Proving Equivalence Relations, Constructing and Defining Operations on Equivalence Classes I think I have an intuitive sense of how ordered pairs can function to specify equivalence classes when used in the construction of integers and rationals, for example. I put the cart before the horse, however, and am less well versed in how to prove an equivalence relation, construct equivalence classes, and define operations on equivalence classes. I have received feedback that one may not define operations on equivalence classes by appealing to individual elements (e.g., it is insufficient to indicate [(a,b)] + [(c,d)] = [(a+c,b+d)]). How does one prove equivalence relations, construct equivalence classes, and define operations on those classes? If the answer is too long for this forum, is there a good demonstration available online?

I found this resource which seems beginner-level and has worked examples specifically regarding integer construction: https://www.math.wustl.edu/~freiwald/310integers.pdf
