CBSE Class 12 Mathematics Notes Chapter 1 Relations and Functions

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The concept of relation refers to the relationship between two objects or values.

A relation R from a non-empty set A to a non-empty set B is a subset of the Cartesian product A x B.
The set of all the first elements of the ordered pair in a relation R is called the domain of the relation R.
The set of all the second elements in a relation R is called the range of the relation.

A function is a relationship that specifies that there should only be one output for each input.

It is a type of relation (a collection of ordered pairs) that follows a rule, which states that each y-value should be related to just one y-value.
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

Class 12 Maths Relations and Functions is of a weightage of 8 marks in the final mathematics examination 2024.

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Additional Resources for Preparation
Relations and Functions	NCERT Solutions for Class 12 Mathematics Chapter 1
Relations and Functions MCQs	Relations and Functions Important Questions

Class 12 Mathematics Chapter 1 Notes – Relations and Functions

Relation

Let A and B be two sets.
Then a relation R from set A to set B is a subset of A x B.
Thus R is a relation from A to B ⇔ R ⊂ A x B.

Domain

Let R be a relation from a set A to a set B.
Then the first set of all the first components or coordinates of the ordered pairs belonging to R is called the domain of R.
Thus, the domain of R = {a : (a, b) ϵ R}.
The domain of R ⊂ A.

Range

Let R be a relation from a set A to a set B.
Then the set of all second components or coordinates of the ordered pairs belonging to R is called the range of R.
Thus, the range of R = {b : (a, b) ϵ R}.
The range of R ⊂ B.

Relation on a Set

Let A be a non-void set.
Then a relation from A to itself i.e. a subset of A x A is called a relation on set A.

The Inverse of a Relation

Let A and B be the two sets, and let R be a relation from a set A to a set B.
Then, the inverse of R, denoted by R-1 is a relation from B to A and is defined by

R-1 = {(b, a) : (a, b) ϵ R}

Clearly, (a, b) ϵ R ⇔ (b, a) ϵ R-1
Also, Domain (R) = Range (R-1) and, Range (R) = Domain (R-1)

Types of Relation

Different types of Relation are
- Void Relation
- Universal Relation
- Identity Relation
- Reflexive Relation
- Symmetric Relation
- Transitive Relation
- Antisymmetric Relation

Void Relation

Let A be a set. Then ɸ ⊂ A x A and so it is a relation on A.
This relation is called the void or empty relation on set A.
A relation R on a set A is called a void relation, if no element of A is related to any element of A.

Universal Relation

Let A be a set. Then A x A ⊂ A x A and so it is a relation on A.
This relation is called the universal relation on set A.
A relation R on a set A is called a universal relation, if each element of A is related to every element of A.

Identity Relation

Let A be a set. Then the relation IA = {(a, a): a ϵ A} on A is called the identity relation on A.
A relation IA on A is called the identity relation if every element of A is related to itself only.

Reflexive Relation

A relation R on a set A is said to be reflexive if every element of A is related to itself.
Thus, R is reflexive ⇔ (a, a) ϵ R for all a ϵ A.
A relation R on a set A is not reflexive if there exists an element a ϵ A such that (a, a) ∉ R.

Symmetric Relation

A relation R on a set A is said to be a symmetric relation if

(a, b) ϵ R ⇒ (b, a) ϵ R for all a, b ϵ A

i.e. aRb ⇒ bRa for all a, b ϵ A

Transitive Relation

A relation R on a set A is said to be a transitive relation if

(a, b) ϵ R and (b, c) ϵ R ⇒ (a, c) ϵ R for all a, b, c ϵ A

i.e. aRb and bRc ⇒ aRc for all a, b, c ϵ A

Antisymmetric Relation

A relation R on a set A is said to be an antisymmetric relation if

(a, b) ϵ R and (b, a) ϵ R ⇒ a = b for all a, b ϵ A

Functions

A function is a relationship that specifies that there should only be one output for each input.
It is a type of relation that follows a rule, which states that each y-value should be related to just one y-value.
Let A and B be two non-empty sets.
A relation f from A to B i.e. a subset of A x B is called a function from A to B if
- For each a ϵ A there exists b ϵ B such that (a, b) ϵ f.
- (a, b) ϵ f and (a, c) ϵ f ⇒ b = c.

Types of Functions

Different types of functions are
- One-one Function (Injection)
- Many-one Function
- Onto Function (Surjection)
- One-one Onto Function (Bijection)

One-one Function (Injection)

A function f: A → B is said to be a one-to-one function or an injection if different elements of A have different images in B.
Thus, f: A → B is one-one.

One-one Function

One-one Function

Many-one Function

A function f: A → B is said to be a many-one function if two or more elements of set A have the same images in B.
Thus, f: A → B is a many-one function if there exists x, y ϵ A such that x ≠ y but f(x) = f(y).

