Skip to content
← Mathematics

Mathematics · Class 11 Science

Ch 1Sets, Relations and Functions — Class 11 Mathematics, concept-first.

Sets, relations and functions sit at the heart of modern mathematical thinking, but the idea of a function did not arrive fully formed. As the mathematician Luzin observed, the concept underwent profound changes over time.

104

Q&A

8

Concepts

~7m

Unit weightage

Start learning — read this chapter →

Key concepts

Hover a concept to preview it and jump to its most relevant Q&A.

Sets and Set Operations

A set is a well-defined, distinguishable collection of objects -- given any object, we must be able to decide for certain whether it belongs.

Most relevant Q&A

Start with this concept →

Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

Introduction

Sets, relations and functions sit at the heart of modern mathematical thinking, but the idea of a function did not arrive fully formed.

1.2

Sets

Sets recap. A set is a well-defined, distinguishable collection of objects: given any object, we must be able to decide definitively whether it belongs to the collection or not.

1.2.1

Properties of Set Operations

These are the algebraic laws that union, intersection and complement obey (all quantifiers are "for all sets inside a fixed universal set "):

1.3

Cartesian Product

Definition. For non-empty sets , the Cartesian product of with is the set of ordered pairs For three sets, is a set of ordered triplets.

+Exercise 1.1i10 questions
  1. Q1Write the following in roster form. (i) $\{x\in N : x^2<121 \text{ and } x \text{ is a prime}\}$. (ii) the set of all positive roots of the…Free
  2. Q2Write the set $\{-1,\ 1\}$ in set builder form.Free
  3. Q3State whether the following sets are finite or infinite. (i) $\{x\in N : x \text{ is an even prime number}\}$. (ii) $\{x\in N : x \text{ is…Free
  4. Q4By taking suitable sets $A,\ B,\ C$, verify the following results: (i) $A\times(B\cap C)=(A\times B)\cap(A\times C)$. (ii) $A\times(B\cup C)…Preview
  5. Q5Justify the trueness of the statement: "An element of a set can never be a subset of itself."Preview
  6. Q6If $n(P(A))=1024$, $n(A\cup B)=15$ and $n(P(B))=32$, then find $n(A\cap B)$.Preview
  7. Q7If $n(A\cap B)=3$ and $n(A\cup B)=10$, then find $n(P(A\Delta B))$.Preview
  8. Q8For a set $A$, $A\times A$ contains 16 elements and two of its elements are $(1,3)$ and $(0,2)$. Find the elements of $A$.Preview
  9. Q9Let $A$ and $B$ be two sets such that $n(A)=3$ and $n(B)=2$. If $(x,1),(y,2),(z,1)$ are in $A\times B$, find $A$ and $B$, where $x,y,z$ are…Preview
  10. Q10If $A\times A$ has 16 elements, $S=\{(a,b)\in A\times A : a<b\}$; $(-1,2)$ and $(0,1)$ are two elements of $S$, then find the remaining elem…Preview
1.4

Constants and Variables, Intervals and Neighbourhoods

Before defining relations and functions rigorously, three ideas need to be pinned down precisely: what counts as a constant versus a variable, and how intervals and neighbourhoods describe subsets of…

1.4.1

Constants and Variables

A constant is a quantity that stays unchanged throughout a mathematical process; a variable is one that changes.

1.4.2

Intervals and Neighbourhoods

The real line. Every real number corresponds to a unique point on a line (and vice versa); we call this line the real line.

1.5

Relations

From everyday relations to mathematics. "How is he related to you?" ("my father", "my teacher", "not related") shows that relation connects one person to another.

1.5.1

Type of Relations

The three basic properties. Let be a relation on a non-empty set .

