Definition: $(a, b) \sim (c, d)$ iff $a + d = b + c$. Explain how this relation represents subtraction. (Do not prove that it's an equivalence relation.)
I'm not sure how to answer this/what the final step looks like so any help would be appreciated. I believe you set each side equal to a single variable first? Could be completely wrong
I'm interpreting this question as charitably as possible. Unfortunately, it seems from the comments that your instructor has provided you with vague instructions.
The relation $\sim$ on $\mathbb{N}\times\mathbb{N}$ allows you to represent subtraction as follows: for a given natural number $n\in\mathbb{N}$, write $n$ as $[(n,0)]$ (the equivalence class of $(n, 0)$). Then for any two natural numbers $n, m$, we can simply define $n-m = [(n, 0)]- [(m, 0)] = [(n, m)]$. One can easily check that this is well-defined. This interpretation of subtraction hinges on the idea of $(a, b)$ being considered "large" if $a$ is much bigger than $b$. It also allows the representation of negative numbers by using pairs where $b>a$.
That's about the best I can give from the context provided; since a relation isn't an operation, all I can really do is show you how the operation is actually defined. But my real advice is to ask for clarification from your instructor directly.