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Mathematics LibreTexts

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.