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Relation R in the set A of human beings in a town at a particular time given by R = {(x, y) : x is exactly 7 cm taller than y}. - Mathematics

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Question

Determine whether the following relation is reflexive, symmetric and transitive:

Relation R in the set A of human beings in a town at a particular time given by R = {(x, y) : x is exactly 7 cm taller than y}.

Sum
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Solution

(i) Reflexive:

R = {(x, y) : x is exactly 7 cm taller than y}

Now, (x, x) ∉ R

Since a human being (x) cannot be taller than himself.

∴ R is not reflexive.

(ii) Symmetric:

Now, let (x, y) ∈ R

⇒ x is exactly 7 cm taller than y.

Then, y is not taller than x.

∴ (y, x) ∉ R

Indeed, if x is exactly 7 cm taller than y, then y is exactly 7 cm shorter than x.

∴ R is not symmetric.

(iii) Transitive:

Now, let (x, y), (y, z) ∈ R

⇒ x is exactly 7 cm taller than y, and y is exactly 7 cm taller than z.

⇒ x is exactly 14 cm taller than z.

∴ (x, z) ∉ R

∴ R is not transitive.

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

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Chapter 1: Relations and Functions - Exercise 1.1 [Page 5]

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NCERT Mathematics Part 1 and 2 [English] Class 12
Chapter 1 Relations and Functions
Exercise 1.1 | Q 1. 5. (c) | Page 5

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