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Mathematics · Class 12 Science

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

For a non-empty set , recall from Class XI that any subset is called a relation in the set (a relation from to itself). If , we write and say " is related to ". Two extreme cases deserve names: the empty relation (no element is related to any element) and the universal relation (every element is related to every elemen…

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Key concepts

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Types of Relations

Imagine you have a set of people in a room. A relation is simply a rule that tells you whether two people are connected in some way.

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In previous exams

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CA Foundation

4 concepts · 11 appearances

2026, 2025, 2024, 2023, 2021

WBJEE

2 concepts · 4 appearances

2026, 2025, 2021

Chapter contents

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

1

Types of Relations

For a non-empty set , recall from Class XI that any subset is called a relation in the set (a relation from to itself). If , we write and say " is related to ".

2

Equivalence Relations and Equivalence Classes

Equivalence relations formalise the everyday idea of "being alike" in some respect — same remainder on division, same length, same shape, and so on.

3

One-One (Injective) Functions

Recall from Class XI that a function assigns to every element of a unique element of . We now classify functions by how they use the codomain .

4

Onto (Surjective) Functions and Bijections

While injectivity is about no repeats, surjectivity is about nothing left out — every target in must receive at least one arrow.

5

Composite Functions

Reading right to left: first apply to , landing in ; then apply to that result, landing in . The composite is only defined when the codomain of the "inner" function matches (or lies inside) the domain…

6

Inverse of a Function

Why bijectivity is exactly what's needed: to define unambiguously for every , two things must hold. First, must be onto — otherwise some has no with , leaving undefined.

Summary

- A relation in a set is reflexive if for every ; symmetric if ; transitive if . The three properties are logically independent.

Sample & Board Papers

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

+Show 22 questions22 questions
  1. Q1Let A = {1, 2, 3}. Define a relation (on A) which is reflexive and symmetric but not transitive.Preview
  2. Q2R1 and R2 are two equivalence relations defined on set A (not equal to phi). Show that R1 intersection R2 is an equivalence relation.Preview
  3. Q3The domain for which the functions f(x) = 3x^2 - 2x and g(x) = 3(3x - 2) are equal, will be (a) {1, 2/3} (b) {1, 3} (c) {2/3, 3} (d) {2/3, 0…Preview
  4. Q4f(x) = (3x + 4)/(5x - 7) (x ≠ 7/5) and g(x) = (7x + 4)/(5x - 3) (x ≠ 3/5), show that f(g(x)) = g(f(x)).Preview
  5. Q5Let, A = {1, 2}, B = {1, 8} and f : A → B, g : A → B are two mappings defined as f(x) = x³ and g(x) = 6x² - 11x + 6, then prove that f = g.Preview
  6. Q6Find the range of the function f(x) = 1/(1 - x²), x is real and x ≠ ±1.Preview
  7. Q7f : R → R is a mapping where f(x) = x³ - 6, for all x ∈ R, R = set of real numbers. Prove that f is a bijective mapping.Preview
  8. Q8The domain in which the functions f(x) = 3x² - 2x and g(x) = 3(3x - 2) will be equal is: **OR** Let A = {1, 2, 3} and R be a relation define…Preview
  9. Q9If the functions f : R → R and g : R → R defined as f(x) = 3x + 2 and g(x) = 2x - 3, [R = set of Real numbers] then (g∘f)(x) is: **OR** The…Preview
  10. Q10State which of the following is total number of relations from Set A = { 1, 2, 3 } to Set B = { a, b }. (a) 2^6 (b) 2^8 (c) 2^4 (d) 2^5Preview
  11. Q11If f(x) = (3x+4)/(5x-7) (x real and x≠7/5) and g(x) = (7x+4)/(5x-3) (x real and x≠3/5), then show that f.g(x) = g.f(x).Preview
  12. Q12A relation R on set of natural numbers N is defined as (x,y)∈R, when x-y is divisible by 10 for all x,y∈N. Prove that R is an equivalence re…Preview
  13. Q13Number of relations on a set with 5 elements is (a) 5 (b) 25 (c) 2^5 (d) 2^25Preview
  14. Q14If g(x) = 2x² + 1 and f(x) = 3x, then find the value of f{g(x)} - g{f(x)}.Preview
  15. Q15Let A = R - {3}, B = R - {1}. Prove that the function f: A → B defined by f(x) = (x-2)/(x-3) is one-one and onto. (f is bijective)Preview
  16. Q16If a*b = a²+b² ∀a,b∈IN, then (4*5)*3 is equal to (a) 50 (b) 60 (c) 1230 (d) 1690Preview
  17. Q17Let IR be the set of all real numbers and for all x∈IR the mapping f:IR→IR is defined by f(x)=ax+3. If fof=I(IR), then find the value of a.…Preview
  18. Q18Let A be the set of all straight lines in a plane. Let a relation R be defined as R = {(x,y): x is perpendicular to y, x,y∈A}. Check whether…Preview
  19. Q19If $\rho$ be a relation on the set of all integers $\mathbb{Z}$ and $\rho = \{(x, y) : |x - y| \le 5,\; x, y \in \mathbb{Z}\}$ then the rela…Preview
  20. Q20Let $\mathbb{R}$ be the set of real numbers and $f : \mathbb{R} \to \mathbb{R}$, $g : \mathbb{R} \to \mathbb{R}$ are two mappings such that…Preview
  21. Q21Statement (Q): $f : \mathbb{R} \to \mathbb{R}$ is a function defined as $f(x) = [x]$, greatest integer function, $f(x)$ is not onto function…Preview
  22. Q22Let $\mathbb{R}$ be the set of real numbers and $f : \mathbb{R} \to \mathbb{R}$ be given by $f(x) = 2x - 3$. Then the value of $f^{-1}(0)$ i…Preview

