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Chapter 5 Sets

5.5 Set Operations with Three Sets

Learning Objectives

By the end of this section, you will be able to:

  • Interpret Venn diagrams with three sets
  • Create Venn diagrams with three sets
  • Apply set operations to three sets
  • Prove equality of sets using Venn diagrams

Have you ever searched for something on the Internet and then soon after started seeing multiple advertisements for that item while browsing other web pages? Large corporations have built their businesses on data collection and analysis. As we start working with larger data sets, the analysis becomes more complex. In this section, we will extend our knowledge of set relationships by including a third set.

A Venn diagram with two intersecting sets breaks up the universal set into four regions; simply adding one additional set will increase the number of regions to eight, doubling the complexity of the problem.

Venn Diagrams with Three Sets

Below is a Venn diagram with two intersecting sets, which breaks up the universal set into four distinct regions.

A Venn diagram with two overlapping circles labeled A and B, showing intersections and complements within a universal set U.
Figure 1. Venn diagram with two intersecting sets
Next, we see a Venn diagram with three intersecting sets, which breaks up the universal set into eight distinct regions.
A Venn diagram with three overlapping circles labeled A, B, and C showing intersections and complements within a universal set U.
Figure 2. Three-set Venn diagram
In the next example, we will explore the three main blood factors, [latex]A, B[/latex] and [latex]Rh[/latex]. The following background information about blood types will help explain the relationships between the sets of blood factors. If an individual has blood factor [latex]A[/latex] or [latex]B[/latex], those will be included in their blood type. The [latex]Rh[/latex] factor is indicated with a [latex]+[/latex] or a [latex]-[/latex]. For example, if a person has all three blood factors, then their blood type would be [latex]AB^+[/latex]. In the Venn diagram, they would be in the intersection of all three sets, [latex]A \cap B \cap Rh^+[/latex]. If a person did not have any of these three blood factors, then their blood type would be [latex]O^-[/latex], and they would be in the set [latex](A \cup B \cup Rh^+)'[/latex], which is the region outside all three circles.

Example 1

Use the Venn diagram below, which shows the blood types of [latex]100[/latex] people who donated blood at a local clinic, to answer the following questions.

Venn diagram showing blood types of 100 people. Categories include A, B, O, Rh factors. Key numbers: O+ 37, A+ 36, B+ 8, AB+ 3.
Figure 3. Blood types of 100 people Venn diagram
  1. How many people with a type [latex]A[/latex] blood factor donated blood?

The number of people who donated blood with a type [latex]A[/latex] blood factor will include the sum of all the values included in the [latex]A[/latex] circle. It will be the union of sets [latex]A^-, A^+, AB^-[/latex], and [latex]AB^+[/latex]
[latex]n(A)=n(A^-)+n(A^+)+n(AB^-)+n(AB^+)=6+36+1+3=46[/latex].

  1. Julio has blood type [latex]B^+[/latex]. If he needs to have surgery that requires a blood transfusion, he can accept blood from anyone who does not have a type [latex]A[/latex] blood factor. How many people donated blood that Julio can accept?

In part a, it was determined that the number of donors with a type [latex]A[/latex] blood factor is [latex]46[/latex]. To determine the number of people who did not have a type [latex]A[/latex] blood factor, use the following property [latex]A'[/latex] union [latex]A[/latex] is equal to [latex]U[/latex], which means [latex]n(A)+n(A')=n(U)[/latex], and [latex]n(A')=n(U)-n(A)=100-46=54[/latex]. Thus, [latex]54[/latex] people donated blood that Julio can accept.

  1. How many people who donated blood do not have the [latex]Rh^+[/latex] blood factor?

This would be everyone outside the [latex]Rh^+[/latex]circle, or everyone with a negative [latex]Rh[/latex] factor, [latex]n(Rh^-)=n(O^-)+n(A^-)+n(AB^-)+n(B^-)=7+6+1+2=16[/latex].

