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Let A Be the Set of All Human Beings in a Town at a Particular Time. Determine Whether Each of the Following Relations Are Reflexive, Symmetric and Transitive : R = {(X, Y) : X And Y Work at the Sam - Mathematics

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Question

Let A be the set of all human beings in a town at a particular time. Determine whether the following relation is reflexive, symmetric and transitive:

 R = {(x, y) : x and y work at the same place}

Sum
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Solution

(i) Reflexivity: 

Let x be an arbitrary element of R.

Then, x R   

x and x work at the same place, which is true since they are the same.

(x, xR

R is a reflexive relation.

(ii) Symmetry:

Let (x, y) R

x and y work at the same place.

y and x work at the same place.

(y, x) R

R is a symmetric relation.

(iii) Transitivity:

Let (x, y) R and (y, z) R.

Then, x and y work at the same place.

y and z also work at the same place.

⇒ x, y and z all work at the same place.

x and z work at the same place.

⇒ (x, z) R

∴ R is a transitive relation.

Hence, R is reflexive, symmetric and transitive.

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Chapter 1: Relations - Exercise 1.1 [Page 10]

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RD Sharma Mathematics [English] Class 12
Chapter 1 Relations
Exercise 1.1 | Q 1.1 | Page 10
NCERT Mathematics Part 1 and 2 [English] Class 12
Chapter 1 Relations and Functions
Exercise 1.1 | Q 1. 5. (a) | Page 5

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