Discrete Math: Unary and binary predicates?

Recall that a natural number x is a multiple of a natural number y if there exists a natural number n such that x=ny. Allowing the variables x and y to range over the domain of all natural numbers, let P(x,y) denote the binary predicate "y is a multiple of x." Is P (x,y) an equivalence relation?|||To show P(x, y) is an equivalence relation we need to show reflexivity, symmetry, and transitivity.





Let x, y, z be natural numbers.





Reflexive: Is P(x, x)? Is "x a multiple of x"? Does there exist a natural number n such that x = x*n? Yes, take n = 1. Thus, P(x, x) is reflexive.





Symmetric: Does P(x, y) imply P(y, x)? The answer is no here. But we need to provide a counterexample. Take x = 4 and y = 2. Need to show that P(4, 2) is true but P(2, 4) is false.





P(4, 2) =%26gt; 4 is a multiple of 2 =%26gt; there exists natural number n such that 4 = 2n We see this holds by taking n = 2.





P(2, 4) =%26gt; 2 is a multiple of 4 =%26gt; there exists natural number n such that 2 = 4n The only solution here is n = 1/2 but that is not a natural number. Thus, this fails to hold.





Therefore, P(x, y) is not symmetric, thus not an equivalence relation.|||It's reflexive and transitive, but it's not symmetric; for example,


P(1,2) but not P(2,1).





Therefore, P does not meet the requirements for an equivalence relation.