Class 12 Mathematics Notes Chapter 1 (Relations and functions) – Mathematics Part-I Book

Detailed Notes with MCQs of Chapter 1: Relations and Functions from your NCERT Class 12 Maths Part-I book. This chapter lays the foundation for many concepts in calculus and higher mathematics, and it's frequently tested in government exams. Pay close attention to the definitions and types, as questions often test your understanding of these core ideas.

Chapter 1: Relations and Functions - Detailed Notes

1. Introduction

• Ordered Pair: A pair of elements grouped together in a particular order, denoted as (a, b). Note that (a, b) ≠ (b, a) unless a = b.
• Cartesian Product: Given two non-empty sets A and B, the Cartesian product A × B is the set of all possible ordered pairs (a, b) such that a ∈ A and b ∈ B.
  • A × B = {(a, b) | a ∈ A and b ∈ B}
  • If n(A) = p and n(B) = q, then n(A × B) = pq.
  • If either A or B is an infinite set, then A × B is an infinite set.
  • A × A × A = {(a, b, c) | a, b, c ∈ A}, called an ordered triplet.

2. Relations

• Definition: A relation R from a non-empty set A to a non-empty set B is a subset of the Cartesian product A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.
  • R ⊆ A × B
  • If (a, b) ∈ R, we say 'a is related to b' under the relation R, written as a R b.
• Relation on a Set: A relation from a set A to itself (i.e., a subset of A × A) is called a relation on set A.
• Domain: The set of all first elements of the ordered pairs in a relation R from set A to set B is called the domain of the relation R. Domain(R) = {a | (a, b) ∈ R for some b ∈ B}. Domain(R) ⊆ A.
• Range: The set of all second elements of the ordered pairs in a relation R from set A to set B is called the range of the relation R. Range(R) = {b | (a, b) ∈ R for some a ∈ A}. Range(R) ⊆ B.
• Codomain: The entire set B is called the codomain of the relation R from A to B.
• Total Number of Relations: If n(A) = p and n(B) = q, then n(A × B) = pq. The total number of possible subsets (and hence relations) from A to B is 2^(pq).

3. Types of Relations (on a set A)

Let R be a relation on a non-empty set A (R ⊆ A × A).

• (i) Empty Relation: A relation R in a set A is called an empty relation if no element of A is related to any element of A, i.e., R = ∅ ⊂ A × A.
  • Example: Let A = {1, 2, 3}. R = {(a, b) | a + b = 10}. Here, no pair (a, b) from A × A satisfies a + b = 10. So, R is empty.
• (ii) Universal Relation: A relation R in a set A is called a universal relation if each element of A is related to every element of A, i.e., R = A × A.
  • Example: Let A = {1, 2}. R = {(a, b) | |a - b| ≥ 0}. Since the absolute difference is always non-negative, all pairs (1,1), (1,2), (2,1), (2,2) are in R. So, R = A × A.
• (iii) Reflexive Relation: R is reflexive if (a, a) ∈ R for every a ∈ A. (Every element must be related to itself).
  • Example: R = {(a, b) | a ≤ b} on the set of integers Z. Since a ≤ a is always true, R is reflexive.
  • Non-Example: R = {(a, b) | a is perpendicular to b} on the set of lines in a plane. A line cannot be perpendicular to itself.
• (iv) Symmetric Relation: R is symmetric if (a, b) ∈ R implies that (b, a) ∈ R for all a, b ∈ A. (If a is related to b, then b must be related to a).
  • Example: R = {(L1, L2) | L1 is parallel to L2} on the set of lines. If L1 || L2, then L2 || L1.
  • Non-Example: R = {(a, b) | a is the father of b}. If a is the father of b, b cannot be the father of a.
• (v) Transitive Relation: R is transitive if (a, b) ∈ R and (b, c) ∈ R implies that (a, c) ∈ R for all a, b, c ∈ A. (If a is related to b, and b is related to c, then a must be related to c).
  • Example: R = {(a, b) | a < b} on the set of real numbers R. If a < b and b < c, then a < c.
  • Non-Example: R = {(a, b) | a is the friend of b}. If A is a friend of B, and B is a friend of C, A is not necessarily a friend of C.
• (vi) Equivalence Relation: A relation R on a set A is an equivalence relation if R is reflexive, symmetric, and transitive.
  • Key Examples:
    • Relation '=' on any set.
    • Relation 'is congruent to' on the set of triangles.
    • Relation 'is similar to' on the set of triangles.
    • Relation R = {(a, b) | a ≡ b (mod m)} (congruence modulo m) on the set of integers Z.
    • Relation R = {(a, b) | a - b is an integer} on the set of real numbers R.
• Equivalence Class: If R is an equivalence relation on A, then for any a ∈ A, the equivalence class of a, denoted by [a], is the set of all elements b ∈ A such that (a, b) ∈ R.
  • [a] = {b ∈ A | (a, b) ∈ R}
  • Equivalence classes are either disjoint or identical.
  • The set of all equivalence classes forms a partition of the set A.

