1.4: Review Exercises

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Set $A$ is the set of letters in the word mississippi. Present set $A$ in roster form and give $n(A)$.

Write a verbal description of the set $V=\{a, e, i, o, u\}$. Then, find $n(V)$.

Present set $B$ in roster notation: $B = \{x | x \in \mathbb{N} \ \text{ and }3 \leq x <10\}$ . Then, find $n(B)$.

Consider sets $A=\{1, 2, 3, 5\}$, $B=\{1, 2,a\}$, $C=\{a, b, c, d\}$ and $D=\{1, a, 2\}$.

a. Determine if the following statements are true or false.

$b \in C$
$1 \notin A$
$4 \notin A$
$B \subseteq D$
$B \subset D$
$B \subset A$
$B = D$
$A \cong C$
$A = C$

b. Make up your own set $E$ so that set $E$ is equivalent but not equal to set $B$.

c. List all subsets of set $B$. How many of these are proper subsets of set $B$?

d. How many subsets does set $C$ have?

$Venn 5.png$

The Venn diagram represents the sets of guests at a dinner party.
1. Lucy is allergic to nuts but not vegetarian. In which region does Lucy belong in this Venn diagram?
2. Darren is a vegetarian who is not allergic to nuts. In which region would Darren belong in this Venn diagram?
3. Describe a guest who belongs in region #6.

Draw and label a 3-set Venn diagram so that it shows how these sets are related:

$U = \{1, 2, 3, 4, 5, 6, a, b, c \}$

$A = \{1, 3, a, b \}$

$B = \{1, 2, a \}$

$C = \{3, 4, 5, b \}$

For the universal set $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9 \}$ , $A = \{3, 4, 5, 7\}$, $B = \{1,3, 4, 5, 8 \}$, and $C = \{6, 7, 9 \}$, find each result below. Be sure to show the steps as needed to justify how you reached the answer.

$A \cap B$
$B \cup C$
$A \cap (B \cup C)$
$A^{\prime} \cap B^{\prime}$
$(C \cup A)^{\prime}$
$A \cap B^{\prime}$
$n(C^{\prime})$ Note this asks for cardinality.
$n(A \cap B \cap C)$ Note this asks for cardinality.

Use the Venn diagram to find the results of the set operations. $Venn 2.png$
1. $B^{\prime}$
2. $A \cup B \cup C$
3. $A \cup B $
4. $A \cap C^{\prime}$
5. $(A \cup B)^{\prime}$

Among 100 students surveyed, 60 are taking a math course, 70 are taking psychology, and 40 are taking both math and psychology.
1. Use the information to enter numbers a Venn diagram so it shows the appropriate cardinality in each of the four regions.
2. How many students are taking math or psychology?
3. How many students are taking math but not psychology?
4. How many students are taking neither math nor psychology?
5. How many students are taking exactly one course?

Use the cardinality principle to solve:
1. If $n(A)=16$, $n(B)=22$, and $n(A \cup B)=30$, find $n(A \cap B)$.
2. If $n(B)=12$, $n(A \cup B)=30$ and $n(A \cap B) = 10$, find $n(A)$.

$Venn 11.png$

This Venn diagram shows the results of a survey that asked if students owned a laptop, cell phone, and iPod. Use the diagram to answer the questions.
1. How many students own an iPod?
2. How many students do not own a lap top?
3. How many students own only a laptop?
4. How many students own an iPod and a laptop?
5. How many students own an iPod or a laptop, but no cell phone?

Fifty (50) moviegoers were asked whether they had ever seen The Matrix (M), Star Wars (SW), and Lord of the Rings (LOTR). The results are below. Use the given information to show the appropriate cardinalities in a 3-set Venn diagram. Be sure to show cardinalities for all eight regions.

18 had seen The Matrix
20 had seen Lord of the Rings
24 had seen Star Wars
14 had seen Lord of the Rings and Star Wars
12 had seen The Matrix and Lord of the Rings
10 had seen The Matrix and Star Wars
6 had seen all three

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