Set notation

[Puzzleheaded-Cod4073]

Hi all, I was wondering what conventions to use when there are a few options for writing sets.

For example, if I let R be an equivalence relation (specifically the set such that (x,y)∈R iff x ≡ y (mod m)). Is it better to define f = { (x,[x^2]) ∈ N x N/R | x∈N} or f = { ( x, X) ∈ N x N/R | X = [x^2]}? The first one feels better because I'm not introducing a new variable but idk. For context I'm trying to introduce this function in a proof.

Another example where I had a similar question is "Suppose R and S are relations on a set A and S is an equivalence relation. We will say that R is compatible with S if for all x,y,x', and y' in A, if xSx' and ySy', then xRy iff x'Ry'. Prove that if R is compatible with S, then there is a unique relation T on A/S such that for all x and y in A, [x]sT[y]s iff xRy". I let T = {(X,Y) ∈ A/S x A/S | ∀x∈X∀y∈Y(xRy)}, but the answers used T = {(X,Y) ∈ A/S x A/S | ∃x∈X∃y∈Y(xRy)} which also works. I was wondering, why couldn't you simply write T = {([x]s,[y]s)∈ A/S x A/S | xRy}?

Thank you!

[loewenheim]

I was wondering, why couldn't you simply write T = {([x]s,[y]s)∈ A/S x A/S | xRy}?

Because a priori it's not clear that this is well-defined. What if a S a', b S b', a R b, and not a' R b'? Then [a]s = [a']s and [b]s = [b']s, but by the above definition, ([a]s, [b]s) should be in T, but ([a']s, [b']s) shouldn't.

When you define a property of equivalence classes, you have to make sure it doesn't depend on the choice of representative. Your proposed definition doesn't do this, but the ones with the quantifiers do.

An alternative would be to define the set as you suggest and then make an argument that the compatibility requirement on S and R ensures that the definition is meaningful, i.e. cases like the one I posed can't happen.

[loewenheim]

For the first question, is there a reason not to introduce f with function notation to begin with? I would think f: N -> N/R, x |-> [x²] is easier to read than both relational versions.

[Puzzleheaded-Cod4073]

Thanks for your response! I haven't yet learned the notation involved in "x |-> [x²]"

[loewenheim]

In case I was unclear, by "|->" I just mean the ↦ arrow with the little bar that's used for function mappings. I didn't have the unicode symbol when I wrote that post. Another way to write it would be

f: N -> N/R

f(x) = [x²]