Note: This is the Solution of review exercise 3 from newly published book by PCTB (Punjab Curriculum and Textbook Board, Pakistan) for new 9th session 2025 Onward.
Review Exercise 3
Question # 01: Four option are given against each statement. Encircle the correct option.
\((i)\) The set builder form of the set \(\{1,\frac{1}{3} ,\frac{1}{5} ,\frac{1}{7} ,\dotsc \}\) is:
\( a) \quad \) \( \{x\mid x=\frac{1}{n} ,n\in W\} \)
\( b) \quad \) \( \{x\mid x=\frac{1}{2n+1} ,n\in W\}\)
\(c) \quad \) \( \{x\mid x=\frac{1}{n+1} ,n\in W\}\)
\( d) \quad \) \( \{x\mid x=2n+1,n\in W\}\)
\((ii)\) If \(A=\{\}\), then \(P(A)\) is:
\( a) \quad \) \( \{\}\)
\( b) \quad \) \( \{1\}\)
\(c) \quad \) \(\{\{\}\} \)
\( d) \quad \) \(\phi \)
\((iii)\) If \( U=\{1,2,3,4,5\}\), \(A=\{1,2,3\}\) and \(B=\{3,4,5\}\), then \(U-( A\cap B)\) is:
\( a) \quad \) \( \{1,2,4,5\}\)
\( b) \quad \) \(\{2,3\} \)
\(c) \quad \) \(\{ 1,3,4,5\} \)
\( d) \quad \) \( \{1,2,3\}\)
\((iv)\)\(\quad\)If \(A\) and \(B\) are overlapping sets, then \(n(A-B)\) is equal to:
\( a) \quad \) \(n(A) \)
\( b) \quad \) \( n(B)\)
\(c) \quad \) \( A\cap B\)
\( d) \quad \) \( n(A)-n(A\cap B)\)
\((v)\) \(\quad\) If \(A\subseteq B\) and \(B-A\neq \phi\), then \(n( B-A)\) is equal to
\( a) \quad \) \(0 \)
\( b) \quad \) \(n(B)\)
\(c) \quad \) \( n(A)\)
\( d) \quad \) \( n(B)-n(A)\)
\((vi)\) If \(n( A\cup B)=50\), \(n( A) =30\) and \(n( B) =35\), then \(n( A\cap B) =\) :
\( a) \quad \) \( 23\)
\( b) \quad \) \( 15\)
\(c) \quad \) \( 9\)
\( d) \quad \) \( 40\)
\((vii)\)\(\quad\)If \(A=\{1,2,3,4\} \) and \(B=\{x,y,z\}\), then cartesian product of \(A\) and \(B\) contains exactly _______ elements.
\( a) \quad \) \(13\)
\( b) \quad \) \(12\)
\(c) \quad \) \(10\)
\( d) \quad \) \(6\)
\((viii)\)\(\quad\)if \(f(x)=x^2-3x+2\), then the value of \(f(a+1)\) is equal to:
\( a) \quad \) \( a+1\)
\( b) \quad \) \( a^2+1\)
\(c) \quad \) \( a^2+2a+1\)
\( d) \quad \) \( a^2-a\)
\((ix)\)\(\quad\)Given that \(f(x)=3x+1\), if \(f(x)=28\), the the value of \(x\) is:
\( a) \quad \) \(9\)
\( b) \quad \) \(27\)
\(c) \quad \) \(3\)
\( d) \quad \) \(18\)
\((x)\) \(\quad\)Let \(A=\{1,2,3\}\) and \(B=\{a,b\}\) two non-empty sets and \(f:A\rightarrow B\) be a function defined as \(f=\{(1,a),(2,b),(3,b)\}\), then which of the following statement is true?
