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Mathematics · Class 12 Science

Ch 1Relations and Functions — Class 12 Mathematics, concept-first.

In Class XI you met the fundamental ideas of relations and functions — domain, co-domain, and range — and studied real-valued functions such as polynomial, rational, trigonometric, exponential, and logarithmic functions along with their graphs. That foundation is the starting point for this chapter.

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Key concepts

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Relation Properties

A relation on a set pairs elements of with one another. Some relations behave in regular, predictable ways, and we name these behaviours properties.

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Chapter contents

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

1.1

Introduction

In Class XI you met the fundamental ideas of relations and functions — domain, co-domain, and range — and studied real-valued functions such as polynomial, rational, trigonometric, exponential, and lo…

1.2

Types of Relations

22 Q

A relation in a set is any subset of . The smallest such subset is the empty set and the largest is the whole product ; these two extremes give the simplest relations.

+Worked Examplesi6 questions
  1. Example 1Let $A$ be the set of all students of a boys school. Show that the relation $R$ in $A$ given by $R = \{(a, b): a \text{ is sister of } b\}$…Free
  2. Example 2Let $T$ be the set of all triangles in a plane with $R$ a relation in $T$ given by $R = \{(T_1, T_2): T_1 \text{ is congruent to } T_2\}$. S…Free
  3. Example 3Let $L$ be the set of all lines in a plane and $R$ be the relation in $L$ defined as $R = \{(L_1, L_2): L_1 \text{ is perpendicular to } L_2…Preview
  4. Example 4Show that the relation $R$ in the set $\{1, 2, 3\}$ given by $R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)\}$ is reflexive but neither symme…Preview
  5. Example 5Show that the relation $R$ in the set $\mathbb{Z}$ of integers given by $R = \{(a, b): 2 \text{ divides } a - b\}$ is an equivalence relatio…Preview
  6. Example 6Let $R$ be the relation defined in the set $A = \{1, 2, 3, 4, 5, 6, 7\}$ by $R = \{(a, b): \text{both } a \text{ and } b \text{ are either o…Preview
+Exercise 1.1i16 questions
  1. Q1Determine whether each of the following relations are reflexive, symmetric and transitive: (i) Relation R in the set $A = \{1, 2, 3, \dots,…Free
  2. Q2Show that the relation R in the set $\mathbf{R}$ of real numbers, defined as $R = \{(a, b) : a \le b^2\}$ is neither reflexive nor symmetric…Free
  3. Q3Check whether the relation R defined in the set $\{1, 2, 3, 4, 5, 6\}$ as $R = \{(a, b) : b = a + 1\}$ is reflexive, symmetric or transitive…Free
  4. Q4Show that the relation R in $\mathbf{R}$ defined as $R = \{(a, b) : a \le b\}$, is reflexive and transitive but not symmetric.Preview
  5. Q5Check whether the relation R in $\mathbf{R}$ defined by $R = \{(a, b) : a \le b^3\}$ is reflexive, symmetric or transitive.Preview
  6. Q6Show that the relation R in the set $A = \{1, 2, 3\}$ given by $R = \{(1, 2), (2, 1)\}$ is symmetric but neither reflexive nor transitive.Preview
  7. Q7Show that the relation R in the set A of all the books in a library of a college, given by $R = \{(x, y) : x$ and $y$ have same number of pa…Preview
  8. Q8Show that the relation R in the set $A = \{1, 2, 3, 4, 5\}$ given by $R = \{(a, b) : |a - b|$ is even$\}$, is an equivalence relation. Show…Preview
  9. Q9Show that each of the relation R in the set $A = \{x \in Z : 0 \le x \le 12\}$, given by (i) $R = \{(a, b) : |a - b|$ is a multiple of $4\}$…Preview
  10. Q10Give an example of a relation. Which is (i) Symmetric but neither reflexive nor transitive. (ii) Transitive but neither reflexive nor symmet…Preview
  11. Q11Show that the relation R in the set A of points in a plane given by $R = \{(P, Q) :$ distance of the point P from the origin is same as the…Preview
  12. Q12Show that the relation R defined in the set A of all triangles as $R = \{(T_1, T_2) : T_1$ is similar to $T_2\}$, is equivalence relation. C…Preview
  13. Q13Show that the relation R defined in the set A of all polygons as $R = \{(P_1, P_2) : P_1$ and $P_2$ have same number of sides$\}$, is an equ…Preview
  14. Q14Let L be the set of all lines in XY plane and R be the relation in L defined as $R = \{(L_1, L_2) : L_1$ is parallel to $L_2\}$. Show that R…Preview
  15. Q15Let R be the relation in the set $\{1, 2, 3, 4\}$ given by $R = \{(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)\}$. Choose the corr…Preview
  16. Q16Let R be the relation in the set N given by $R = \{(a, b) : a = b - 2, b > 6\}$. Choose the correct answer. (A) $(2, 4) \in R$ (B) $(3, 8) \…Preview
1.3