  1. How many people had type [latex]A[/latex] and type [latex]B[/latex] blood?

Exercise 1

Solution
  1. [latex]n(B)=n(AB^+)+n(AB^-)+n(B^+)+n(B^-)=14[/latex]
  2. [latex]n(B')=n(U)-n(B)=86[/latex]
  3. [latex]n(B \cup Rh^+)=87[/latex]
Creating Venn Diagrams with Three Sets

In general, when creating Venn diagrams from data involving three subsets of a universal set, the strategy is to work from the inside out. Start with the intersection of the three sets, then address the regions that involve the intersection of two sets. Next, complete the regions that involve a single set, and finally address the region in the universal set that does not intersect with any of the three sets. This method can be extended to any number of sets. The key is to start with the region involving the most overlap, working your way from the center out.

Example 2

A teacher surveyed her class of [latex]43[/latex] students to find out how they prepared for their last test. She found that [latex]24[/latex] students made flash cards, [latex]14[/latex] studied their notes, and [latex]27[/latex] completed the review assignment. Of the entire class of [latex]43[/latex] students, [latex]12[/latex] completed the review and made flash cards, [latex]9[/latex] completed the review and studied their notes, and [latex]7[/latex] made flash cards and studied their notes, while only [latex]5[/latex] students completed all three of these tasks. The remaining students did not do any of these tasks. Create a Venn diagram with subsets labeled “Notes,” “Flash Cards,” and “Review” to represent how the students prepared for the test.

Step 1: First, draw a Venn diagram with three intersecting circles to represent the three intersecting sets: Notes, Flash Cards, and Review. Label the universal set with the cardinality of the class.

A three-set Venn diagram labeled Notes, Flash Card, and Review. Outside the Venn diagram, 'U equals Class equals 43' is marked.
Figure 4. Venn diagram for Step 1
Step 2: Next, in the region where all three sets intersect, enter the number of students who completed all three tasks.
A three-set Venn diagram labeled Notes, Flash Card, and Review. Outside the Venn diagram, 'U equals Class equals 43' is marked. 5 is in the center.
Figure 5. Venn diagram for Step 2

Step 3: Next, calculate the value and label the three sections where just two sets overlap.

  1. Review and flash card overlap. A total of [latex]12[/latex] students completed the review and made flash cards, but 5 of these 12 students did all three tasks, so we need to subtract [latex]12-5=7[/latex]. This is the value for the region where the flash card set intersects with the review set.
  2. Review and notes overlap. A total of [latex]9[/latex] students completed the review and studied their notes, but again, 5 of these 9 students completed all three tasks. So we subtract [latex]9-5=4[/latex]. This is the value for the region where the review set intersects with the notes set.
  3. Flash card and notes overlap. A total of [latex]7[/latex] students made flash cards and studied their notes; subtracting the 5 students that did all three tasks from this number leaves [latex]2[/latex] students who only studied their notes and made flash cards. Add these values to the Venn diagram.
Venn diagram of a class of 43, showing overlaps: Notes (yellow) with 4; Flash Cards (red) with 7; Review (blue) with 7; all three overlap with 5, others with 2.
Figure 6. Venn diagram for Step 3
Step 4: Now repeat this process to find the number of students who only completed one of these three tasks.
  1. A total of [latex]24[/latex] students completed flash cards, but we have already accounted for [latex]2+5+7=14[/latex] of these. Thus, [latex]24−14=10[/latex] students who just made flash cards.
  2. A total of [latex]14[/latex] students studied their notes, but we have already accounted for [latex]4+5+2=11[/latex] of these. Thus, [latex]14-11=3[/latex] students only studied their notes.
  3. A total of [latex]27[/latex] students completed the review assignment, but we have already accounted for [latex]4+5+7=16[/latex] of these, which means [latex]27-16=11[/latex] students only completed the review assignment.
  4. Add these values to the Venn diagram.
Venn diagram with three circles: Notes, Flash Cards, Review. Overlaps show numbers: Notes/Flash Cards 2, all three 5, Flash Cards 10, Review 11.
Figure 7. Venn diagram for Step 4

Step 5: Finally, compute how many students did not do any of these three tasks. To do this, we add together each value that we have already calculated for the separate and intersecting sections of our three sets: [latex]3+2+4+5+10+7+11=42[/latex]. Because there are [latex]43[/latex] students in the class and [latex]43-42=1[/latex], this means only one student did not complete any of these tasks to prepare for the test. Record this value somewhere in the rectangle, but outside of all the circles, to complete the Venn diagram.