+Exercise 1.2i9 questions
  1. Q1Discuss the following relations for reflexivity, symmetricity and transitivity: (i) The relation $R$ defined on the set of all positive inte…Free
  2. Q2Let $X=\{a,b,c,d\}$ and $R=\{(a,a),(b,b),(a,c)\}$. Write down the minimum number of ordered pairs to be included to $R$ to make it (i) refle…Free
  3. Q3Let $A=\{a,b,c\}$ and $R=\{(a,a),(b,b),(a,c)\}$. Write down the minimum number of ordered pairs to be included to $R$ to make it (i) reflexi…Free
  4. Q4Let $P$ be the set of all triangles in a plane and $R$ be the relation defined on $P$ as $aRb$ if $a$ is similar to $b$. Prove that $R$ is a…Preview
  5. Q5On the set of natural numbers let $R$ be the relation defined by $aRb$ if $2a+3b=30$. Write down the relation by listing all the pairs. Chec…Preview
  6. Q6Prove that the relation "friendship" is not an equivalence relation on the set of all people in Chennai.Preview
  7. Q7On the set of natural numbers let $R$ be the relation defined by $aRb$ if $a+b\le6$. Write down the relation by listing all the pairs. Check…Preview
  8. Q8Let $A=\{a,b,c\}$. What is the equivalence relation of smallest cardinality on $A$? What is the equivalence relation of largest cardinality…Preview
  9. Q9In the set $Z$ of integers, define $mRn$ if $m-n$ is divisible by 7. Prove that $R$ is an equivalence relation.Preview
1.6

Functions

Motivating example. Suppose is the set of students who wrote a test and is the set of possible marks; relate a student to a mark if scored .

1.6.1

Ways of Representing Functions

(a) Tabular representation. When the domain elements are explicitly listed, a function can be written as a two-row table of arguments against values .

1.6.2

Some Elementary Functions

Some function "shapes" are common enough to deserve their own names.

1.6.3

Types of Functions

One-to-one and onto. For :

1.6.4

Operations on Functions

Composition. Given and , define by first applying , then : . This is the composition of with , written (read " composite with ", applied right-to-left: do first, then ).

1.6.5

Inverse of a Function

Definition. For a bijection , the inverse is the function defined by whenever ; write . A function with an inverse is invertible.

1.6.6

Algebra of Functions

A function whose co-domain is (or a subset of ) is a real-valued function. When share the same domain , we can combine their output values using ordinary real-number arithmetic, defining new functions…

1.6.7

Some Special Functions

Named function families.

