Understanding an example of equivalence relation On $\mathbb{N}\times\mathbb{N}$, Let $\left( a,b\right) \equiv \left( c,d\right) \Leftrightarrow a+d=c+b$.
Show that it is equivalence relation. Find equivalence of class of this.
My answer.
For all $a,b\in\mathbb{N}$, we have $a+b=a+b$. Clear. So, The relation is reflexivity.
For all $a,b,c,d\in\mathbb{N}$, we assume $a+b=c+d$. Hence, also we know $c+d=a+b$. Thus, the relation is symetric.
For all $a,b,c,d,e,f\in\mathbb{N}$, we assume $a+d=c+b$ and $c+f=e+d$. We will show that $a+f=e+b$. Sum of both of sides $a+b=c+d$ and $c+f=e+d$, then we obtain $(a+f)+(c+d)=(c+d)+(e+b)$. Hence, by the cancelation, we obtain $(a+f)=(e+b)$. Thus, The relation $R$ is transitivity.
Therefore, the relation $R$ is equivalence relation. Can you check my answer,so, what is the equivalence classes of this?

Note that $a+d=c+b$ is true iff $a-b=c-d$. Let us interpret a pair such as $(a,b)$ as: $a$ is the amount of income this month, $b$ as the expenditure this month for a single person. We say two persons are equivalent if the money they saved this month (that is income minus the expenses)  is the same for these two persons.
Now have fun solving your problem  without using plus and minus signs or brackets.
