Sets in themselves do not have "equivalence" or "inverses", and they cannot be "reflexive", "symmetrical" or "transitive". We must take the set under a relation. For instance, let us examine the set you have given under the relation "=". Clearly, for we have that (reflexive), if then (symmetric), and if and then . Thus, "=" is called an equivalence relation (an absurdly simple example, I know).
I'm not entirely sure what you mean by inverses. If we take a set under an operation, for instance the set under addition modulo 3 then you have an identity element, (an identity element is a neutral element - ), and every element has an inverse (an element that will take the element back to the inverse): the inverse of 1 is 2 and the inverse of 2 is 1 as .
Is that what you are looking for?