4. Functions

• Definition: A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B.
  • Notation: f: A → B, where A is the domain and B is the codomain.
  • If (a, b) ∈ f, then b is called the image of a under f (b = f(a)), and a is called the pre-image of b.
  • Range: The set of all images of elements of A under f. Range(f) = {f(a) | a ∈ A}. Range(f) ⊆ Codomain(B).
• Key Conditions:
  1. Every element in the domain A must have an image in B.
  2. No element in the domain A can have more than one image in B.
• Real Function: A function whose domain and codomain are subsets of the set of real numbers R.

5. Types of Functions

Let f: A → B be a function.

• (i) One-one (Injective) Function: A function f is one-one if distinct elements of A have distinct images in B.
  • Condition: For every x₁, x₂ ∈ A, f(x₁) = f(x₂) implies x₁ = x₂.
  • Alternatively: x₁ ≠ x₂ implies f(x₁) ≠ f(x₂).
  • Graphical Test: A horizontal line intersects the graph of a one-one function at most once.
  • Example: f: R → R, f(x) = 2x + 3. If f(x₁) = f(x₂), then 2x₁ + 3 = 2x₂ + 3 ⇒ x₁ = x₂.
• (ii) Many-one Function: A function f is many-one if it is not one-one. That is, there exist at least two distinct elements in A which have the same image in B.
  • Example: f: R → R, f(x) = x². Here f(-2) = 4 and f(2) = 4. Distinct elements -2 and 2 have the same image.
• (iii) Onto (Surjective) Function: A function f is onto if every element of the codomain B is the image of at least one element of the domain A.
  • Condition: For every y ∈ B, there exists an element x ∈ A such that f(x) = y.
  • Key Property: Range(f) = Codomain(B).
  • Example: f: R → R, f(x) = x + 1. For any y ∈ R (codomain), we can find x = y - 1 ∈ R (domain) such that f(x) = (y - 1) + 1 = y.
• (iv) Into Function: A function f is into if it is not onto. That is, there exists at least one element in the codomain B which is not the image of any element in the domain A.
  • Key Property: Range(f) is a proper subset of Codomain(B).
  • Example: f: R → R, f(x) = x². The codomain is R, but the range is [0, ∞). Negative real numbers have no pre-image.
• (v) Bijective Function (One-one and Onto): A function f is bijective if it is both one-one (injective) and onto (surjective).
  • Bijective functions are essential for defining inverse functions.

6. Composition of Functions

• Let f: A → B and g: B → C be two functions. The composition of f and g, denoted by g o f (read as 'g of f'), is a function defined from A to C.
  • (g o f)(x) = g(f(x)), for all x ∈ A.
  • For g o f to be defined, the Range(f) must be a subset of the Domain(g).
• Properties:
  • Composition is not necessarily commutative: f o g ≠ g o f (in general).
  • Composition is associative: If f: A → B, g: B → C, h: C → D, then h o (g o f) = (h o g) o f.
  • If f and g are both one-one, then g o f is one-one.
  • If f and g are both onto, then g o f is onto.
  • If f and g are both bijective, then g o f is bijective.

7. Invertible Functions

• Definition: A function f: A → B is said to be invertible if there exists a function g: B → A such that:
  • g o f = I<0xE2><0x82><0x90> (Identity function on A, i.e., g(f(x)) = x for all x ∈ A)
  • f o g = I<0xE2><0x82><0x91> (Identity function on B, i.e., f(g(y)) = y for all y ∈ B)
  • The function g is called the inverse of f and is denoted by f⁻¹.
• Condition for Invertibility: A function f: A → B is invertible if and only if f is bijective (one-one and onto).
• Finding the Inverse:
  1. Write y = f(x).
  2. Solve the equation for x in terms of y. (x = g(y))
  3. Replace y with x to get the formula for f⁻¹(x). (f⁻¹(x) = g(x))
  4. Verify that f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.
• Property: If f is invertible, then f⁻¹ is also invertible and (f⁻¹)⁻¹ = f.
• Property: If f: A → B and g: B → C are invertible, then g o f: A → C is also invertible and (g o f)⁻¹ = f⁻¹ o g⁻¹.

