I have a relation where xβy if and only if x is a multiple of y. β is defined on the set of integers, The book says it is not antisymmetric. However when i test it, i came out with this:
Let x = ky and y = lx, which leads to x = klx. Divide both side by x, i get 1=kl, which therefore prove that x is equal to y. and i thought it is a anti-symmetric, But the book says its not antisymmetric, is there something wrong for my answer? thank You!
The problem is that $kl=1$ doesn't imply k=l=1 but holds for $k=l=-1$ too. In fact $x=-2$ and $y=2$ leads to a counterexample, try it.