

Wie man: Set Operations with Two Sets

1. Determine the intersection of two sets.
2. Determine the union of two sets.
3. Determine the cardinality of the union of two sets.
4. Apply the concepts of AND and OR to set operations.
5. Draw conclusions from Venn diagrams with two sets.
6. The complement of set
7. In the text we do not cover set difference between two sets
8. How many guests had cake or ice cream?

Fragen, die die Leute stellen

Are some infinities bigger than others?

Yes. The integers and the rationals can be listed; the real numbers cannot (Cantor's diagonal argument), so there are strictly more reals than integers.

What is the difference between a relation and a function?

A relation pairs inputs with outputs freely; a function is a relation in which every input gets exactly one output.

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