Meet Bob. Bob works in a warehouse gathering items to ship out for fulfillment of online orders. Customers hate to pay for extra shipping, so companies like Bob’s often group items going to the same destination together into the same box to save on shipping costs. The criteria for grouping sound simple: an item can only be placed in a box with another item if both items are going to the same destination. Whether Bob realizes it or not, he is performing an equivalence relation to the shipping orders he fills.
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Zone of Mr. Frosti 
Um, when you say “This class would need to hold at the very least the information that is necessary for us to determine its equivalence to other items of this class. This will serve as our equivalence class,” you are mixing up terminology at best. You may have defined a Ruby class that defines an equivalence relation[1], but you have not necessarily defined an equivalence class. To do so, you must first define you super set S (say, all items in the warehouse) and then show that all a and b in S are symmetric, transitive, and reflexive under you equivalence relation. Only then can you say that you have defined an equivalence class.
[1] — http://planetmath.org/?method=l2h&id=349&op=getobj&from=objects
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Further, a more rigorous definition of equivalence classes can be found at [2]. You should always refer to Planet Math over Wikipedia when discussing mathematics at such a level. The Wikipedia articles are not worth much of anything.
[2] — http://planetmath.org/?method=l2h&id=468&op=getobj&from=objects
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