Venn diagram with three circles: Notes, Flash Cards, Review. Overlaps show numbers: Notes/Flash Cards 2, all three 5, Flash Cards 10, Review 11. 1 on outside.
Figure 8. Venn diagram for Step 5

Exercise 2

A group of [latex]50[/latex] people attending a conference who preordered their lunch were able to select their choice of soup, salad, or sandwich. A total of [latex]17[/latex] people selected soup, [latex]29[/latex] people selected salad, and [latex]35[/latex] people selected a sandwich. Of these orders, [latex]11[/latex] attendees selected soup and salad, [latex]10[/latex] attendees selected soup and a sandwich, and [latex]18[/latex] selected a salad and a sandwich, while [latex]8[/latex] people selected a soup, a salad, and a sandwich. Create a Venn diagram with subsets labeled “Soup,” “Salad,” and “Sandwich,” and label the cardinality of each section of the Venn diagram as indicated by the data.
Solution
A Venn diagram shows conference attendees' meal preferences. Three intersecting circles represent soup, salad, and sandwich choices.
Figure 9. Attendees at a conference with sets: Soup, Sandwich, Salad—Complete Venn diagram solution

Applying Set Operations to Three Sets

Set operations are applied between two sets at a time. Parentheses indicate which operation should be performed first. As with numbers, the innermost parentheses are applied first. Next, find the complement of any sets, then perform any union or intersections that remain.

Example 3

Perform the set operations as indicated on the following sets.

[latex]U=\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\},[/latex]

[latex]A= \{0, 1, 2, 3, 4, 5, 6\},[/latex]

[latex]B=\{0, 2, 4, 6, 8, 10, 12\},[/latex]

[latex]C=\{0, 3, 6, 9, 12\}[/latex]

  1. Find [latex](A \cap B)\cap C[/latex].

Parentheses first, [latex]A[/latex] intersection [latex]B[/latex] equals [latex]A \cap B=\{0, 2, 4, 6\}[/latex], the elements common to both [latex]A[/latex] and [latex]B[/latex]. [latex](A \cap B)\cap C=\{0, 2, 4, 6\} \cap \{0, 3, 6, 9, 12\}=\{0, 6\}[/latex], because the only elements that are in both sets are [latex]0[/latex] and [latex]6[/latex].

  1. Find [latex]A \cap (B \cup C)[/latex].

Parentheses first, [latex]B[/latex] union [latex]C[/latex] equals [latex]B \cup C=\{0, 2, 3, 4, 6, 8, 9, 10, 12\}[/latex], the collection of all elements in set [latex]B[/latex] or set [latex]C[/latex] or both. [latex]A \cap (B \cup C)=\{0, 1, 2, 3, 4, 5, 6\}\cap\{0, 2, 3, 4, 6, 8, 9, 10, 12\}=\{0, 2, 3, 4, 6\}[/latex]because the intersection of these two sets is the set of elements that are common to both sets.

  1. Find [latex](A \cap B) \cup C'[/latex].

Exercise 3

Notice that the answers to Exercise 3 are the same as those in Example 3. This is not a coincidence. The following equivalences hold true for sets:

  • [latex]A\cap(B\cup C)=(A\cap B)\cap C[/latex] and [latex]A\cup(B\cup C)=(A\cup B)\cup C[/latex]. These are the associative property for set intersection and set union.
  • [latex]A \cap B=B \cap A[/latex] and [latex]A\cup B=B\cup A[/latex]. These are the commutative property for set intersection and set union.
  • [latex]A\cap (B\cup C)=(A\cap B)\cup(A\cap C)[/latex] and [latex]A\cup (B\cap C)=(A\cup B)\cap(A\cup C)[/latex]. These are the distributive property for sets over union and intersection, respectively.

Proving Equality of Sets Using Venn Diagrams

To prove set equality using Venn diagrams, the strategy is to draw a Venn diagram to represent each side of the equality, then look at the resulting diagrams to see if the regions under consideration are identical.

Augustus De Morgan was an English mathematician known for his contributions to set theory and logic. De Morgan’s law for set complement over union states that [latex](A \cup B)'=A' \cap B'[/latex]. In the next example, we will use Venn diagrams to prove De Morgan’s law for set complement over union is true. But before we begin, let us confirm De Morgan’s law works for a specific example. While showing something is true for one specific example is not a proof, it will provide us with some reason to believe that it may be true for all cases.

