Give 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.

Solution:

i. A = {5, 6, 7}

R = {(5, 6), (6, 5)}

(5, 5), (6, 6), (7, 7) ∉ R

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

(5, 6), (6, 5) ∈ R and (6, 5) ∈ R, R is symmetric.

⇒ (5, 6), (6, 5) ∈ R, but (5, 5) ∉ R.

Therefore,

R is not transitive.

Relation R is symmetric but not reflexive or transitive.

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

a ∈ R, (a, a) ∉ R [since a cannot be less than itself]

R is not reflexive.

(1, 2) ∈ R (as 1 < 2)

But 2 is not less than 1

∴ (2, 1) ∈ R

R is not symmetric.

(a, b), (b, c) ∈ R

⇒ a < b and b < c

⇒ a < c

⇒ (a, c) ∈ R

Therefore,

R is transitive.

Relation R is transitive but not reflexive and symmetric.

iii. A = {4, 6, 8}

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

R is reflexive since a ∈ A, (a, a) ∈ R

R is symmetric since(a, b) ∈ R

⇒ (b, a) ∈ R for a, b ∈ R

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

R is reflexive and symmetric but not transitive.

iv. R = {(a, b) : a³ > b³}

(a, a) ∈ R

Therefore,

R is reflexive.

(2, 1) ∈ R

But (1, 2) ∉ R

Therefore,

R is not symmetric.

(a, b), (b, c) ∈ R

⇒ a³ ≥ b³ and b³ < c³

⇒ a³ < c³

⇒ (a, c) ∈ R

Therefore,

 R is transitive.

R is reflexive and transitive but not symmetric.

v. Let A = {- 5, - 6}

R = {(- 5, -6), (- 6, - 5), (- 5, - 5)}

R is not reflexive as (- 6, - 6) ∉ R

(- 5, - 6), (- 6, - 5) ∈ R

R is symmetric.

(- 5, - 6), (- 6, - 5) ∈ R

(- 5, - 5) ∈ R

R is transitive.

Therefore,

R is symmetric and transitive but not reflexive

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NCERT Solutions for Class 12 Maths - Chapter 1 Exercise 1.1 Question 10

Give an example of a relation, which is (i). Symmetric but neither reflexive nor (ii). Transitive but neither reflexive nor (iii). Reflexive and symmetric but not (iv). Reflexive and transitive but not v. Symmetric and transitive but not reflexive.

Summary:

(i). A = {5, 6, 7} R = {(5, 6), (6, 5)} .Relation R is symmetric but not reflexive or transitive. (v) Let A = {- 5, - 6} R = {(- 5, -6), (- 6, - 5), (- 5, - 5)}  R is symmetric and transitive but not reflexive