Suppose $R_1$ is a $n$-ary relation which is a subset of $\prod_{i=1}^n A_i$ and $R_2$ is a $m$-ary relation which is a subset of $\prod_{i=1}^m B_i$. The equality for two relations is defined as, $R_1=R_2$ if $n=m$ and $A_i=B_i$ for all $1\le i\le n$.
So, if we consider $A=\{1,2\}$ and $B=\{a,b\}$ and $C=\{a,b,c\}$. Now, suppose $R_1$ is the relation, which is a subset of $A\times B$ is $\{(1,a),(2,b)\}$ and $R_2$ is a subset of $A\times C$ and also $\{(1,a),(2,b)\}$. Here as $B\neq C$, we don't have $R_1=R_2$ from the definition! But as a set they are same!
Can anyone explain me why equality of relation is defined like that? In other word if we have defined $R_1=R_2$ if they are equal as set then where we are doing wrong? am I missing something about definition of relation?
Your are correct that as sets $R_1$ and $R_2$ are the same. As you say, our definition of equality in relations demands that the domain and range sets be the same. A relation is defined as a triple of domain, range, and the pairs in the relation. In your example $R_1$ is surjective and $R_2$ is not.