+Exercise 1.3i20 questions
  1. Q1Suppose that 120 students are studying in 4 sections of eleventh standard in a school. Let $A$ denote the set of students and $B$ denote the…Free
  2. Q2Write the values of $f$ at $-4,\ 1,\ -2,\ 7,\ 0$ if $$f(x)=\begin{cases}-x+4 & \text{if } -\infty<x\le-3\\ x+4 & \text{if } -3<x<-2\\ x^2-x…Free
  3. Q3Write the values of $f$ at $-3,\ 5,\ 2,\ -1,\ 0$ if $$f(x)=\begin{cases}x^2+x-5 & \text{if } x\in(-\infty,0)\\ x^2+3x-2 & \text{if } x\in(3,…Free
  4. Q4State whether the following relations are functions or not. If it is a function check for one-to-oneness and ontoness. If it is not a functi…Preview
  5. Q5Let $A=\{1,2,3,4\}$ and $B=\{a,b,c,d\}$. Give a function from $A\to B$ for each of the following: (i) neither one-to-one nor onto. (ii) not…Preview
  6. Q6Find the domain of $\dfrac{1}{1-2\sin x}$.Preview
  7. Q7Find the largest possible domain of the real valued function $f(x)=\dfrac{\sqrt{4-x^2}}{\sqrt{x^2-9}}$.Preview
  8. Q8Find the range of the function $\dfrac{1}{2\cos x-1}$.Preview
  9. Q9Show that the relation $xy=-2$ is a function for a suitable domain. Find the domain and the range of the function.Preview
  10. Q10If $f,g:R\to R$ are defined by $f(x)=|x|+x$ and $g(x)=|x|-x$, find $g\circ f$ and $f\circ g$.Preview
  11. Q11If $f,g,h$ are real valued functions defined on $R$, then prove that $(f+g)\circ h=f\circ h+g\circ h$. What can you say about $f\circ(g+h)$?…Preview
  12. Q12If $f:R\to R$ is defined by $f(x)=3x-5$, prove that $f$ is a bijection and find its inverse.Preview
  13. Q13The weight of the muscles of a man is a function of his body weight $x$ and can be expressed as $W(x)=0.35x$. Determine the domain of this f…Preview
  14. Q14The distance of an object falling is a function of time $t$ and can be expressed as $s(t)=-16t^2$. Graph the function and determine if it is…Preview
  15. Q15The total cost of airfare on a given route is comprised of the base cost $C$ and the fuel surcharge $S$ in rupee. Both $C$ and $S$ are funct…Preview
  16. Q16A salesperson whose annual earnings can be represented by the function $A(x)=30{,}000+0.04x$, where $x$ is the rupee value of the merchandis…Preview
  17. Q17The function for exchanging American dollars for Singapore Dollar on a given day is $f(x)=1.23x$, where $x$ represents the number of America…Preview
  18. Q18The owner of a small restaurant can prepare a particular meal at a cost of Rupees 100. He estimates that if the menu price of the meal is $x…Preview
  19. Q19The formula for converting from Fahrenheit to Celsius temperatures is $y=\dfrac{5x}{9}-\dfrac{160}{9}$. Find the inverse of this function an…Preview
  20. Q20A simple cipher takes a number and codes it, using the function $f(x)=3x-4$. Find the inverse of this function, determine whether the invers…Preview
1.7

Graphing Functions using Transformations

33 Q

"A picture is worth a thousand words" -- rather than plotting many points from scratch, it is far faster to recognise a complicated curve as a transformation of a simpler, already-known one.

