Two relations defined on same sets be equal? We say two relations $R_1$ and $R_2$ are equal if:
$$R_1\subseteq \prod_{i=1}^n A_i$$
$$R_2\subseteq \prod_{i=1}^m B_i$$
then,
$$n = m ,$$
$$A_i=B_i\quad \forall i,\,1\le i \le n,$$
$$let,s_j\,\,and\,\,t_j\,\,be\,\,tuples\,\,of\,\,R_1\,\,and\,\,R_2\,\,respectively$$
$$s.t.,\,\,s_j=&lt;a_1,a_2,a_3,\ldots,a_j&gt;\epsilon\,R_1\land$$
$$t_j=&lt;b_1,b_2,b_3,\ldots,b_j&gt;\epsilon\,R_2$$
$$\Rightarrow\,(a_1=b_1\,\land a_2=b_2\,\land\ldots\land a_n=b_n)$$
$$\forall \,j\,\epsilon \mathbb N)$$
where $A_i$ and $B_i$ are sets.
So, will the below relations be equal?
$$Name = \{x,y\}\,\,and\,\,City = \{a,b\}$$
let, $$R_1=\{&lt;p,q&gt;|\quad p\,\epsilon\,Name\land q\,\epsilon\,City,\quad p\,\,was\,\,born\,\,in\,\,city\,\,q\}$$
$$R_2=\{&lt;p,q&gt;|\quad p\,\epsilon\,Name\land q\,\epsilon\,City,\quad p\,\,lives\,\,in\,\,city\,\,q\}$$
suppose we get,
$$R_1=\{&lt;x,a&gt;,&lt;y,b&gt;\}$$
$$R_2=\{&lt;x,a&gt;,&lt;y,b&gt;\}$$
That is, x was born in the same city which he/she lives in.
Can we say,$$R_1=R_2$$
The relations $R_1$ and $R_2$ are equal if and only if both $x$ and $y$ live in the same city as they were born in. That is, for all $p\in\{x,y\}$ and all $q\in\{a,b\}$ we have
$$
\text{$p$ was born in city $q$} \quad\Longleftrightarrow\quad \text{$p$ lives in city $q$.}
$$
You wrote down the same set of pairs to define $R_1$ and $R_2$, so indeed they are equal.
Equality of relations is a special case of equality of sets, and the sets you wrote down are equal.
The relations $R_1$ and $R_2$ are equal if and only if both $x$ and $y$ live in the same city as they were born in. That is, for all $p\in\{x,y\}$ and all $q\in\{a,b\}$ we have
$$
\text{$p$ was born in city $q$} \quad\Longleftrightarrow\quad \text{$p$ lives in city $q$.}
$$
You wrote down the same set of pairs to define $R_1$ and $R_2$, so indeed they are equal.
Equality of relations is a special case of equality of sets, and the sets you wrote down are equal.