2.E: Basic Concepts of Sets (Exercises)

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May 20, 2022
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- 2.5: Properties of Sets
- 3: Number Patterns

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Exercise $\PageIndex{1}$ : Set Operations

Let $A = \{1, 5, 31, 56, 101\}$ , $B = \{22, 56, 5, 103, 87\}$ , $C = 41, 13, 7, 101, 48\}$ , and $D = \{1, 3, 5, 7...\}$

Give the sets resulting from:

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

Answer

1. $A \cap B =\{ 5, 56 \}$

2. $C \cup A=\{1, 5, 7, 13, 31, 41, 48, 56, 101\}$

3. $C \cap D = \{ 7, 13, 41, 101\}$

4. $(A \cup B) \cup (C \cup D)$

Exercise $\PageIndex{2}$ : True or False

$7 \in \{6, 7, 8, 9\}$
$5 \notin \{6, 7, 8, 9\}$
$\{2\} \nsubseteq \{1, 2\}$
$\emptyset \nsubseteq \{\alpha, \beta, x\}$
$\emptyset = \{\emptyset\}$

Answer: $T, T, F, F, F$

Exercise $\PageIndex{3}$ : Subsets

List all the subsets of:

$\{1, 2, 3\}$
$\{\phi, \lambda, \Delta, \mu\}$
$\{\emptyset\}$

Answer

1. $\{\{1, 2, 3\}, \{1, 2\}, \{1, 3\}, \{ 2, 3\}, \{1\}, \{2\}, \{ 3\}, \emptyset \}$

3. $\{\{\emptyset\},\emptyset \}$

Exercise $\PageIndex{4}$ : Venn Diagram

A survey of 100 university students found the following data on their food preferences:

54 preferred Italian cuisine
29 preferred Asian-style cooking
16 preferred both Italian and Asian-style foods
19 preferred both Asian-style and Indian dishes
10 preferred both Italian and Indian cuisines
5 liked them all
11 did not like any of the options

How many students preferred:

Only Indian food?
Only Italian food?
Only one food?

Exercise $\PageIndex{5}$ : Symbols

Assume that the universal set is the set of all integers.
Let
$A=\{-7,-5,-3,-1,1,3,5,7\}$
$B =\{ x \in {\bf Z}| x^2 <9 \}$
$C= \{2,3,4,5,6\}$
$D=\{x \in {\bf Z}| x \leq 9 \}$

In each of the following fill in the blank with most appropriate symbol from $\in, \notin, \subset, =,\neq,\subseteq$ , so that resulting statement is true.

A-----D
3-----B
9-----D

{2}----- $C^c$
$\emptyset$ -----D
A-----C
B-----C
C-----D
0----- $A \cap D$
0----- $A \cup D$

Exercise $\PageIndex{6}$ : Prove or disprove

Given subsets $A,B,C$ of a universal set $U$ , prove the statements that are true and give counter examples to disprove those that are false.

$A-(B \cap C)=(A-B) \cup(A-C).$
If $A \cap B= A \cap C$ then $B= C$ .
If $A \cup B= A \cup C$ then $B= C$ .
$A-(B - C)=(A-B)-C.$
If $A \times B \subseteq C \times D$ then $A\subseteq C$ and $B \subseteq D.$
If $A\subseteq C$ and $B \subseteq D$ then $A \times B \subseteq C \times D.$

Exercise $\PageIndex{7}$ : Set operations

Let $A = \{ r,e,a,s,o,n,i,g\}, B = \{m,a,t,h,e,t,i,c,l\}$ and $C$ = the set of vowels. Calculate:

$A \cup B \cup C.$
$A \cap B.$
${C}^c$ .

Exercise $\PageIndex{8}$ : Prove or disprove

Given subsets $A,B,C$ of a universal set $U$ , prove the statements that are true and give counter examples to disprove those that are false.

$P(A \cup B) = P(A) \cup P(B).$
$P(A \cap B) = P(A) \cap P(B).$
$P(A^c)=(P(A))^c$
$P(A - B) = P(A) - P(B).$

Exercise $\PageIndex{9}$ : Equal Sets

Consider the following sets:

$A=\{x \in \mathbb{Z}| x= 2m, m \in \mathbb{Z}\}$ and $B=\{x \in \mathbb{Z}| x= 2(n-1), n \in \mathbb{Z}\}$ .

Are $A$ and $B$ equal? Justify your answer.

Exercise $\PageIndex{10}$ : Product of Sets

Let $A=\{1,3,5\}$ , and
$B =\{ a,b \}$ .

Then

Find $A \times B$ and $B \times A$ .
Are $A \times B$ and $B \times A$ equal? Justify your answer.

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Exercise 2.E.1\PageIndex{1}: Set Operations

Exercise 2.E.2\PageIndex{2}: True or False

Exercise 2.E.3\PageIndex{3}: Subsets

Exercise 2.E.4\PageIndex{4}: Venn Diagram

Exercise 2.E.5\PageIndex{5}: Symbols

Exercise 2.E.6\PageIndex{6}: Prove or disprove

Exercise 2.E.7\PageIndex{7}: Set operations

Exercise 2.E.8\PageIndex{8}: Prove or disprove

Exercise 2.E.9\PageIndex{9}: Equal Sets

Exercise 2.E.10\PageIndex{10}: Product of Sets

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