+Exercise 1.4i8 questions
  1. Q1For the curve $y=x^3$, draw (i) $y=-x^3$ (ii) $y=x^3+1$ (iii) $y=x^3-1$ (iv) $y=(x+1)^3$ with the same scale.Free
  2. Q2For the curve $y=x^{1/3}$, draw (i) $y=-x^{1/3}$ (ii) $y=x^{1/3}+1$ (iii) $y=x^{1/3}-1$ (iv) $y=(x+1)^{1/3}$Free
  3. Q3Graph the functions $f(x)=x^3$ and $g(x)=\sqrt[3]{x}$ on the same coordinate plane. Find $f\circ g$ and graph it on the plane as well. Expla…Free
  4. Q4Write the steps to obtain the graph of the function $y=3(x-1)^2+5$ from the graph $y=x^2$.Preview
  5. Q5From the curve $y=\sin x$, graph the functions (i) $y=\sin(-x)$ (ii) $y=-\sin(-x)$ (iii) $y=\sin\left(\dfrac{\pi}{2}+x\right)$ which is $\co…Preview
  6. Q6From the curve $y=x$, draw (i) $y=-x$ (ii) $y=2x$ (iii) $y=x+1$ (iv) $y=\dfrac12x+1$ (v) $2x+y+3=0$.Preview
  7. Q7From the curve $y=|x|$, draw (i) $y=|x-1|+1$ (ii) $y=|x+1|-1$ (iii) $y=|x+2|-3$.Preview
  8. Q8From the curve $y=\sin x$, draw $y=\sin|x|$ (Hint: $\sin(-x)=-\sin x$.)Preview
+Exercise 1.5i25 questions
  1. Q1If $A=\{(x,y):y=e^x,\ x\in R\}$ and $B=\{(x,y):y=e^{-x},\ x\in R\}$ then $n(A\cap B)$ is (1) Infinity (2) $0$ (3) $1$ (4) $2$Free
  2. Q2If $A=\{(x,y):y=\sin x,\ x\in R\}$ and $B=\{(x,y):y=\cos x,\ x\in R\}$ then $A\cap B$ contains (1) no element (2) infinitely many elements (…Free
  3. Q3The relation $R$ defined on a set $A=\{0,-1,1,2\}$ by $xRy$ if $|x^2+y^2|\le2$, then which one of the following is true? (1) $R=\{(0,0),(0,-…Free
  4. Q4If $f(x)=|x-2|+|x+2|,\ x\in R$, then (1) $f(x)=\begin{cases}-2x & \text{if } x\in(-\infty,-2]\\ 4 & \text{if } x\in(-2,2]\\ 2x & \text{if }…Preview
  5. Q5Let $R$ be the set of all real numbers. Consider the following subsets of the plane $R\times R$: $$S=\{(x,y):y=x+1 \text{ and } 0<x<2\} \tex…Preview
  6. Q6Let $A$ and $B$ be subsets of the universal set $N$, the set of natural numbers. Then $A'\cup[(A\cap B)\cup B']$ is (1) $A$ (2) $A'$ (3) $B$…Preview
  7. Q7The number of students who take both the subjects Mathematics and Chemistry is 70. This represents 10% of the enrollment in Mathematics and…Preview
  8. Q8If $n((A\times B)\cap(A\times C))=8$ and $n(B\cap C)=2$, then $n(A)$ is (1) $6$ (2) $4$ (3) $8$ (4) $16$Preview
  9. Q9If $n(A)=2$ and $n(B\cup C)=3$, then $n[(A\times B)\cup(A\times C)]$ is (1) $2^3$ (2) $3^2$ (3) $6$ (4) $5$Preview
