Equivalence Relations and Equivalence Classes We will define the relation ~ on $\mathbb N \times \mathbb N$ by $(a,b)\sim (c,d)$ iff $a + d = b + c$.
Prove that the operation given by: $[(a,b)][(c,d)] \stackrel{\text{def}}= [(ac + bd, ad + bc)]$ is well-defined.
My attempt at this proof:
Let $(a,b)$ and $(a',b')$ be elements of $[(a,b)]$, and similarly, $(c,d)$ and $(c',d')$ be elements of $[(c,d)]$.
My guess is that we need to show that:
$(a'c' + b'd', a'd' + b'c')$ is an element of $[(ac + bd, ad + bc)]$.
So we know the following:
$$aa' + bb' = ab' + ba' \\ cc' + dd' = cd' + bc'$$
But now I'm wondering what to work with. It would be great if someone could give some assistance. Thank you!
Edit: I am also seeking assistance on the following problems:
Q: Prove that $\mathbb N \times \mathbb N$/~ contains an additive identity, i.e. find an element [(i, j)] ∈ $\mathbb N \times \mathbb N$/~ with the property that
$$[(i,j)] + [(c,d)] = [(c,d)] $$
Q: Prove that every element of $\mathbb N \times \mathbb N$/~ has an additive inverse, i.e. for any $[(a,b)] \in \mathbb N \times \mathbb N/~$, show that there exists $[(c,d)] \in \mathbb N \times \mathbb N/~$ such that
$$[(a,b)] + [(c,d)] = [(i,j)]$$ where [(i,j)] is the additive identity.
 A: What is it all about? We want to introduce the negative numbers, constructed somehow using only the naturals. An integer is represented by $(a,b)$ and it wants to be $a-b$ which not necessarily exist yet (in $\mathbb N$), and now we are only allowed to use $+$ and natural numbers.
How did you get the last 2 equations? They should be $a+b' = b+a'$ and $c+d' = d+c'$..
When $(a,b)\sim(a',b')$, either $a\le a'$ or $a'<a$, because of symmetry, we may also assume the former, if it makes you more comfortable. Then $a'=a+n$ for some $n\in\mathbb N$, and hence, using the defn. of $\sim$, we also have $b'=b+n$. Similarly, we can assume that $c\le c'$ and hence $c'=c+m$ and $d'=d+m$. And.. probably the best is to make one step at one time:
First assume that $C:=(c,d)$ is fixed, and $A:=(a,b)\sim (a',b')=:A'$, then show that $AC\sim A'C$. Finally this same step applies for $A'C\sim A'C'$.
For the rest two questions, guess what integers are represented by the following pairs: $(149,149)$, $(2,7)$, $(7,2)$.. hope it helps
