
2.E: Basic Concepts of Sets (Exercises)

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:

1. $$A \cap B$$
2. $$C \cup A$$
3. $$C \cap D$$
4. $$(A \cup B) \cup (C \cup D)$$

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

1. $$7 \in \{6, 7, 8, 9\}$$
2. $$5 \notin \{6, 7, 8, 9\}$$
3. $$\{2\} \nsubseteq \{1, 2\}$$
4. $$\emptyset \nsubseteq \{\alpha, \beta, x\}$$
5. $$\emptyset = \{\emptyset\}$$

Exercise $$\PageIndex{3}$$: Subsets

List all the subsets of:

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

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:

1. Only Indian food?
2. Only Italian food?
3. 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 $$\PageIndex{6}$$:

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.

1.  $$A-(B \cap C)=(A-B) \cup(A-C).$$
2. If $$A \cap B= A \cap C$$ then $$B= C$$.
3. If $$A \cup B= A \cup C$$ then $$B= C$$.
4.  $$A-(B - C)=(A-B)-C.$$
5.   If $$A \times B \subseteq C \times D$$ then $$A\subseteq C$$ and $$B \subseteq D.$$
6.  If $$A\subseteq C$$ and $$B \subseteq D$$ then $$A \times B \subseteq C \times D.$$

Exercise $$\PageIndex{7}$$:

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:

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

Exercise $$\PageIndex{8}$$:

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.

1. $$P(A \cup B) = P(A) \cup P(B).$$
2. $$P(A \cap B) = P(A) \cap P(B).$$
3. $$P(A^c)=(P(A))^c$$
4. $$P(A - B) = P(A) - P(B).$$

Exercise $$\PageIndex{9}$$:

Consider the following sets:

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

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

Exercise $$\PageIndex{10}$$:

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

Then

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