Types of Functions

20 Q

In Class XI you studied the identity, constant, polynomial, rational, modulus, and signum functions, and how to add, subtract, multiply, and divide functions.

+Worked Examplesi8 questions
  1. Example 7Let $A$ be the set of all 50 students of Class X in a school. Let $f: A \to \mathbb{N}$ be function defined by $f(x) = $ roll number of the…Free
  2. Example 8Show that the function $f: \mathbb{N} \to \mathbb{N}$, given by $f(x) = 2x$, is one-one but not onto.Free
  3. Example 9Prove that the function $f: \mathbb{R} \to \mathbb{R}$, given by $f(x) = 2x$, is one-one and onto. ![Graph of the line y = f(x) = 2x through…Free
  4. Example 10Show that the function $f: \mathbb{N} \to \mathbb{N}$, given by $f(1) = f(2) = 1$ and $f(x) = x - 1$, for every $x > 2$, is onto but not one…Preview
  5. Example 11Show that the function $f: \mathbb{R} \to \mathbb{R}$, defined as $f(x) = x^2$, is neither one-one nor onto. ![The image of 1 and –1 under f…Preview
  6. Example 12Show that $f: \mathbb{N} \to \mathbb{N}$, given by $f(x) = \begin{cases} x + 1, & \text{if } x \text{ is odd} \\ x - 1, & \text{if } x \text…Preview
  7. Example 13Show that an onto function $f: \{1, 2, 3\} \to \{1, 2, 3\}$ is always one-one.Preview
  8. Example 14Show that a one-one function $f: \{1, 2, 3\} \to \{1, 2, 3\}$ must be onto.Preview
+Exercise 1.2i12 questions
  1. Q1Show that the function $f: \mathbf{R}_* \to \mathbf{R}_*$ defined by $f(x) = \frac{1}{x}$ is one-one and onto, where $\mathbf{R}_*$ is the s…Free
  2. Q2Check the injectivity and surjectivity of the following functions: (i) $f: \mathbf{N} \to \mathbf{N}$ given by $f(x) = x^2$ (ii) $f: \mathbf…Free
  3. Q3Prove that the Greatest Integer Function $f: \mathbf{R} \to \mathbf{R}$, given by $f(x) = [x]$, is neither one-one nor onto, where $[x]$ den…Free
  4. Q4Show that the Modulus Function $f: \mathbf{R} \rightarrow \mathbf{R}$, given by $f(x) = |x|$, is neither one-one nor onto, where $|x|$ is $x…Preview
  5. Q5Show that the Signum Function $f: \mathbf{R} \rightarrow \mathbf{R}$, given by $f(x) = \begin{cases} 1, \text{ if } x > 0 \\ 0, \text{ if }…Preview
  6. Q6Let $A = \{1, 2, 3\}$, $B = \{4, 5, 6, 7\}$ and let $f = \{(1, 4), (2, 5), (3, 6)\}$ be a function from A to B. Show that $f$ is one-one.Preview
  7. Q7In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer. (i) $f: \mathbf{R} \rightarro…Preview
  8. Q8Let A and B be sets. Show that $f: A \times B \rightarrow B \times A$ such that $f(a, b) = (b, a)$ is a bijective function.Preview
  9. Q9Let $f: \mathbf{N} \rightarrow \mathbf{N}$ be defined by $f(n) = \begin{cases} \frac{n+1}{2}, \text{ if } n \text{ is odd} \\ \frac{n}{2}, \…Preview
  10. Q10Let $A = \mathbf{R} - \{3\}$ and $B = \mathbf{R} - \{1\}$. Consider the function $f: A \rightarrow B$ defined by $f(x) = \frac{x-2}{x-3}$. I…Preview
  11. Q11Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be defined as $f(x) = x^4$. Choose the correct answer. (A) $f$ is one-one onto (B) $f$ is many-on…Preview
  12. Q12Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be defined as $f(x) = 3x$. Choose the correct answer. (A) $f$ is one-one onto (B) $f$ is many-one…Preview
1.4

Composition of Functions and Invertible Function

Two functions that can be applied one after the other can be combined into a single function — their composition.