  10. Q10If two sets $A$ and $B$ have 17 elements in common, then the number of elements common to the set $A\times B$ and $B\times A$ is (1) $2^{17}…Preview
  11. Q11For non-empty sets $A$ and $B$, if $A\subset B$ then $(A\times B)\cap(B\times A)$ is equal to (1) $A\cap B$ (2) $A\times A$ (3) $B\times B$…Preview
  12. Q12The number of relations on a set containing 3 elements is (1) $9$ (2) $81$ (3) $512$ (4) $1024$Preview
  13. Q13Let $R$ be the universal relation on a set $X$ with more than one element. Then $R$ is (1) not reflexive (2) not symmetric (3) transitive (4…Preview
  14. Q14Let $X=\{1,2,3,4\}$ and $R=\{(1,1),(1,2),(1,3),(2,2),(3,3),(2,1),(3,1),(1,4),(4,1)\}$. Then $R$ is (1) reflexive (2) symmetric (3) transitiv…Preview
  15. Q15The range of the function $\dfrac{1}{1-2\sin x}$ is (1) $(-\infty,-1)\cup\left(\dfrac13,\infty\right)$ (2) $\left(-1,\dfrac13\right)$ (3) $\…Preview
  16. Q16The range of the function $f(x)=|\lfloor x\rfloor-x|,\ x\in R$ is (1) $[0,1]$ (2) $[0,\infty)$ (3) $[0,1)$ (4) $(0,1)$Preview
  17. Q17The rule $f(x)=x^2$ is a bijection if the domain and the co-domain are given by (1) $R,\ R$ (2) $R,\ (0,\infty)$ (3) $(0,\infty),\ R$ (4) $[…Preview
  18. Q18The number of constant functions from a set containing $m$ elements to a set containing $n$ elements is (1) $mn$ (2) $m$ (3) $n$ (4) $m+n$Preview
  19. Q19The function $f:[0,2\pi]\to[-1,1]$ defined by $f(x)=\sin x$ is (1) one-to-one (2) onto (3) bijection (4) cannot be definedPreview
  20. Q20If the function $f:[-3,3]\to S$ defined by $f(x)=x^2$ is onto, then $S$ is (1) $[-9,9]$ (2) $R$ (3) $[-3,3]$ (4) $[0,9]$Preview
  21. Q21Let $X=\{1,2,3,4\}$, $Y=\{a,b,c,d\}$ and $f=\{(1,a),(4,b),(2,c),(3,d),(2,d)\}$. Then $f$ is (1) an one-to-one function (2) an onto function…Preview
  22. Q22The inverse of $f(x)=\begin{cases}x & \text{if } x<1\\ x^2 & \text{if } 1\le x\le4\\ 8\sqrt x & \text{if } x>4\end{cases}$ is (1) $f^{-1}(x)…Preview
  23. Q23Let $f:R\to R$ be defined by $f(x)=1-|x|$. Then the range of $f$ is (1) $R$ (2) $(1,\infty)$ (3) $(-1,\infty)$ (4) $(-\infty,1]$Preview
  24. Q24The function $f:R\to R$ is defined by $f(x)=\sin x+\cos x$ is (1) an odd function (2) neither an odd function nor an even function (3) an ev…Preview
  25. Q25The function $f:R\to R$ is defined by $$f(x)=\dfrac{(x^2+\cos x)(1+x^4)}{(x-\sin x)(2x-x^3)+e^{-|x|}}$$ is (1) an odd function (2) neither a…Preview