Let [latex]U=\{1, 2, 3, 4, 5, 6, 7\}, A=\{2, 3, 4\}[/latex], and [latex]B=\{3, 4, 5, 6\}[/latex]. We will use these sets in the equation [latex](A \cup B)'=A' \cap B'[/latex].

To begin, find the value of the set defined by each side of the equation.

Step 1: [latex]A\cup B[/latex] is the collection of all unique elements in set [latex]A[/latex] or set [latex]B[/latex] or both. [latex]A\cup B=\{2, 3, 4, 5, 6\}[/latex]. The complement of [latex]A[/latex] union [latex]B[/latex], [latex](A\cup B)'[/latex], is the set of all elements in the universal set that are not in [latex]A \cup B[/latex]. So the left side of the equation [latex](A \cup B)'[/latex] is equal to the set [latex]\{1, 7\}[/latex].

Step 2: The right side of the equation is [latex]A' \cap B'[/latex]. [latex]A'[/latex] is the set of all members of the universal set [latex]U[/latex] that are not in set [latex]A[/latex].

[latex]A'=\{1, 5, 6, 7\}[/latex]. Similarly, [latex]B'=\{1, 2, 7\}[/latex].

Step 3: Finally, [latex]A'\cap B'[/latex] is the set of all elements that are in both [latex]A'[/latex] and [latex]B'[/latex]. The numbers [latex]1[/latex] and [latex]7[/latex] are common to both sets; therefore, [latex]A' \cap B'=\{1, 7\}[/latex]. Because [latex]\{1, 7\}=\{1, 7\}[/latex], we have demonstrated that De Morgan’s law for set complement over union works for this particular example.

The Venn diagram below depicts this relationship.

Set A shows 2. Set B shows 5, 6. The intersection of the sets shows 3, 4. Outside 1 and 7 are shown.
Figure 10. Two-set Venn diagram with sets A and B

Example 4

De Morgan’s law for the complement of the union of two sets [latex]A[/latex] and [latex]B[/latex] states that

[latex](A \cup B)' = A' \cap B'[/latex]. Use a Venn diagram to prove that De Morgan’s law is true.

Step 1: First, draw a Venn diagram representing the left side of the equality. The regions of interest are shaded to highlight the sets of interest. [latex]A \cup B[/latex] is shaded on the left, and [latex](a\cup B)'[/latex] is shaded on the right.

Venn diagram showing sets A and B. Left: union A ∪ B shaded. Right: complement (A ∪ B)' shaded outside the circles. Arrow indicates transformation.
Figure 11. Left Side of De Morgan's law
Step 2: Next, draw a Venn diagram to represent the right side of the equation. [latex]A'[/latex] is shaded and [latex]B'[/latex] is shaded. Because [latex]A'[/latex] and [latex]B'[/latex] mix to form the intersection, [latex]A' \cap B'[/latex] is also shaded.
Venn diagram illustration showing set operations. Blue set A', yellow set B'. Intersection in green highlights overlap, conveying set relationships.
Figure 12. Right side of the equation for De Morgan's law
Step 3: Verify the conclusion. Because the shaded region in the Venn diagram for [latex](A\cup B)'[/latex] matches the shaded region in the Venn diagram for [latex]A' \cap B'[/latex], the two sides of the equation are equal, and the statement is true.
This completes the proof that De Morgan’s law is valid.

Exercise 4

De Morgan’s law for the complement of the intersection of two sets [latex]A[/latex] and [latex]B[/latex] states that [latex](A \cup B)' = A' \cap B'[/latex]. Use a Venn diagram to prove that De Morgan’s law is true.

Solution

The left side of the equation is:

Two Venn diagrams illustrating set operations. The left shows the intersection of sets A and B in blue. The right shows its complement highlighted in blue.
Figure 13. Venn diagram of intersection of two sets and its complement

 

The right side of the equation is given by:
Three overlapping rectangles, labeled \(A'\), \(B'\), and \(A' \cup B'\), each contain a circle, labeled \(A\) and \(B\), representing a set union concept.
Figure 14. Venn diagram of the union of the complement of two sets

Extra Practice

A gamers club at Baily Middle School consisting of 25 members was surveyed to find out who played board games, card games, or video games. Use the results depicted in the Venn diagram below to answer the following exercises.

