2.E: Basic Concepts of Sets (Exercises)
- Page ID
- 4911
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.