More questions

27 Q
+Show 2 questions2 questions
  1. Q26Let $A = \{1, 2, 3, 4, 5\}$ and $R = \{(a,b) : a, b \in A,\ |a-b| \leq 1\}$. Determine whether $R$ is reflexive, symmetric and transitive. I…Free
  2. Q27Let $f, g : \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = |x|$ and $g(x) = [x]$ (the greatest integer function). Examine whether $f$ and…Preview
+Show 7 questions7 questions
  1. Example 1Let $A = \{1, 2, 3\}$ and $R = \{(1,1), (2,2), (3,3), (1,2), (2,1)\}$. Determine whether $R$ is reflexive, symmetric and transitive.Free
  2. Example 2Let $R$ be the relation on $\mathbb{Z}$ defined by $a\,R\,b$ iff $5$ divides $a-b$. Show that $R$ is an equivalence relation and find its eq…Free
  3. Example 3Let $R$ be the relation on the set of all straight lines in a plane defined by $l_1\,R\,l_2$ iff $l_1$ is perpendicular to $l_2$. Determine…Free
  4. Example 4Show that the function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = 3x - 2$ is one-one and onto.Preview
  5. Example 5Show that the function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x^2$ is neither one-one nor onto. Suggest a restriction of the dom…Preview
  6. Example 6If $f(x) = 2x + 3$ and $g(x) = x^2 - 1$, find $(g \circ f)(x)$ and $(f \circ g)(x)$. Are they equal?Preview
  7. Example 7Show that $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = \dfrac{7x-3}{4}$ is invertible, and find $f^{-1}(x)$.Preview
+Show 4 questions4 questions
  1. Q8Let $R$ be the relation on $\mathbb{Z}$ defined by $a\,R\,b$ iff $a < b$. Determine whether $R$ is reflexive, symmetric and transitive.Free
  2. Q9Let $R$ be the relation on the set $\mathbf{N}$ of positive integers defined by $a\,R\,b$ iff $a$ divides $b$. Determine whether $R$ is refl…Free
  3. Q10Let $A = \{1, 2, 3, 4\}$ and $R = \{(1,3), (3,1), (2,4), (4,2), (1,1), (2,2)\}$. Determine whether $R$ is reflexive, symmetric and transitiv…Preview
  4. Q11Let $A = \{1, 2, 3\}$ and $R = \{(1,1), (2,2), (3,3), (1,2), (2,3)\}$. Determine whether $R$ is reflexive, symmetric and transitive.Preview
+Show 4 questions4 questions
  1. Q12Show that the relation $R$ on $\mathbb{Z}$ defined by $a\,R\,b$ iff $4$ divides $a - b$ is an equivalence relation, and list its equivalence…Free
  2. Q13Let $R$ be the relation on the set of all triangles in a plane defined by $T_1\,R\,T_2$ iff $T_1$ is similar to $T_2$. Show that $R$ is an e…Free
  3. Q14Let $A = \{1, 2, 3, 4, 5, 6\}$ and let $R$ be the relation on $A$ given by $R = \{(a,b) : a, b \in A,\ a \text{ and } b \text{ leave the sam…Preview
  4. Q15On the set of all books in a library, define $R$ by: book$_1\,R\,$book$_2$ iff book$_1$ and book$_2$ have the same number of pages. Show tha…Preview
+Show 4 questions4 questions
  1. Q16Show that the function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = 5x - 7$ is one-one and onto.Free
  2. Q17Show that the function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x^3$ is both one-one and onto.Free
  3. Q18Show that the function $f : \mathbf{N} \to \mathbf{N}$ defined by $f(x) = 2x$ is one-one but not onto.Preview
  4. Q19Let $f : \mathbb{Z} \to \mathbb{Z}$ be defined by $f(x) = x^2$. Determine whether $f$ is one-one and whether it is onto.Preview
+Show 3 questions3 questions
  1. Q20If $f(x) = 2x + 1$ and $g(x) = x^2 - 2$, find $(f \circ g)(x)$ and $(g \circ f)(x)$, and verify that they are not equal.Free
  2. Q21Let $f(x) = \sqrt{x}$ (domain $x \geq 0$) and $g(x) = x + 4$ (domain $\mathbb{R}$). Find $(g \circ f)(x)$ and its domain, and $(f \circ g)(x…Preview
  3. Q22If $f(x) = 3x - 5$ and $(f \circ g)(x) = 6x + 1$, find $g(x)$.Preview
+Show 3 questions3 questions
  1. Q23Show that $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = 4x + 3$ is invertible, and find $f^{-1}(x)$.Free
  2. Q24Show that $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x^2$ is not invertible. Restrict the domain and codomain suitably to obtain an…Preview
  3. Q25If $f(x) = \dfrac{2x-1}{3}$, $x \in \mathbb{R}$, show that $f$ is invertible, find $f^{-1}(x)$, and verify that $f(f^{-1}(x)) = x = f^{-1}(f…Preview