Venn diagram of 25 Gamers Club members: 1 board, 0 card, 5 video, 4 board-video, 3 card-video, 2 board-card, 10 all games; shows overlap and exclusivity.
Figure 15. Venn diagram of Gamers Club Members
  1. How many gamers club members play all three types of games: board games, card games, and video games?
  2. How many gamers are in the set [latex]\text{Board} \cap \text{Video}[/latex]?
  3. If Javier is in the region with a total of three members, what type of games does he play?
  4. How many gamers play video games?
  5. How many gamers are in the set [latex]\text{Board} \cup \text{Card}[/latex]?
  6. How many members of the gamers club do not play video games?
  7. How many members of this club only play board games?
  8. How many members of this club only play video games?
  9. How many members of the gamers club play video and card games?
  10. How many members of the gamers club are in the set [latex]\text{C⁢a⁢r⁢d}'[/latex]?

A blood drive at City Honors High School recently collected blood from 140 students, staff, and faculty. Use the results depicted in the Venn diagram below to answer the following exercises.

Venn diagram with three circles labeled A, B, and Rh+. Overlapping areas show numbers: A only has 8, B only 3, Rh+ only 52, and all three share 4.
Figure 16. Venn diagram of donors
  1. Blood type [latex]AB^{+}[/latex] is the universal acceptor. Of the 140 people who donated at City Honors, how many had blood type [latex]AB^{+}[/latex]?
  2. Blood type [latex]O^{−}[/latex] is the universal donor. Anyone needing a blood transfusion can receive this blood type. How many people who donated blood during this drive had [latex]O^{−}[/latex] blood?
  3. How many people donated with a type A blood factor?
  4. How many people donated with a type A and type B blood factor (that is, they had type AB blood).
  5. How many donors were [latex]O^{+}[/latex]?
  6. How many donors were not [latex]R⁢h^{+}[/latex]?
  7. Opal has blood type [latex]A^{+}[/latex]. If she needs to have surgery that requires a blood transfusion, she can accept blood from anyone who does not have a type B blood factor. How many people donated blood during this drive at City Honors that Opal can accept?
  8. Find [latex]𝑛⁢(𝐴 \cap R⁢h^{+})[/latex].
  9. Find [latex]𝑛⁢(𝐴 \cup R⁢h^{+})[/latex].
  10. Find [latex]𝑛⁢(𝐴 \cap 𝐵 \cap R⁢h^{−})[/latex].
For the following exercises, create a three circle Venn diagram to represent the relationship between the described sets.
  1. The number of elements in the universal set, [latex]𝑈[/latex], is [latex]𝑛⁡(𝑈) =48[/latex]. Sets [latex]𝐴[/latex], [latex]𝐵[/latex], and [latex]𝐶[/latex] are subsets of [latex]𝑈: 𝑛⁡(𝐴) =23, 𝑛⁡(𝐵) =25,[/latex]and [latex]𝑛⁡(𝐶) =17[/latex]. Also, [latex]𝑛⁢(𝐴 \cap 𝐵) =15[/latex], [latex]𝑛⁢(𝐵 \cap 𝐶) =12[/latex], [latex]𝑛⁢(𝐶 ∩𝐴) =11,[/latex] and [latex]𝑛⁢(𝐴 \cap 𝐵 \cap 𝐶) =8[/latex].
  2. The number of elements in the universal set, [latex]𝑈[/latex], is [latex]𝑛⁡(𝑈) =88[/latex]. Sets [latex]𝐴[/latex], [latex]𝐵[/latex], and [latex]𝐶[/latex] are subsets of [latex]𝑈: 𝑛⁡(𝐴) =31[/latex], [latex]𝑛⁡(𝐵) =46,[/latex] [latex]𝑛⁡(𝐶) =33[/latex]. Also, [latex]𝑛⁢(𝐴 \cap 𝐵) =24,[/latex] [latex]𝑛⁢(𝐵 \cap 𝐶) =24,[/latex] [latex]𝑛⁢(𝐶 \cap 𝐴) =26,[/latex] and [latex]𝑛⁢(𝐴 \cap 𝐵 \cap 𝐶) =22[/latex].
  3. The number of elements in the universal set, [latex]𝑈[/latex], is [latex]𝑛⁡(𝑈) =52[/latex]. Sets [latex]𝐴[/latex], [latex]𝐵[/latex], and [latex]𝐶[/latex] are subsets of [latex]𝑈: 𝑛⁡(𝐴) =23, 𝑛⁡(𝐵) =27,[/latex] and [latex]𝑛⁡(𝐶) =29[/latex]. Also, [latex]𝑛⁢(𝐴 \cap 𝐵) =22,[/latex] [latex]𝑛⁢(𝐵 \cap 𝐶) =21,[/latex] [latex]𝑛⁢(𝐶 \cap 𝐴) =19,[/latex] and [latex]𝑛⁢(𝐴 \cap 𝐵 \cap 𝐶) =18[/latex].