+Worked Examplesi3 questions
  1. Example 15Let $f: \{2, 3, 4, 5\} \to \{3, 4, 5, 9\}$ and $g: \{3, 4, 5, 9\} \to \{7, 11, 15\}$ be functions defined as $f(2) = 3$, $f(3) = 4$, $f(4) =…Free
  2. Example 16Find $gof$ and $fog$, if $f: \mathbb{R} \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}$ are given by $f(x) = \cos x$ and $g(x) = 3x^2$. S…Preview
  3. Example 17Let $f: \mathbb{N} \to Y$ be a function defined as $f(x) = 4x + 3$, where $Y = \{y \in \mathbb{N}: y = 4x + 3 \text{ for some } x \in \mathb…Preview

Miscellaneous Examples

+Miscellaneous Examplesi9 questions
  1. Example 18If $R_1$ and $R_2$ are equivalence relations in a set $A$, show that $R_1 \cap R_2$ is also an equivalence relation.Free
  2. Example 19Let $R$ be a relation on the set $A$ of ordered pairs of positive integers defined by $(x, y) \, R \, (u, v)$ if and only if $xv = yu$. Show…Free
  3. Example 20Let $X = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$. Let $R_1$ be a relation in $X$ given by $R_1 = \{(x, y): x - y \text{ is divisible by } 3\}$ and $R…Free
  4. Example 21Let $f: X \to Y$ be a function. Define a relation $R$ in $X$ given by $R = \{(a, b): f(a) = f(b)\}$. Examine whether $R$ is an equivalence r…Preview
  5. Example 22Find the number of all one-one functions from set $A = \{1, 2, 3\}$ to itself.Preview
  6. Example 23Let $A = \{1, 2, 3\}$. Then show that the number of relations containing $(1, 2)$ and $(2, 3)$ which are reflexive and transitive but not sy…Preview
  7. Example 24Show that the number of equivalence relation in the set $\{1, 2, 3\}$ containing $(1, 2)$ and $(2, 1)$ is two.Preview
  8. Example 25Consider the identity function $I_{\mathbb{N}}: \mathbb{N} \to \mathbb{N}$ defined as $I_{\mathbb{N}}(x) = x \; \forall x \in \mathbb{N}$. S…Preview
  9. Example 26Consider a function $f: \left[0, \frac{\pi}{2}\right] \to \mathbb{R}$ given by $f(x) = \sin x$ and $g: \left[0, \frac{\pi}{2}\right] \to \ma…Preview

Miscellaneous Exercise on Chapter 1

+Miscellaneous Exercisei7 questions
  1. Q1Show that the function $f: \mathbf{R} \to \{x \in \mathbf{R} : -1 < x < 1\}$ defined by $f(x) = \frac{x}{1+|x|}$, $x \in \mathbf{R}$ is one…Free
  2. Q2Show that the function $f: \mathbf{R} \to \mathbf{R}$ given by $f(x) = x^3$ is injective.Free
  3. Q3Given a non empty set $X$, consider $P(X)$ which is the set of all subsets of $X$. Define the relation $\mathbf{R}$ in $P(X)$ as follows: Fo…Free
  4. Q4Find the number of all onto functions from the set $\{1, 2, 3, ..., n\}$ to itself.Preview
  5. Q5Let $A = \{-1, 0, 1, 2\}$, $B = \{-4, -2, 0, 2\}$ and $f, g: A \to B$ be functions defined by $f(x) = x^2 - x$, $x \in A$ and $g(x) = 2\left…Preview
  6. Q6Let $A = \{1, 2, 3\}$. Then number of relations containing $(1, 2)$ and $(1, 3)$ which are reflexive and symmetric but not transitive is (A)…Preview
  7. Q7Let $A = \{1, 2, 3\}$. Then number of equivalence relations containing $(1, 2)$ is (A) 1 (B) 2 (C) 3 (D) 4Preview

Summary

- Relation: A subset of . Types: reflexive (), symmetric (), transitive (). An equivalence relation satisfies all three.

