Download PDF

Class 9 Mathematics Notes – Chapter 3: Sets and Functions

Punjab Board · 2026

Exercise 3.1

Q1. Write the following sets in set-builder notation.

(i) \(\{1,4,9,16,25,36,\ldots,484\}\)

Answer:

\[ \{x\mid x=n^2,\ n\in\mathbb N,\ n\leq22\} \]

(ii) \(\{2,4,8,16,32,64,\ldots,150\}\)

Answer:

\[ \{x\mid x=2^n,\ n\in\mathbb N,\ 2^n\leq150\} \]

(iii) \(\{0,\pm1,\pm2,\ldots,\pm1000\}\)

Answer:

\[ \{x\mid x\in\mathbb Z,\ -1000\leq x\leq1000\} \]

(iv) \(\{6,12,18,\ldots,120\}\)

Answer:

\[ \{x\mid x=6n,\ n\in\mathbb N,\ n\leq20\} \]

(v) \(\{100,102,104,\ldots,400\}\)

Answer:

\[ \{x\mid x=2n,\ n\in\mathbb N,\ 50\leq n\leq200\} \]

(vi) \(\{1,3,9,27,81,\ldots\}\)

Answer:

\[ \{x\mid x=3^n,\ n\in\mathbb W\} \]

(vii) \(\{1,2,4,5,10,20,25,50,100\}\)

Answer:

\[ \{x\mid x\in\mathbb N,\ x\text{ is a divisor of }100\} \]

(viii) \(\{5,10,15,\ldots,100\}\)

Answer:

\[ \{x\mid x=5n,\ n\in\mathbb N,\ n\leq20\} \]

(ix) The set of all integers between \(-100\) and \(1000\).

Answer:

\[ \{x\mid x\in\mathbb Z,\ -100<x<1000\} \]

Q2. Write each of the following sets in tabular form.

(i) \(\{x\mid x\text{ is a multiple of }3,\ x\leq35\}\)

Answer:

\[ \{3,6,9,12,15,18,21,24,27,30,33\} \]

(ii) \(\{x\mid x\in\mathbb R,\ 2x+1=0\}\)

Solution:

\[ 2x+1=0 \]

\[ 2x=-1 \]

\[ x=-\frac12 \]

Answer:

\[ \left\{-\frac12\right\} \]

(iii) \(\{x\mid x\in\mathbb P,\ x<12\}\)

Answer:

\[ \{2,3,5,7,11\} \]

(iv) \(\{x\mid x\text{ is a divisor of }128\}\)

Answer:

\[ \{1,2,4,8,16,32,64,128\} \]

(v) \(\{x\mid x=2^n,\ n\in\mathbb N,\ n<8\}\)

Answer:

\[ \{2,4,8,16,32,64,128\} \]

(vi) \(\{x\mid x\in\mathbb N,\ x+4=0\}\)

Answer:

Since \(x=-4\) is not a natural number,

\[ \boxed{\varnothing} \]

(vii) \(\{x\mid x\in\mathbb N,\ x=x\}\)

Answer:

\[ \boxed{\mathbb N} \]

(viii) \(\{x\mid x\in\mathbb Z,\ 3x+1=0\}\)

Solution:

\[ 3x+1=0 \]

\[ x=-\frac13 \]

Since \(-\frac13\notin\mathbb Z\),

Answer:

\[ \boxed{\varnothing} \]

Q3. Write two proper subsets of each of the following sets.

(i) \(\{a,b,c\}\)

Answer:

\[ \boxed{\varnothing,\ \{a\}} \]

(ii) \(\{0,1\}\)

Answer:

\[ \boxed{\varnothing,\ \{0\}} \]

(iii) \(\mathbb N\)

Answer:

\[ \boxed{\{1\},\ \{1,2\}} \]

(iv) \(\mathbb Z\)

Answer:

\[ \boxed{\{0\},\ \{-1,1\}} \]

(v) \(\mathbb Q\)

Answer:

\[ \boxed{\{0\},\ \left\{\frac12\right\}} \]

(vi) \(\mathbb R\)

Answer:

\[ \boxed{\{0\},\ \{1\}} \]

(vii) \(\{x\mid x\in\mathbb Q,\ 0<x\leq2\}\)

Answer:

\[ \boxed{\{1\},\ \{1,2\}} \]

Q4. Is there any set which has no proper subset? If so, name that set.

