Let S=Z\{0}. Define a relation ~ on S by a~b if ab>0. Is ~ an equivalence relation? If so, describe the equivalence classes. My solution: For reflexive, we want to show that a~a. If a s a negative integer, then aa>0. If a is a positive integer, then aa>0. So, a~a. For symmetric, we want to show that a~b implies b~a. Since multiplication over the integers is commutative, ba>0 since neither a nor b can be 0. I am stuck on the transitive property. Is what I have correct so far, and will you help with the last property? I think overall ~ is an equivalence relation, but I am also not sure how to get the equivalence classes. Thanks!