Exemplar Problems

Higher-order thinking / exemplar-style practice problems.

+Show 36 questions36 questions
  1. Q1Let $A = \{a, b, c\}$ and the relation $R$ be defined on $A$ as follows: $R = \{(a, a), (b, c), (a, b)\}$. Then, write minimum number of ord…Free
  2. Q2Let $D$ be the domain of the real valued function $f$ defined by $f(x) = \sqrt{25 - x^2}$. Then, write $D$.Free
  3. Q3Is $g = \{(1, 1), (2, 3), (3, 5), (4, 7)\}$ a function? If $g$ is described by $g(x) = \alpha x + \beta$, then what value should be assigned…Free
  4. Q4Are the following set of ordered pairs functions? If so, examine whether the mapping is injective or surjective. (i) $\{(x, y) : x \text{ is…Preview
  5. Q5Let $\mathbb{C}$ be the set of complex numbers. Prove that the mapping $f : \mathbb{C} \to \mathbb{R}$ given by $f(z) = |z|$, $\forall\, z \…Preview
  6. Q6Let the function $f : \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = \cos x$, $\forall\, x \in \mathbb{R}$. Show that $f$ is neither one-o…Preview
  7. Q7Let $X = \{1, 2, 3\}$ and $Y = \{4, 5\}$. Find whether the following subsets of $X \times Y$ are functions from $X$ to $Y$ or not. (i) $f =…Preview
  8. Q8Let $f : \mathbb{R} \to \mathbb{R}$ be the function defined by $f(x) = \dfrac{1}{2 - \cos x}$, $\forall\, x \in \mathbb{R}$. Then, find the…Preview
  9. Q9Let $n$ be a fixed positive integer. Define a relation $R$ in $\mathbb{Z}$ as follows: $\forall\, a, b \in \mathbb{Z}$, $aRb$ if and only if…Preview
  10. Q10If $A = \{1, 2, 3, 4\}$, define relations on $A$ which have properties of being: (a) reflexive, transitive but not symmetric; (b) symmetric…Preview
  11. Q11Let $R$ be a relation defined on the set of natural numbers $\mathbb{N}$ as follows: $R = \{(x, y) : x \in \mathbb{N}, \ y \in \mathbb{N}, \…Preview
  12. Q12Given $A = \{2, 3, 4\}$, $B = \{2, 5, 6, 7\}$. Construct an example of each of the following: (a) an injective mapping from $A$ to $B$; (b)…Preview
  13. Q13Give an example of a map (i) which is one-one but not onto; (ii) which is not one-one but onto; (iii) which is neither one-one nor onto.Preview
  14. Q14Let $A = \mathbb{R} - \{3\}$, $B = \mathbb{R} - \{1\}$. Let $f : A \to B$ be defined by $f(x) = \dfrac{x - 2}{x - 3}$, $\forall\, x \in A$.…Preview
  15. Q15Let $A = [-1, 1]$. Then, discuss whether the following functions defined on $A$ are one-one, onto or bijective: (i) $f(x) = \dfrac{x}{2}$ (i…Preview
  16. Q16Each of the following defines a relation on $\mathbb{N}$: (i) $x$ is greater than $y$, $x, y \in \mathbb{N}$; (ii) $x + y = 10$, $x, y \in \…Preview
  17. Q17Let $A = \{1, 2, 3, \ldots, 9\}$ and $R$ be the relation in $A \times A$ defined by $(a, b) \, R \, (c, d)$ if $a + d = b + c$ for $(a, b),…Preview
  18. Q18Let $T$ be the set of all triangles in the Euclidean plane, and let a relation $R$ on $T$ be defined as $aRb$ if $a$ is congruent to $b$, $\…Preview
  19. Q19Consider the non-empty set consisting of children in a family and a relation $R$ defined as $aRb$ if $a$ is brother of $b$. Then $R$ is (A)…Preview
  20. Q20The maximum number of equivalence relations on the set $A = \{1, 2, 3\}$ are (A) 1 (B) 2 (C) 3 (D) 5Preview
  21. Q21If a relation $R$ on the set $\{1, 2, 3\}$ be defined by $R = \{(1, 2)\}$, then $R$ is (A) reflexive (B) transitive (C) symmetric (D) none o…Preview
  22. Q22Let us define a relation $R$ in $\mathbb{R}$ as $aRb$ if $a \geq b$. Then $R$ is (A) an equivalence relation (B) reflexive, transitive but n…Preview
  23. Q23Let $A = \{1, 2, 3\}$ and consider the relation $R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)\}$. Then $R$ is (A) reflexive but not…Preview
  24. Q24If the set $A$ contains 5 elements and the set $B$ contains 6 elements, then the number of one-one and onto mappings from $A$ to $B$ is (A)…Preview
  25. Q25Let $A = \{1, 2, 3, \ldots n\}$ and $B = \{a, b\}$. Then the number of surjections from $A$ into $B$ is (A) $^{n}P_2$ (B) $2^n - 2$ (C) $2^n…Preview
  26. Q26Let $f : \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = \dfrac{1}{x}$, $\forall\, x \in \mathbb{R}$. Then $f$ is (A) one-one (B) onto (C)…Preview
  27. Q27Which of the following functions from $\mathbb{Z}$ into $\mathbb{Z}$ are bijections? (A) $f(x) = x^3$ (B) $f(x) = x + 2$ (C) $f(x) = 2x + 1$…Preview
  28. Q28Let $f : [2, \infty) \to \mathbb{R}$ be the function defined by $f(x) = x^2 - 4x + 5$, then the range of $f$ is (A) $\mathbb{R}$ (B) $[1, \i…Preview
  29. Q29Let $f : \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = \begin{cases} 2x, & x > 3 \\ x^2, & 1 < x \leq 3 \\ 3x, & x \leq 1 \end{cases}$. T…Preview
  30. Q30Let the relation $R$ be defined in $\mathbb{N}$ by $aRb$ if $2a + 3b = 30$. Then $R = $ ______.Preview
  31. Q31Let the relation $R$ be defined on the set $A = \{1, 2, 3, 4, 5\}$ by $R = \{(a, b) : |a^2 - b^2| < 8\}$. Then $R$ is given by ______.Preview
  32. Q32Let $R = \{(3, 1), (1, 3), (3, 3)\}$ be a relation defined on the set $A = \{1, 2, 3\}$. Then $R$ is symmetric, transitive but not reflexive…Preview
  33. Q33Every relation which is symmetric and transitive is also reflexive.Preview
  34. Q34An integer $m$ is said to be related to another integer $n$ if $m$ is an integral multiple of $n$. This relation in $\mathbb{Z}$ is reflexiv…Preview
  35. Q35Let $A = \{0, 1\}$ and $\mathbb{N}$ be the set of natural numbers. Then the mapping $f : \mathbb{N} \to A$ defined by $f(2n - 1) = 0$, $f(2n…Preview
  36. Q36The relation $R$ on the set $A = \{1, 2, 3\}$ defined as $R = \{(1, 1), (1, 2), (2, 1), (3, 3)\}$ is reflexive, symmetric and transitive.Preview