  4. The number of elements in the universal set, [latex]𝑈[/latex], is [latex]𝑛⁡(𝑈) =144[/latex]. Sets [latex]𝐴[/latex], [latex]𝐵[/latex], and [latex]𝐶[/latex] are subsets of [latex]𝑈: 𝑛⁡(𝐴) =36[/latex] [latex]𝑛⁡(𝐵) =64,[/latex] and [latex]𝑛⁡(𝐶) =81[/latex]. Also, [latex]𝑛⁢(𝐴 \cap 𝐵) =26,[/latex] [latex]𝑛⁢(𝐵 \cap 𝐶) =61,[/latex] [latex]𝑛⁢(𝐶 \cap 𝐴) =29,[/latex] and [latex]𝑛⁢(𝐴 \cap 𝐵 \cap 𝐶) =25[/latex].
  5. The universal set, [latex]𝑈[/latex], has a cardinality of 36. [latex]𝑛⁡(𝐴) =12,[/latex] [latex]𝑛⁡(𝐵) =12[/latex], [latex]𝑛⁡(𝐶) =15[/latex], [latex]𝑛⁡(𝐴 \text{ and } 𝐵) =3[/latex], [latex]𝑛⁡(𝐵 \text{ and }𝐶) =4[/latex], [latex]𝑛⁡(𝐶 \text{ and } 𝐴) =5,[/latex] and [latex]𝑛⁡(𝐴 \text{ and } 𝐵 \text{ and } 𝐶) =1[/latex].
  6. The universal set, [latex]𝑈[/latex], has a cardinality of 63. [latex]𝑛⁡(𝐴) =29,[/latex] [latex]𝑛⁡(𝐵) =31[/latex], [latex]𝑛⁡(𝐶) =41[/latex], [latex]𝑛⁡(𝐴 \text{ and } 𝐵) =12[/latex], [latex]𝑛⁡(𝐵 \text{ and } 𝐶) =16[/latex], [latex]𝑛⁡(𝐶 \text{ and } 𝐴) =18,[/latex] and [latex]𝑛⁡(𝐴 \text{ and } 𝐵 \text{ and } 𝐶) =5[/latex].
  7. The universal set, [latex]𝑈[/latex], has a cardinality of 72. [latex]𝑛⁡(𝐴) =32,[/latex] [latex]𝑛⁡(𝐵) =32[/latex], [latex]𝑛⁡(𝐶) =44[/latex], [latex]𝑛⁡(𝐴 \text{ and } 𝐵) =18[/latex], [latex]𝑛⁡(𝐵 \text{ and } 𝐶) =22[/latex], [latex]𝑛⁡(𝐶 \text{ and } 𝐴) =26,[/latex] and [latex]𝑛⁡(𝐴 \text{ and } 𝐵 \text{ and } 𝐶) =14[/latex].
  8. The universal set, [latex]𝑈[/latex], has a cardinality of 81. [latex]𝑛⁡(𝐴) =54,[/latex] [latex]𝑛⁡(𝐵) =41[/latex], [latex]𝑛⁡(𝐶) =52[/latex], [latex]𝑛⁡(𝐴 \text{ and } 𝐵) =32[/latex], [latex]𝑛⁡(𝐵 \text{ and } 𝐶) =28[/latex], [latex]𝑛⁡(𝐶 \text{ and } 𝐴) =30,[/latex] and [latex]𝑛⁡(𝐴 \text{ and } 𝐵 \text{ and } 𝐶) =21[/latex].
  9. The anime drawing club at Pratt Institute conducted a survey of its 42 members and found that 23 of them sketched with pastels, 28 used charcoal, and 17 used colored pencils. Of these, 10 club members used all three mediums, 18 used charcoal and pastels, 11 used colored pencils and charcoal, and 12 used colored pencils and pastels. The remaining club members did not use any of these three mediums.
  10. A new SUV is selling with three optional packages: a sport package, a tow package, and an entertainment package. A dealership gathered the following data for all 31 of these vehicles sold during the month of July. A total of 18 SUVs included the entertainment package, 11 included the tow package, and 16 included the sport package. Of these, five SUVs included all three packages, seven were sold with both the tow package and sport package, 11 were sold with the entertainment and sport package, and eight were sold with the tow package and entertainment package. The remaining SUVs sold did not include any of these optional packages.
For the following exercises, perform the set operations as indicated on the following sets: [latex]𝑈 =\{\text{red, orange, yellow, green, blue, indigo, violet}\},[/latex] [latex]𝐴 =\{\text{red, yellow, blue}\},[/latex] [latex]𝐵 =\{\text{orange, green, violet}\},[/latex] and [latex]𝐶 =\{\text{red, green, indigo}\}.[/latex]
  1. Find [latex](𝐴 \cup 𝐵) \cap 𝐶[/latex].
  2. Find [latex](𝐴 \cap 𝐶) \cup 𝐵[/latex].
  3. Find [latex]𝑈 \cap (𝐵 \cup 𝐶)[/latex].
  4. Find [latex](𝐵 \cap 𝐴) \cap 𝑈[/latex].
  5. Find [latex]𝐴 \cap (𝐵 \cap 𝐶)'[/latex].
  6. Find [latex]𝐴' \cap (𝐵 \cup 𝐶)[/latex].

