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Defines a relation on N: xy is square of an integer, x, y ∈ N Determine the above relation is reflexive, symmetric and transitive.

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Question

Defines a relation on N:

xy is square of an integer, x, y ∈ N

Determine the above relation is reflexive, symmetric and transitive.

Sum
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Solution

On natural numbers N:

xRy if xy is a perfect square

Check Each Property:

Reflexive: 
x ⋅ x = x2 is always a perfect square.
So, the relation is reflexive.

Symmetric:

If xy is a perfect square, then yx is the same.
So, the relation is symmetric.

Transitive (Not always):

Even if xy and yz are perfect squares,

xz may not be a perfect square.

(Example: x = 1, y = 4, z = 2:
xy = 4, yz = 8 → but yz is not a perfect square)

The relation is reflexive and symmetric, but not transitive.

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Chapter 1: Relations - Exercise 1.1 [Page 11]

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R.D. Sharma Mathematics Volume 1 and 2 [English] Class 12
Chapter 1 Relations
Exercise 1.1 | Q 18.3 | Page 11

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