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Each of the following defines a relation on N: x y is square of an integer x, y ∈ N Determine whether above relation is reflexive, symmetric and transitive.

 Each of the following defines a relation on N: 
  x y is square of an integer x, y ∈ N 
 
Determine  whether above relation is  reflexive, symmetric and transitive.

Grade:12

1 Answers

Harshit Singh
askIITians Faculty 5963 Points
2 years ago
Dear Student

Given, xy is square of an integer x, y∈N

R = {(x, y) : xy is a square of an integer x, y∈N}

It’s clearly (x, x)∈R,∀x∈N

As x^2is square of an integer for any x∈N

Thus, R is reflexive.

If (x, y)∈R⇒(y, x)∈R

So, R is symmetric.

Now, if xy is square of an integer and yz is square of an integer.

Then,let xy=m^2and yz= n^2for some m,n∈Z

x = m^2/y and z = x^2/y


xz = m^2 n^2 / y^2 , which is square of an integer.

Thus, R is transitive.

Thanks

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