For the following exercises, perform the set operations as indicated on the following sets: [latex]𝑈 =\{20,21,22,…,29\},[/latex] [latex]𝐴 =\{21,24,27\},[/latex] [latex]𝐵 =\{20,22,24,28\},[/latex] and [latex]𝐶 =\{21,23,25,27\}.[/latex]

  1. Find [latex]𝐴 \text{ and } 𝐵 \text{ and } 𝐶'[/latex].
  2. Find [latex]𝐴' \text{ or } 𝐵 \text{ or } 𝐶[/latex].
  3. Find [latex](𝐴 \text{ or } 𝐵) \text{ and } 𝐶'[/latex].
  4. Find [latex](𝐴 \text{ or } 𝐵) or 𝐶'[/latex].
  5. Find [latex](𝐴 \text{ and } 𝐶) \text{ and } 𝐵'[/latex].
  6. Find [latex](𝐴 \text{ or } 𝐵)' \text{ and } 𝐶[/latex].
For the following exercises, use Venn diagrams to prove the following properties of sets:
  1. Commutative property for the union of two sets: [latex]𝐴 \cup 𝐵 =𝐵 \cup 𝐴[/latex].
  2. Commutative property for the intersection of two sets: [latex]𝐴 \cap 𝐵 =𝐵 \cap 𝐴[/latex].
  3. Associative property for the intersection of three sets: [latex](𝐴 \cap 𝐵) \cap 𝐶 =𝐴 \cap (𝐵 \cap 𝐶)[/latex].
  4. Associative property for the union of three sets: [latex]𝐴 \cup (𝐵 \cup 𝐶) =(𝐴 \cup 𝐵) \cup 𝐶[/latex].
  5. Distributive property for set intersection over set union: [latex]𝐴 \cap (𝐵 \cup 𝐶) =(𝐴 \cap 𝐵) \cup (𝐴 \cap 𝐶)[/latex].
  6. Distributive property for set union over set intersection: [latex]𝐴 \cup (𝐵 \cap 𝐶) =(𝐴 \cup 𝐵) \cap (𝐴 \cup 𝐶)[/latex].

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