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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.
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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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