Sample & Board Papers

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

+Show 7 questions7 questions
  1. Q1Traffic flows from 𝐷 to 𝐸 and 𝐷 to 𝐶. The department wants to represent and analyze this data using relations and functions. Use the given d…Preview
  2. Q2An organization conducted a bike race under 2 different categories — boys and girls. In all, there were 250 participants. Among all of them…Preview
  3. Q3A relation R is defined on the set of real numbers $\mathbb{R}$ as $R = \{(x, y) : x \cdot y$ is an irrational number$\}$. Check whether R i…Preview
  4. Q4Examine whether the operation $*$ defined on $\mathbb{R}$, the set of all real numbers, by $a * b = \sqrt{a^2 + b^2}$ is a binary operation…Preview
  5. Q5Examine whether the operation $*$ defined on $R$ by $a * b = ab + 1$ is (i) a binary or not, (ii) if a binary operation, is it associative o…Preview
  6. Q6If $a * b$ denotes the larger of '$a$' and '$b$' and if $a \circ b = (a * b) + 3$, then write the value of $(5) \circ (10)$, where $*$ and $…Preview
  7. Q7Let $A = \{x \in Z : 0 \le x \le 12\}$. Show that $R = \{(a,b) : a, b \in A,\ |a - b|\ \text{is divisible by 4}\}$ is an equivalence relatio…Preview