Many-one Function

Many-one Function

Onto Function (Surjection)

A function f: A → B is said to be a onto function or a surjection if every element of set B is the f-image of some element of A i.e. f(A) = B or range of f is the co-domain of f.
Thus, f: A → B is a surjection if, for each b ϵ B, there exists a ϵ A such that f(a) = b.

Surjection Function

Surjection Function

One-one Onto Function (Bijection)

A function f: A → B is a bijection if it is one-one as well as onto.
In other words, a function f: A → B is a bijection, if it is
- One-one i.e. f(x) = f(y) ⇒ x = y for all x, y ϵ A.
- Onto i.e. for all y ϵ B, there exists x ϵ A such that f(x) = y.

Bijection Function

Bijection Function

Composite of Functions

Let f: A → B and g: B → C be two functions.
Then the composition of f and g is defined as the function of g o f: A → C given by

g o f(x) = g (f(x)), ∀ x ∈ A

Invertible Function

A function f: X → Y is defined to be invertible if there exists a function g: Y → X such that g o f = IX and f o g = IY.
Then the function is called the inverse of f and is denoted by f-1.

Binary Operations

A binary operation ∗ on a set A is a function ∗: A x A → A.
We denote ∗ (a, b) by a ∗ b.
A binary operation ∗ on the set X is called commutative if a ∗ b = b ∗ a for every a, b ϵ X.

There are Some important List Of Top Mathematics Questions On Relations And Functions Asked In CBSE CLASS XII

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CBSE Class 12 Mathematics Notes Chapter 1 Relations and Functions

Class 12 Mathematics Chapter 1 Notes – Relations and Functions

CBSE CLASS XII Related Questions

1.
For \[ f(x)=x+\frac{1}{x}, \quad x\neq 0. \]

2.
The least value of \[ f(x)=e^{-x} \] in the interval \[ [0,3] \] is:

3.
For a square matrix \(A\), \[ (3A)^{-1}= \]

4.
Find: \[ \int \frac{x^2}{(x^2+9)(x^2+16)}\,dx \]

5.
For two vectors \(\vec{a}\) and \(\vec{b}\):

Assertion (A): \[ |\vec{a}\times\vec{b}|^2+(\vec{a}\cdot\vec{b})^2 = |\vec{a}|^2|\vec{b}|^2 \] Reason (R): \[ |\vec{a}\times\vec{b}| = (\vec{a}\cdot\vec{b})\tan\theta, \quad \theta\neq\frac{\pi}{2}. \]

6.
Check whether the function \[ f(x)= \begin{cases} \dfrac{|x-3|}{2(x-3)}, & x<3,\\[6pt] \dfrac{x-6}{6}, & x\geq 3 \end{cases} \] is continuous at \(x=3\) or not.

Similar Mathematics Concepts

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CBSE Class 12 Mathematics Notes Chapter 1 Relations and Functions

Class 12 Mathematics Chapter 1 Notes – Relations and Functions

CBSE CLASS XII Related Questions

1. For \[ f(x)=x+\frac{1}{x}, \quad x\neq 0. \]

2. The least value of \[ f(x)=e^{-x} \] in the interval \[ [0,3] \] is:

3. For a square matrix \(A\), \[ (3A)^{-1}= \]

4. Find: \[ \int \frac{x^2}{(x^2+9)(x^2+16)}\,dx \]

5.For two vectors \(\vec{a}\) and \(\vec{b}\): Assertion (A): \[ |\vec{a}\times\vec{b}|^2+(\vec{a}\cdot\vec{b})^2 = |\vec{a}|^2|\vec{b}|^2 \] Reason (R): \[ |\vec{a}\times\vec{b}| = (\vec{a}\cdot\vec{b})\tan\theta, \quad \theta\neq\frac{\pi}{2}. \]

6.Check whether the function \[ f(x)= \begin{cases} \dfrac{|x-3|}{2(x-3)}, & x<3,\\[6pt] \dfrac{x-6}{6}, & x\geq 3 \end{cases} \] is continuous at \(x=3\) or not.

Similar Mathematics Concepts

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
For \[ f(x)=x+\frac{1}{x}, \quad x\neq 0. \]

2.
The least value of \[ f(x)=e^{-x} \] in the interval \[ [0,3] \] is:

3.
For a square matrix \(A\), \[ (3A)^{-1}= \]

4.
Find: \[ \int \frac{x^2}{(x^2+9)(x^2+16)}\,dx \]

5.
For two vectors \(\vec{a}\) and \(\vec{b}\):

Assertion (A): \[ |\vec{a}\times\vec{b}|^2+(\vec{a}\cdot\vec{b})^2 = |\vec{a}|^2|\vec{b}|^2 \] Reason (R): \[ |\vec{a}\times\vec{b}| = (\vec{a}\cdot\vec{b})\tan\theta, \quad \theta\neq\frac{\pi}{2}. \]

6.
Check whether the function \[ f(x)= \begin{cases} \dfrac{|x-3|}{2(x-3)}, & x<3,\\[6pt] \dfrac{x-6}{6}, & x\geq 3 \end{cases} \] is continuous at \(x=3\) or not.