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Question
The following defines a relation on N:
x y is square of an integer x, y ∈ N
Determine which of the above relations are reflexive, symmetric and transitive.
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Solution
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 x2 is 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 = m2 and yz = n2 for 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.
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