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 32 questions32 questions
  1. Q1The range of the function $f: R-\{3\}\to R$ defined by $f(x) = \dfrac{|x-3|}{(x-3)}$ is: (a) $\{0, 1\}$ (b) $\{1, -1\}$ (c) $\{3, -3\}$ (d)…Preview
  2. Q2If $f: R \to R$ be defined by $f(x) = \begin{cases} x, & x < 1 \\ x^2, & x \ge 1 \end{cases}$ then $f^{-1}(x)$ is: (a) $\begin{cases} x, & x…Preview
  3. Q3The domain of the function $f(x) = \dfrac{1}{\sqrt{|x|-x}}$ is: (a) $(-\infty, 0)$ (b) $(-\infty, \infty)$ (c) $(-1, 1)$ (d) $(0, \infty)$Preview
  4. Q4If $A=\{(x,y)/y=e^x, x\in[0,\infty)\}$ and $B=\{(x,y)/y=\sin x, x\in[0,\infty)\}$ then $n(A\cap B)$ is: (a) $\infty$ (b) 1 (c) $\phi$ (d) 0Preview
  5. Q5If $f: \mathbb{R}\to\mathbb{R}$ is defined by $f(x)=|x|-5$, then the range of $f$ is: (a) $(-\infty, -5)$ (b) $(-\infty, 5)$ (c) $[-5, \inft…Preview
  6. Q6Write the use of horizontal line test.Preview
  7. Q7Is it correct to say $A\times A=\{(a,a): a\in A\}$? Justify your answer.Preview
  8. Q8Construct a suitable domain X such that $f: X\to N$ defined by $f(n)=n+3$ to be one to one and onto.Preview
  9. Q9The function $f:[0,2\pi] \to [-1,1]$ defined by $f(x) = \sin x$ is: (a) one-to-one (b) onto (c) bijection (d) cannot be definedPreview
  10. Q10If $A = \{1, 2, 3\}$, $B = \{1, 4, 6, 9\}$ and $R$ is a relation from $A$ to $B$ defined by '$x$ is greater than $y$'. The range of $R$ is:…Preview
  11. Q11If $A = \{1, 2, 3, 4\}$ and $B = \{3, 4, 5, 6\}$, find $n[(A \cup B) \times (A \cap B) \times (A \Delta B)]$.Preview
  12. Q12In the set $\mathbf{Z}$ of integers, define $mRn$ if $m-n$ is divisible by 7. Prove that R is an equivalence relation.Preview
  13. Q13(a) Let $f, g: \mathbf{R} \to \mathbf{R}$ be defined as $f(x) = 2x - |x|$ and $g(x) = 2x + |x|$. Find $f \circ g$. **OR** (b) Suppose the ch…Preview
  14. Q14Let A and B be subsets of the universal set N, the set of natural numbers. Then $A' \cup [(A \cap B) \cup B']$ is: (a) B (b) A (c) N (d) $A'…Preview
  15. Q15Find the number of subsets of A if $A=\{x : x=4n+1,\ 2 \le n \le 5,\ n \in N\}$.Preview
  16. Q16(a) In a survey of 5000 persons in a town, it was found that 45% of the persons know language A, 25% know language B, 10% know language C, 5…Preview
  17. Q17The rule $f(x) = x^2$ is a bijection if the domain and the co-domain are given by: (a) $(0, \infty), R$ (b) $R, R$ (c) $[0, \infty), [0, \in…Preview
  18. Q18The number of relations on a set containing 3 elements is: (a) 512 (b) 9 (c) 1024 (d) 81Preview
  19. Q19If $A = \{1, 2, 3, 4\}$; $B = \{3, 4, 5, 6\}$ find $n((A\cup B) \times (A\cap B) \times (A\Delta B))$.Preview
  20. Q20$f(x) = \begin{cases}-x+4 & -\infty<x\leq -3\\ x+4 & -3<x<-2\\ x^2-x & -2\leq x<1\\ x-x^2 & 1\leq x<7\\ 0 & \text{otherwise}\end{cases}$ Wri…Preview
  21. Q21If $n((A\times B)\cap(A\times C))=8$ and $n(B\cap C)=2$ then $n(A)$ is: (a) $8$ (b) $6$ (c) $16$ (d) $4$Preview
  22. Q22If the function $f:[-3,3]\to S$ defined by $f(x)=x^2$ is onto, then S is: (a) $[-3,3]$ (b) $[-9,9]$ (c) $[0,9]$ (d) $\mathbf{R}$Preview
  23. Q23Find the domain of $\dfrac{1}{1-2\sin x}$Preview
  24. Q24Write the values of $f$ at $-3, 5, 0$ if $f(x)=\begin{cases}x^2+x-5 & \text{if } x\in(-\infty,0)\\ x^2+3x-2 & \text{if } x\in(3,\infty)\\ x^…Preview
  25. Q25The inverse function of $y = \log_e x$ is: (a) $y = e^x$ (b) $y = \log_e x$ (c) $y = e^{-x}$ (d) $y = -\log_e x$Preview
  26. Q26The number of relations on a set containing 3 elements is: (a) $512$ (b) $9$ (c) $1024$ (d) $81$Preview
  27. Q27(a) In a survey of 5000 persons in a town, it was found that 45% of the persons know Language A, 25% know Language B, 10% know Language C, 5…Preview
  28. Q28The number of relations on a set containing 3 elements is: (a) 512 (b) 9 (c) 1024 (d) 81Preview
  29. Q29If the function $f : [-3, 3] \to S$ defined by $f(x) = x^2$ is onto, then $S$ is: (a) $[-3, 3]$ (b) $[-9, 9]$ (c) $[0, 9]$ (d) $\mathbb{R}$Preview
  30. Q30If $n(A\cap B)=3$ and $n(A\cup B)=10$, then find $n(P(A\Delta B))$.Preview
  31. Q31Write the values of $f$ at $-4, 1, -2, 7, 0$ if $$f(x)=\begin{cases}-x+4 & ; -\infty<x\le -3\\ x+4 & ; -3<x<-2\\ x^2-x & ; -2\le x<1\\ x-x^2…Preview
  32. Q32In the set $\mathbb{Z}$ of integers, define $mRn$ if $m-n$ is divisible by 11. Prove that $R$ is an equivalence relation. **OR** Prove that…Preview