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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.
Dear StudentGiven, xy is square of an integer x, y∈NR = {(x, y) : xy is a square of an integer x, y∈N}It’s clearly (x, x)∈R,∀x∈NAs x^2is square of an integer for any x∈NThus, R is reflexive.If (x, y)∈R⇒(y, x)∈RSo, 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∈Zx = m^2/y and z = x^2/yxz = m^2 n^2 / y^2 , which is square of an integer.Thus, R is transitive.Thanks
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