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?

Equivalence relations can be defined on any set $X$.
For a relation $R$ on a set $X$, $R$ is an equivalence relation if $R$ is reflexive, symmetric, and transitive. So to check/prove that a relation $R$ on $X$ is an equivalence relation, you need to check that $R$ satisfies those three properties.
Now given an equivalence relation $R$ on $X$, for any $x\in X$, the equivalence class of $x$ is $[x]_R=\{y\in X:yRx\}$. So the equivalence class of some $x\in X$ is the set of all $y\in X$ that are related to $x$. These equivalence classes are not really "constructed" as much as determined by $R$.
And you can define whatever operation you want on equivalence classes.
EXAMPLE: Let $X$ be the set of strings of length 1 to 4 formed by the English alphabet and define a relation $R$ on $X$ by $xRy$ iff $x$ has the same length as $y$. You can show that $R$ is an equivalence relation.  
Now take any string from $X$, say 'xyza', then the equivalence class of 'xyza' is $[xyza]_R=\{y\in X:yRxyza\}$, the set of strings with the same length as 'xyza', i.e. the set of strings with length 4.
There are many operations you can define on the equivalence classes but the important thing is that any such operation can only have one output. You could say, given two equivalence classes $[x]$ and $[y]$, define the operation $+$, by $[x]+[y]=[xy]$. So $+$ outputs the set of all strings with the same length as 'xy', the string formed by concatenating 'x' and 'y'.
