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Given an example of a relation. Which is reflexive and transitive but not symmetric.

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Question

Given an example of a relation. Which is reflexive and transitive but not symmetric.

Sum
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Solution

Define a relation R in R as:

R = {a, b): a3 ≥ b3}

Clearly (a, a) ∈ R as a3 = a3.

∴ R is reflexive.

Now,

(2, 1) ∈ R (as 23 ≥ 13)

But,

(1, 2) ∉ R (as 13 < 23)

∴ R is not symmetric.

Now,

Let (a, b), (b, c) ∈ R.

⇒ a3 ≥ b3 and b3 ≥ c3

⇒ a3 ≥ c3

⇒ (a, c) ∈ R

∴ R is transitive.

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

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Chapter 1: Relations and Functions - EXERCISE 1.1 [Page 6]

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NCERT Mathematics Part 1 and 2 [English] Class 12
Chapter 1 Relations and Functions
EXERCISE 1.1 | Q 10. (iv) | Page 6

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