8. Binary Operations (Less frequently emphasized in some exams, but part of NCERT)

• Definition: A binary operation * on a set A is a function *: A × A → A. It takes two elements from A and produces a unique element in A. We denote *(a, b) by a * b.
• Properties:
  • (i) Commutativity: * is commutative if a * b = b * a for all a, b ∈ A.
  • (ii) Associativity: * is associative if (a * b) * c = a * (b * c) for all a, b, c ∈ A.
  • (iii) Identity Element: An element e ∈ A is the identity element for * if a * e = e * a = a for all a ∈ A. The identity element, if it exists, is unique.
  • (iv) Inverse Element: An element a ∈ A is invertible with respect to * if there exists an element b ∈ A such that a * b = b * a = e, where e is the identity element. The element b is called the inverse of a, denoted by a⁻¹.

Multiple Choice Questions (MCQs)

1. Let A = {1, 2, 3}. Which of the following relations on A is reflexive?
   A) R = {(1, 1), (2, 2)}
   B) R = {(1, 1), (2, 2), (3, 3), (1, 2)}
   C) R = {(1, 2), (2, 1)}
   D) R = {(1, 1), (2, 2), (3, 1)}

2. Let R be the relation on the set N of natural numbers defined by R = {(a, b) : a + 3b = 12, a ∈ N, b ∈ N}. The Range of R is:
   A) {1, 2, 3}
   B) {3, 6, 9}
   C) {1, 2}
   D) {3, 2, 1}

3. Let f: R → R be defined by f(x) = x². The function f is:
   A) One-one and onto
   B) One-one but not onto
   C) Onto but not one-one
   D) Neither one-one nor onto

4. Let A = {a, b, c} and B = {1, 2}. The number of possible functions from A to B is:
   A) 6
   B) 8
   C) 9
   D) 4

5. Consider the relation R = {(x, y) | x, y are real numbers and x = wy for some rational number w} on the set R. Then R is:
   A) Reflexive only
   B) Symmetric only
   C) Transitive only
   D) An equivalence relation

6. Let f: R → R be defined by f(x) = 3x - 4. The inverse function f⁻¹(x) is:
   A) (x + 4) / 3
   B) (x - 4) / 3
   C) 3x + 4
   D) 4x - 3

7. Let f(x) = x + 7 and g(x) = x - 7, x ∈ R. Find (f o g)(x).
   A) x
   B) 0
   C) 2x
   D) 14

8. A relation R on a set A is called an equivalence relation if it is:
   A) Reflexive and Symmetric
   B) Symmetric and Transitive
   C) Reflexive and Transitive
   D) Reflexive, Symmetric and Transitive

9. Let * be a binary operation on the set Z of integers defined by a * b = a + b - 5. The identity element for * is:
   A) 0
   B) -5
   C) 5
   D) 1

10. A function f: A → B is invertible if and only if it is:
    A) Injective (One-one)
    B) Surjective (Onto)
    C) Bijective (One-one and Onto)
    D) Many-one

Answer Key for MCQs:

1. B (Must contain (1,1), (2,2), (3,3))
2. D (Pairs are (9,1), (6,2), (3,3). Range is {1, 2, 3})
3. D (f(-1)=f(1)=1, so not one-one. Range is [0, ∞) ≠ R, so not onto)
4. B (Each of the 3 elements in A can map to 2 elements in B. Total functions = 2³ = 8)
5. D (Reflexive: x = 1x (w=1 is rational). Symmetric: If x = wy, then y = (1/w)x (1/w is rational if w≠0). If x=0, then y=0, so y=1x. Transitive: If x = w₁y and y = w₂z, then x = (w₁w₂)z (w₁w₂ is rational). Hence, equivalence.)
6. A (Let y = 3x - 4. Then 3x = y + 4, so x = (y + 4) / 3. Thus f⁻¹(x) = (x + 4) / 3)
7. A ((f o g)(x) = f(g(x)) = f(x - 7) = (x - 7) + 7 = x)
8. D (Definition of Equivalence Relation)
9. C (We need a * e = a. So a + e - 5 = a ⇒ e - 5 = 0 ⇒ e = 5. Check: e * a = 5 + a - 5 = a)
10. C (Fundamental theorem for invertible functions)

Study these notes carefully, focusing on the precise definitions and conditions for each type of relation and function. Practice identifying these types with various examples. Good luck with your preparation!