Equivalence classes for $\mathbb{Z}\times \mathbb{Z}$ where $((a,b),(c,d)) ∈ R$ if and only if $a−d = c−b$

Question

Let $R$ be the relation on $\mathbb{Z}\times \mathbb{Z}$ where $((a,b),(c,d) ∈ R$ if and only if $a−d = c−b$.

(a) Define a function $f$ such that $f(a,b) = f(c,d)$ if and only if $((a,b),(c,d))$ exist in $R$.

Edit: I discovered that $f(x,y) = x+y$ satisfies the function requirement. I no longer need help with this part!

(b) Describe the equivalence classes. How many classes are there and how many elements in each class?

I am completely lost as to how I would even start to approach these problems, especially the equivalence classes because all examples I saw had an ordered pair or a smaller set with them instead of all integers. Any guidance is appreciated!

Answer 1

(I assume that $Z$ means $\mathbb{Z} :=$ the set of integers.)

You have figured out that two pairs of integers are equivalent if they have the same sum.

So, for example, the following are equivalent: $(0,0), (1,-1), (-1,1), (2,-2),(-2,2), \ldots$

They are equivalent because they all have the same sum: $0$

Similarly, you can make a list of all elements that have sum $1$; these will all be equivalent.

Then you could do the same for sum $-1$, or sum $2$, or sum $-2$, or ... well, for any integer.

Hopefully this helps in figuring out part (b) of your question.