Answer:

Yes, the empty set (or null set), denoted by \(\varnothing\) or \(\{\}\), has no proper subsets.

Q5. What is the difference between \(\{a,b\}\) and \(\{\{a,b\}\}\)?

Answer:

\(\{a,b\}\) is a set containing two elements, \(a\) and \(b\).
\(\{\{a,b\}\}\) is a set containing only one element, which is the set \(\{a,b\}\) itself.

Q6. What is the number of elements of the power set of each of the following sets?

Formula:

\[ \boxed{n(P(A))=2^n} \]

where \(n\) is the number of elements in the original set.

(i) \(\varnothing\)

\[ n=0\Rightarrow 2^0=1 \]

Answer: \(\boxed{1}\)

(ii) \(\{0,1\}\)

\[ n=2\Rightarrow 2^2=4 \]

Answer: \(\boxed{4}\)

(iii) \(\{1,2,3,4,5,6,7\}\)

\[ n=7\Rightarrow 2^7=128 \]

Answer: \(\boxed{128}\)

(iv) \(\{0,1,2,3,4,5,6,7\}\)

\[ n=8\Rightarrow 2^8=256 \]

Answer: \(\boxed{256}\)

(v) \(\{a,\{b,c\}\}\)

There are two elements: \(a\) and \(\{b,c\}\).

\[ n=2\Rightarrow 2^2=4 \]

Answer: \(\boxed{4}\)

(vi) \(\{\{a,b\},\{c,d\},\{d,e\}\}\)

There are three elements in this set.

\[ n=3\Rightarrow 2^3=8 \]

Answer: \(\boxed{8}\)

Q7. Write down the power set of each of the following sets.

(i) \(\{9,11\}\)

Answer:

\[ P(A)=\{\varnothing,\{9\},\{11\},\{9,11\}\} \]

(ii) \(\{+,-,\times,\div\}\)

Answer:

The power set contains \(2^4=16\) subsets.

\[ \begin{aligned} P(A)=\{&\varnothing,\{+\},\{-\},\{\times\},\{\div\},\\ &\{+,-\},\{+,\times\},\{+,\div\},\\ &\{-,\times\},\{-,\div\},\{\times,\div\},\\ &\{+,-,\times\},\{+,-,\div\},\\ &\{+,\times,\div\},\{-,\times,\div\},\\ &\{+,-,\times,\div\}\} \end{aligned} \]

(iii) \(\{\varnothing\}\)

Answer:

\[ P(A)=\{\varnothing,\{\varnothing\}\} \]

(iv) \(\{a,b,c\}\)

Answer:

\[ \begin{aligned} P(A)=\{&\varnothing,\{a\},\{b\},\{c\},\\ &\{a,b\},\{a,c\},\{b,c\},\\ &\{a,b,c\}\} \end{aligned} \]

Exercise 3.2 – Selected Key Problems

Q1. Given

\[ U=\{x\mid x\text{ is a multiple of }2,\ 0<x\leq30\} \]

\[ A=\{x\mid x\text{ is a multiple of }6\} \]

\[ B=\{x\mid x\text{ is a multiple of }8\} \]

(i) Write the sets in tabular form.

Answer:

\[ \begin{aligned} U=\{&2,4,6,8,10,12,14,16,18,20,\\ &22,24,26,28,30\} \end{aligned} \]

Taking the elements of \(A\) and \(B\) within the universal set \(U\):

\[ A=\{6,12,18,24,30\} \]

\[ B=\{8,16,24\} \]

(ii) Find \(A\cap B\).

The intersection contains the elements common to both sets.

\[ \boxed{A\cap B=\{24\}} \]

(iii) Draw a Venn diagram.

