Definition: The set of integers $\mathbb{Z}$ is defined via the set $\mathbb{N} \times \mathbb{N}$ where we specify the ordered pairs $(a,b)$ and $(c,d)$ are equivalent iff $a+b=c+d$ where addition and multiplication is defined as $(a,b)+(c,d)=(a+c,b+d)$ and $(a,b)(c,d)=(ac+bd,ad+bc)$
Show that the relation $R$ defined as $(a,b)R(c,d)$ iff $a+d=b+c$ is an equivalence relation. And that multiplication and addition are well defined.
To show its reflexive I need to show that $(a,b)R(a,b)$ well clearly $a+b=a+b$ so the relation is reflexive.
Now to show its symmetric I need to show that $(a,b)R(b,a)$ thus $a+b=b+a$ since addition is commutative this is the same as $a+b=a+b$ thus the relation is symmetric.
To show it is transitive I need to show that if $(a,b)R(b,c)$ and $(b,c)R(c,d)$ then $(a,b)R(c,d)$. Well then we know that $a+b=b+c$ and $b+c=c+d$ thus by substitution $a+b=c+d$ so $(a,b)R(c,d)$. Since the relation is reflexive, symmetric and transitive then it is an equivalence relation.
Now to show its well defined I need to show that the result does not depend on the representative from each equivalence class.
Thus to show addition is well defined if $(a,b)$ and $(k,l)$ are equivalent and pairs $(c,d)$ and $(m,n)$ are equivalent, then $(a,b)+(c,d)$ and $(k,l)+(m,n)$ are equivalent.
Thus if $(a,b)$ and $(k,l)$ are equivalent then $a+b=k+l$ likewise if $(c,d)$ and $(m,n)$ are equivalent, then $c+d=m+n$. Then $(a,b)+(c,d)=(a+c,b+d)$ and $(k,l)+(m,n)=(k+m,l+n)$
Now when trying to do the algebra I keep running into problems I tried to separate the variables and add them together like for the first coordinate I got that $a+b=k+l+m+n-(b+d)$ and $k+m=k+l+m+n-(l+m)$ which would work if the second coordinate was the same but they trying to show that I end up in the same issue that it holds if the first coordinates are the same.
Now for multiplication I need to show that $(a,b)(c,d)$ and $(k,l)(m,n)$ are equivalent then $(ac+bd,ad+bc)=(km+lm,kn+lm)$ again the algebra was leading to lots of issues so I figure this is not the best approach.
Any help with the well defined portion would be appreciated or if I just need to mess with the equations more or if there is a better approach.