Skip to main content

Exercises 2.6 Chapter Test

Exercise Group.
Draw a bag diagram and find the cardinal number of each set.
3.

\(\{ \emptyset, 5, \{2, 7\} \}\)

Answer.
3
Exercise Group.
Determine whether each statement is true or false. Write a sentence to explain your answer.
4.

\(\{3\} \in \{1, 2, 3, 4\}\)

Answer.
False. There is an element of 3 in the set, but {3} is not an element of that set.
5.

\(\emptyset \subseteq \{1, 2, 3, 4\}\)

Answer.
True. The empty set is always a possible subset.
6.

\(\{3\} \subset \{1, 2, 3, 4\}\)

Answer.
True. {3} is a proper subset because it can be created from the elements of {1, 2, 3, 4}, but it is not equal to {1, 2, 3, 4}.
7.

\(\emptyset \in \{1, 2, 3, 4\}\)

Answer.
False. The empty set is not an element of the set {1, 2, 3, 4}.
Exercise Group.
Let U = {2, 4, 6, 8, 10, 12, 14, 16, 18}, A = {2, 6, 10}, B = {6, 10, 16, 18} and C = {14, 16}. Find each of the following.
10.

\(A \cup (B \cap C)\)

Answer.
{2, 6, 10, 16}
11.

\(B \cup C^c\)

Answer.
{2, 4, 6, 8, 10, 12, 16, 18}
Exercise Group.
Draw a Venn Diagram to represent the following:
14.

\((A \cup B) \cap C^c\)

Answer.
15.

The following table gives information about students in the Math Club:

Table 2.6.1.
Student Year Major GPA
Alice Freshman Computer Science 3.0
Bob Junior Math 2.5
Carol Sophomore Math 4.0
Dave Junior Engineering 3.5
Elisabeth Freshman Engineering 3.0
Frank Senior Computer Science 4.0
George Freshman Math 2.0
Holly Sophomore Engineering 3.5

Let, U = all students in the math club, M = students majoring in math, G = students with a GPA \(\ge\) 3.0, and F = students who are Freshman.

Describe each set in words. Then find the set.

  1. \(\displaystyle M \cap G\)

  2. \(\displaystyle M \cup F\)

  3. \(\displaystyle G^c \cap F\)

Answer.

  1. Students majoring in math and students with a GPA greater than or equal to 3.0. {Carol}

  2. Students majoring in math or Freshman. {Alice, Bob, Carol, Elisabeth, George}

  3. Students without a GPA less than 3.0 and students who are Freshman. {George}

16.

An athletic director of a school has surveyed 200 students. The results are:

  • 140 students like volleyball

  • 120 students like basketball

  • 85 students like both volleyball and basketball

  1. How many students like volleyball, but not basketball?

  2. How many like basketball but not volleyball?

  3. How many do not like either of these sports?

Answer.

  1. 55

  2. 35

  3. 25

Solution.
17.

A music teacher has surveyed 495 students. The results are below:

  • 320 like rap music

  • 395 like rock music

  • 295 like heavy metal

  • 280 like both rap and rock

  • 190 like both rap and heavy metal

  • 245 like both rock and heavy metal

  • 160 like all three

  1. How many student like exactly two of the three types of music?

  2. How many like only rock music?

  3. How many like only one of the three types of music?

Answer.

  1. 235

  2. 30

  3. 60

Solution.
18.

The Dean of Academic Services surveyed a group of students about which support services they were using. The results were:

  • 5 use office hours, tutoring and online study groups

  • 16 use office hours and tutoring

  • 28 use tutoring

  • 14 use tutoring and online study groups

  • 8 use office hours and online study groups but not tutoring

  • 23 use office hours but not tutoring

  • 18 use only online study groups

  • 37 use none of these services

  1. How many students were surveyed?

  2. How many were using only office hours?

  3. How many were using online study groups?

Answer.

  1. 106

  2. 15

  3. 40

Solution.