Answer:

Draw a rectangle representing the universal set \(U\), with two overlapping circles representing \(A\) and \(B\). Place the common element \(24\) in the intersection of the circles.

Q7. In a class of 55 students, 34 like to play cricket and 30 like to play hockey. Each student likes at least one game. How many like both games?

Solution:

Given:

\[ n(U)=55,\quad n(C)=34,\quad n(H)=30 \]

Since every student likes at least one game:

\[ n(C\cup H)=55 \]

Using the formula:

\[ n(C\cup H)=n(C)+n(H)-n(C\cap H) \]

Substituting the values:

\[ 55=34+30-n(C\cap H) \]

\[ 55=64-n(C\cap H) \]

\[ n(C\cap H)=64-55 \]

\[ \boxed{n(C\cap H)=9} \]

Answer: 9 students like both games.

Exercise 3.3

Q1. For \(A=\{1,2,3,4\}\), find the following relations and state the domain and range.

(i) \(\{(x,y)\mid y=x\}\)

Answer:

\[ R=\{(1,1),(2,2),(3,3),(4,4)\} \]

Domain:

\[ \{1,2,3,4\} \]

Range:

\[ \{1,2,3,4\} \]

(ii) \(\{(x,y)\mid y+x=5\}\)

Answer:

\[ R=\{(1,4),(2,3),(3,2),(4,1)\} \]

Domain:

\[ \{1,2,3,4\} \]

Range:

\[ \{1,2,3,4\} \]

(iii) \(\{(x,y)\mid y+x<5\}\)

Answer:

\[ R=\{(1,1),(1,2),(1,3),(2,1),(2,2),(3,1)\} \]

Domain:

\[ \{1,2,3\} \]

Range:

\[ \{1,2,3\} \]

(iv) \(\{(x,y)\mid y+x>5\}\)

Answer:

\[ R=\{(2,4),(3,3),(3,4),(4,2),(4,3),(4,4)\} \]

Domain:

\[ \{2,3,4\} \]

Range:

\[ \{2,3,4\} \]

Q3. If \(g(x)=3x+2\) and \(h(x)=x^2+1\), find the following.

(i) \(g(0)\)

\[ g(0)=3(0)+2 \]

\[ \boxed{g(0)=2} \]

(ii) \(g(-3)\)

\[ g(-3)=3(-3)+2=-9+2 \]

\[ \boxed{g(-3)=-7} \]

(iii) \(g\left(-\frac23\right)\)

\[ g\left(-\frac23\right)=3\left(-\frac23\right)+2 \]

\[ =-2+2 \]

\[ \boxed{g\left(-\frac23\right)=0} \]

(iv) \(h(1)\)

\[ h(1)=1^2+1 \]

\[ \boxed{h(1)=2} \]

(v) \(h(-4)\)

\[ h(-4)=(-4)^2+1=16+1 \]

\[ \boxed{h(-4)=17} \]

(vi) \(h\left(-\frac12\right)\)

\[ \begin{aligned} h\left(-\frac12\right) &=\left(-\frac12\right)^2+1\\ &=\frac14+1\\ &=\frac54 \end{aligned} \]

\[ \boxed{h\left(-\frac12\right)=\frac54} \]

Q4. Given \(f(x)=ax+b+1\), where \(a,b\) are constants. If \(f(3)=8\) and \(f(6)=14\), find \(a\) and \(b\).

Solution:

Since \(f(3)=8\):

\[ 3a+b+1=8 \]

\[ 3a+b=7 \qquad (1) \]

Since \(f(6)=14\):

\[ 6a+b+1=14 \]

\[ 6a+b=13 \qquad (2) \]

Subtract equation (1) from equation (2):

\[ (6a+b)-(3a+b)=13-7 \]

\[ 3a=6 \]

\[ \boxed{a=2} \]

Substitute \(a=2\) into equation (1):

\[ 3(2)+b=7 \]

\[ 6+b=7 \]

\[ \boxed{b=1} \]

Answer:

\[ \boxed{a=2,\quad b=1} \]

Review Exercise 3

Q1. Multiple Choice Questions (MCQs)

(i) The set-builder form of the set \(\left\{1,\frac13,\frac15,\frac17,\ldots\right\}\) is:

Correct Answer: (b)

\[ \left\{x\mid x=\frac{1}{2n+1},\ n\in\mathbb W\right\} \]

Note: This answer depends on whether the textbook defines \(\mathbb W\) to include zero. If zero is included, the set-builder expression also generates \(1\), as required.

