CBSE Class 12 Mathematics Chapter 1 Relations and Functions Revision Notes: With the 2024 board exams around the corner, the time to put down the books and begin revising the topics has arrived. Mathematics is a subject that requires regular practice and constant revision. In the final stage of the session year, it’s advised to students to focus on revising what they already know instead of channelling their energy into learning new concepts.
The CBSE Class 12 exams will commence from February 15, 2024, and the mathematics paper is due on March 9. The first chapter in the Class 12 math books is Relations and Functions. It’s an important chapter in the course and holds a significant weightage in the final exam. You can check out the CBSE Class 12 chapter 1 relations and functions revision notes here, along with supporting study material like mind maps and multiple choice questions.
CBSE Class 12 Maths Chapter 1 Relations and Functions Revision Notes
Basic Definitions and Summary
Relation
If A and B are two non-empty sets, then a relation R from A to B is a subset of A x B.
If R ⊆ A x B and (a, b) ∈ R, then we say that a is related to b by the relation R, written as aRb.
Domain and Range of a Relation
Let R be a relation from a set A to set B. Then, set of all first components or coordinates of the ordered pairs belonging to R is called the domain of R, while the set of all second components or coordinates = of the ordered pairs belonging to R is called the range of R.
Thus, domain of R = {a : (a , b) ∈ R} and range of R = {b : (a, b) ∈ R}
Empty relation is the relation R in X given by R = φ ⊂ X × X.
Universal relation is the relation R in X given by R = X × X.
Reflexive relation R in X is a relation with (a, a) ∈ R ∀ a ∈ X.
Symmetric relation R in X is a relation satisfying (a, b) ∈ R implies (b, a) ∈ R.
Transitive relation R in X is a relation satisfying (a, b) ∈ R and (b, c) ∈ R implies that (a, c) ∈ R.
Equivalence relation R in X is a relation that is reflexive, symmetric and transitive.
Equivalence class [a] containing a ∈ X for an equivalence relation R in X is the subset of X containing all elements b related to a.
A function f : X → Y is one-one (or injective) if
f(x1) = f(x2) ⇒ x1 = x2 ∀ x1, x2 ∈ X.
A function f : X → Y is onto (or subjective) if given any y ∈ Y, ∃ x ∈ X such that f(x) = y.
A function f : X → Y is one-one and onto (or bijective), if f is both one-one and onto.
Given a finite set X, a function f : X → X is one-one (respectively onto) if and only if f is onto (respectively one-one). This is the characteristic property of a finite set. This is not true for infinite set.
Practice Questions:
Question 1: Find out whether each of the following relations are reflexive, symmetric and transitive.
(i) Relation R in the set A = {1, 2, 3…13, 14} defined as
R = {(x, y): 3x − y = 0}
(ii) Relation R in the set N of natural numbers defined as
R = {(x, y): y = x + 5 and x < 4}
(iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as
R = {(x, y): y is divisible by x}
Answer:
(i) A = {1, 2, 3 … 13, 14}
R = {(x, y): 3x − y = 0}
∴R = {(1, 3), (2, 6), (3, 9), (4, 12)}
R is not reflexive since (1, 1), (2, 2) … (14, 14) ∉ R.
Also, R is not symmetric as (1, 3) ∈R, but (3, 1) ∉ R. [3(3) − 1 ≠ 0]
Also, R is not transitive as (1, 3), (3, 9) ∈R, but (1, 9) ∉ R.
[3(1) − 9 ≠ 0]
Hence, R is neither reflexive, nor symmetric, nor transitive.
(ii) R = {(x, y): y = x + 5 and x < 4} = {(1, 6), (2, 7), (3, 8)}
It is seen that (1, 1) ∉ R.
∴R is not reflexive. (1, 6) ∈R
But,(6, 1) ∉ R.
∴R is not symmetric.
Now, since there is no pair in R such that (x, y) and (y, z) ∈R, then (x, z) cannot belong to R.
∴ R is not transitive.
Hence, R is neither reflexive, nor symmetric, nor transitive.
(iii) A = {1, 2, 3, 4, 5, 6}
R = {(x, y): y is divisible by x}
We know that any number (x) is divisible by itself.
(x, x) ∈R
∴R is reflexive.
Now, (2, 4) ∈R [as 4 is divisible by 2]
But, (4, 2) ∉ R. [as 2 is not divisible by 4]
∴R is not symmetric.
Let (x, y), (y, z) ∈ R. Then, y is divisible by x and z is divisible by y.
∴z is divisible by x.
⇒ (x, z) ∈R
∴R is transitive.
Hence, R is reflexive and transitive but not symmetric.
CBSE Class 12 Maths Chapter 1 Relations and Functions Revision Notes PDF |
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NCERT Solutions for Class 12 Maths Chapter 1 Relations and Functions
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