NCERT solution Class 12 relations and functions

NCERT Solutions

Class 12 Maths

Relations and Functions

NCERT Questions

Ex. 1.1 Q1
Ex. 1.1 Q2
Ex. 1.1 Q3
Ex. 1.1 Q4
Ex. 1.1 Q5
Ex. 1.1 Q6
Ex. 1.1 Q7
Ex. 1.1 Q8
Ex. 1.1 Q9
Ex. 1.1 Q10
Ex. 1.1 Q11
Ex. 1.1 Q12
Ex. 1.1 Q13
Ex. 1.1 Q14
Ex. 1.1 Q15
Ex. 1.1 Q16
Ex. 1.2 Q1
Ex. 1.2 Q2
Ex. 1.2 Q3
Ex. 1.2 Q4
Ex. 1.2 Q5
Ex. 1.2 Q6
Ex. 1.2 Q7
Ex. 1.2 Q8
Ex. 1.2 Q9
Ex. 1.2 Q10
Ex. 1.2 Q11
Ex. 1.2 Q12
Ex. 1.3 Q1
Ex. 1.3 Q2
Ex. 1.3 Q3
Ex. 1.3 Q4
Ex. 1.3 Q5
Ex. 1.3 Q6
Ex. 1.3 Q7
Ex. 1.3 Q8
Ex. 1.3 Q9
Ex. 1.3 Q10
Ex. 1.3 Q11
Ex. 1.3 Q12
Ex. 1.3 Q13
Ex. 1.3 Q14
Ex. 1.4 Q1
Ex. 1.4 Q2
Ex. 1.4 Q3
Ex. 1.4 Q4
Ex. 1.4 Q5
Ex. 1.4 Q6
Ex. 1.4 Q7
Ex. 1.4 Q8
Ex. 1.4 Q9
Ex. 1.4 Q10
Ex. 1.4 Q11
Ex. 1.4 Q12
Ex. 1.4 Q13
Misc. Ex. Q1
Ex. 1.4 Q2
Ex. 1.4 Q3
Misc. Ex. Q4
Misc. Ex. Q5
Misc. Ex. Q6
Misc. Ex. Q7
Misc. Ex. Q8
Misc. Ex. Q9
Misc. Ex. Q10
Misc. Ex. Q11
Misc. Ex. Q12
Misc. Ex. Q13
Misc. Ex. Q14
Misc. Ex. Q15
Misc. Ex. Q16
Misc. Ex. Q17
Misc. Ex. Q18
Misc. Ex. Q19

Ex. 1.1 Q10

Given an example of a relation. Which is

(i) Symmetric but neither reflexive nor transitive.

(ii) Transitive but neither reflexive nor symmetric.

(iii) Reflexive and symmetric but not transitive.

(iv) Reflexive and transitive but not symmetric.

(v) Symmetric and transitive but not reflexive.

(i) Let A = {5, 6, 7}.

Define a relation R on A as R = {(5, 6), (6, 5)}.

Relation R is not reflexive as (5, 5), (6, 6), (7, 7) ∉ R.

Now, as (5, 6) ∈ R and also (6, 5) ∈ R, R is symmetric.

⇒ (5, 6), (6, 5) ∈ R, but (5, 5) ∉ R

So, R is not transitive.

Hence, relation R is symmetric but not reflexive or transitive.

(ii) Consider a relation R in R defined as:

R = {(a, b): a < b}

For any a ∈ R, we have (a, a) ∉ R since a cannot be strictly less than a itself.

In fact, a = a.

So, R is not reflexive.

Now, (1, 2) ∈ R (as 1 < 2)

But, 2 is not less than 1

So, (2, 1) ∉ R

Hence, R is not symmetric.

Now, let (a, b), (b, c) ∈ R.

⇒ a < b and b < c

⇒ a < c

⇒ (a, c) ∈ R

So, R is transitive.

Hence, relation R is transitive but not reflexive and symmetric.

(iii) Let A = {4, 6, 8}

Define a relation R on A as

A = {(4, 4), (6, 6), (8, 8), (4, 6), (6, 4), (6, 8), (8, 6)}

Relation R is reflexive since for every a ∈ A, (a, a) ∈ R

i.e. {(4, 4), (6, 6), (8, 8)} ∈ R.

Relation R is symmetric since (a, b) ∈ R ⇒ (b, a) ∈ R for all a, b ∈ R

Relation R is not transitive since (4, 6), (6, 8) ∈ R, but (4, 8) ∉ R

Hence, relation R is reflexive and symmetric but not transitive.

(iv) Define a relation R in R as:

R = {a, b): a³ ≥ b³}

Clearly (a, a) ∈ R as a³ = a³

So, R is reflexive.

Now, (2, 1) ∈ R [as 2³ ≥ 1³]

But, (1, 2) ∉ R [Since 1³ < 2³]

So, R is not symmetric.

Now, Let (a, b), (b, c) ∈ R.

⇒ a³ ≥ b³ and b³ ≥ c³

⇒ a³ ≥ c³

⇒ (a, c) ∈ R

So, R is transitive.

Hence, relation R is reflexive and transitive but not symmetric.

(v) Let A = {−5, −6}

Define a relation R on A as

R = {(−5, −6), (−6, −5), (−5, −5)}

Relation R is not reflexive as (−6, −6) ∉ R

Relation R is symmetric as (−5, −6) ∈ R and (−6, −5) ∈ R

It is seen that (−5, −6), (−6, −5) ∈ R

Also, (−5, −5) ∈ R

So, the relation R is transitive.

Hence, relation R is symmetric and transitive but not reflexive.

Relations and Functions Class 12 Maths NCERT Chapter 1 NCERT Solutions

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Review and Revise: Regularly review the concepts and solutions you've learned. Revise your notes and practice regularly to retain the knowledge effectively. You may refer Relations and Functions Class 12 Maths Notes.
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NCERT solutions of Relations and Functions Class 12 Maths - all questions of all exercises are solved. Q. No. 1, 2, 3, and so on. Complete explanation and detailed solution makes them the best videos for Class 12 Maths Relations and Functions NCERT solutions.