\( a) \quad \) \(f\) is injective
\( b) \quad \) \(f \) is surjective
\(c) \quad \) \(f\) is bijective
\( d) \quad \) \(f\) is into only
Question 2: Write each of the following sets in tabular forms:
\((i)\) \(\quad\) \(\{x|x=2n,n\in N \}\)
\((ii)\) \(\quad\) \(\{x|x=2m+1, m \in N\} \)
\((iii)\) \(\;\) \(\{x|x=11n, n\in W \wedge n<11\} \)
\((iv)\) \(\;\) \(\{x|x \in E \wedge 4<x<6\} \)
\((v)\) \(\quad\) \(\{x|x \in O \wedge 5\le x<7\} \)
\((vi)\) \(\quad\) \(\{x|x \in Q \wedge x^2=2\} \)
\((vii)\) \(\;\) \(\{x|x \in Q \wedge x=-x\} \)
\((viii)\) \(\;\) \(\{x|x \in R \wedge x \notin {Q}’\} \)
Question 3: Let \(U=\{1,2,3,4,5,6,7,8,9,10\}\), \(A=\{2,4,6,8,10\}\), \(B=\{1,2,3,4,5\}\) and \(C=\{1,3,5,7,9\}\)
List the members of each of the following sets:
\((i)\) \(\quad\) \({A}’ \)
\((ii)\) \(\quad\) \({B}’ \)
\((iii)\) \(\quad\) \(A\cup B \)
\((iv)\) \(\quad\) \(A- B \)
\((v)\) \(\quad\) \(A\cap C \)
\((vi)\) \(\quad\) \({A}’\cup {C}’ \)
\((vii)\) \(\quad\) \({A}’\cup C \)
\((viii)\) \(\quad\) \({U}’ \)
Question 4: Using Venn diagrams, if necessary, find the single sets equal to the following:
\((i)\) \(\quad\) \({A}’ \)
\((ii)\) \(\quad\) \(A\cap U \)
\((iii)\) \(\quad\) \(A\cup U \)
\((iv)\) \(\quad\) \(A\cup \phi \)
\((v)\) \(\quad\) \(\phi \cap \phi \)
Question 5: Using Venn diagrams to verify the following:
\((i)\) \(\quad\) \(A-B=A\cup {B}’ \)
\((ii)\) \(\quad\) \({(A-B)}’\cap B=B\)
Question 6: Verify the properties for the sets \(A\), \(B\) and \(C\) given below:
\((i)\) Associative of Union
\((ii)\) Associativity of intersection
\((iii)\) Distributive of Union over intersection
\((iv)\) Distributive of intersection over union
\((a)\) \(\;\) \(A=\{1,2,3,4\}\), \(B=\{3,4,5,6,7,8\}\), \(C=\{5,6,7,9,10\}\)
\((b)\) \(\;\) \(A=\phi\), \(B=\{0\}\), \(C=\{0,1,2\}\)
\((c)\) \(\;\) \(A=N\), \(B=Z\), \(C=Q\}\)
Question 7: Verify De Morgan’s Laws for the following sets:
\(U=\{1,2,3, \ldots, 20\}\), \(A=\{2,4,6, \ldots, 20\) and \(B=\{1,3,5, \ldots, 19\}\).
Question 8: Consider the set \(P=\{x|x=5m,m\in N\}\) and \(Q=\{x|x=2m,m\in N\}\). Find \(P\cap Q\)
Question 9: From suitable properties of union and intersection, deduce the following results:
\((i)\) \(\quad\) \(A\cap (A\cup B)=A\cup (A\cap B)\)
\((ii)\) \(\quad\) \(A\cup (A\cap B)=A\cap (A\cup B)\)
Question 10: If \(g(x)=7x-2\) and \(s(x)=8x^2-3\) find:
\((i)\) \(\quad\) \(g(0) \)
\((ii)\) \(\quad\) \(g(-1) \)
\((iii)\) \(\quad\) \(g(-\frac{5}{3}) \)
\((iv)\) \(\quad\) \(s(1) \)
\((v)\) \(\quad\) \(s(-9) \)
\((vi)\) \(\quad\) \(s(\frac{7}{2}) \)
Question 11: Given that \(f(x)=ax+b\), where \(a\) and \(b\) are constant numbers. If \(f(-2)=3\) and \(f(4)=10\), then find the values of \(a\) and \(b\).
Question 12: Consider the function defined by \(k(x) =7x-5\). If \(k(x)=100\), find the value of \(x\).
Question 13: Consider the function \(g(x)=mx^2+n\), where \(m\) and \(n\) are constant numbers. If \(g(4)=20\) and \(g(0)=5\) , find the values of \(m\) and \(n\).
Question 14: A shopping mall has 100 products from various categories labeled 1 to 100, representing the universal set \(U\). The product are categorized as follows:
\(\quad\bullet\quad\) Set \(A\): Electronics, consisting of 30 prodcts labeled from 1 to 30.
\(\quad\bullet\quad\) Set \(B\): Clothing comprises 25 products labeled from 31 to 55.
\(\quad\bullet\quad\) Set \(C\): Beauty Products, comprising 25 products labeled from 76 to 100.
Write each set in tabular form, and find the union of all three sets.
Question 15: Out of the 180 students who appeard in the annual examanination, 120 passed the math test, 90 passed the science test, and 60 passed both the math and science tests.
\((a)\) \(\;\) How many passed either the math or science test?
\((b)\) \(\;\) How many did not pass either of the two tests?
\((c)\) \(\;\) How many passed the science test but not the math test?
\((d)\) \(\;\) How many failed the science test?
Question 16: In a software house of a city with 300 software developers, a survey was conducted to determie which programming languages are liked more. The survey revealed the following statistics:
\(\bullet\quad\) 150 developers like Python.
\(\bullet\quad\) 130 developers like Java.
\(\bullet\quad\) 120 developers like PHP.
\(\bullet\quad\) 70 developers like both Python and Java.
\(\bullet\quad\) 60 developers like both Python and PHP.
\(\bullet\quad\) 50 developers like both Java and PHP.
\(\bullet\quad\) 40 developers like all three languages: Python, Java and PHP.
\(\quad(a)\) \(\;\) How many developers use at least one of these languages?
\(\quad(b)\) \(\;\) How many developers use only one of these languages?
\(\quad(c)\) \(\;\) How many deveopers do not use any of these languages?
\(\quad(d)\) \(\;\) How many developers use only PHP?