(ii) If \(A=\varnothing\), then \(P(A)\) is:

Correct Answer: (b)

\[ \{\varnothing\} \]

(iii) If \(U=\{1,2,3,4,5\}\), \(A=\{1,2,3\}\), and \(B=\{3,4,5\}\), then \(U-(A\cap B)\) is:

\[ A\cap B=\{3\} \]

\[ U-(A\cap B)=\{1,2,4,5\} \]

Correct Answer: (a) \(\{1,2,4,5\}\)

(iv) If \(A\) and \(B\) are overlapping sets, then \(n(A-B)\) is equal to:

Correct Answer: (d)

\[ n(A)-n(A\cap B) \]

(v) If \(A\subseteq B\) and \(B-A\neq\varnothing\), then \(n(B-A)\) is equal to:

Correct Answer: (d)

\[ n(B)-n(A) \]

(vi) If \(n(A\cup B)=50\), \(n(A)=30\), and \(n(B)=35\), then \(n(A\cap B)\) is:

Using the formula:

\[ n(A\cup B)=n(A)+n(B)-n(A\cap B) \]

\[ 50=30+35-n(A\cap B) \]

\[ n(A\cap B)=65-50 \]

Correct Answer: (b) 15

(vii) The Cartesian product of two sets having 4 and 3 elements contains how many elements?

\[ 4\times3=12 \]

Correct Answer: (b) 12

(viii) If \(f(x)=x^2-3x+2\), then \(f(a+1)\) is equal to:

\[ f(a+1)=(a+1)^2-3(a+1)+2 \]

\[ =a^2+2a+1-3a-3+2 \]

\[ =a^2-a \]

Correct Answer: (d) \(a^2-a\)

(ix) Given \(f(x)=3x+1\), if \(f(x)=28\), then the value of \(x\) is:

\[ 3x+1=28 \]

\[ 3x=27 \]

\[ \boxed{x=9} \]

Correct Answer: (a) 9

(x) Let \(A=\{1,2,3\}\) and \(B=\{a,b\}\), and let \(f:A\to B\) be defined as \(f=\{(1,a),(2,b),(3,b)\}\). Then \(f\) is:

Correct Answer: (d) Into only

Q6. Verify De Morgan’s laws for the sets

\[ U=\{1,2,3,\ldots,20\} \]

\[ A=\{2,4,6,\ldots,20\} \]

\[ B=\{1,3,5,\ldots,19\} \]

Law 1: \((A\cup B)’=A’\cap B’\)

Since \(A\) contains all even numbers in \(U\) and \(B\) contains all odd numbers in \(U\):

\[ A\cup B=U \]

Therefore:

\[ (A\cup B)’=U’=\varnothing \]

Also:

\[ A’=B,\qquad B’=A \]

Thus:

\[ A’\cap B’=B\cap A=\varnothing \]

Hence:

\[ \boxed{(A\cup B)’=A’\cap B’} \]

Law 1 is verified.

Law 2: \((A\cap B)’=A’\cup B’\)

Since \(A\) contains even numbers and \(B\) contains odd numbers, they have no common elements.

\[ A\cap B=\varnothing \]

Therefore:

\[ (A\cap B)’=\varnothing’=U \]

Also:

\[ A’\cup B’=B\cup A=U \]

Hence:

\[ \boxed{(A\cap B)’=A’\cup B’} \]

Law 2 is verified.

Notes by Sir Dilawar

For more notes, visit QuranEducationPoint.com