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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, $$n = m ,$$ $$A_i=B_i\quad \forall i,\,1\le i \le n,$$ $$let,s_j\,\,and\,\,t_j\,\,be\,\,tuples\,\,of\,\,R_1\,\,and\,\,R_2\,\,respectively$$ $$s.t.,\,\,s_j=<a_1,a_2,a_3,\ldots,a_j>\epsilon\,R_1\land$$ $$t_j=<b_1,b_2,b_3,\ldots,b_j>\epsilon\,R_2$$ $$\Rightarrow\,(a_1=b_1\,\land a_2=b_2\,\land\ldots\land a_n=b_n)$$ $$\forall \,j\,\epsilon \mathbb N)$$

where $A_i$ and $B_i$ are sets.

So, will the below relations be equal? $$Name = \{x,y\}\,\,and\,\,City = \{a,b\}$$ let, $$R_1=\{<p,q>|\quad p\,\epsilon\,Name\land q\,\epsilon\,City,\quad p\,\,was\,\,born\,\,in\,\,city\,\,q\}$$ $$R_2=\{<p,q>|\quad p\,\epsilon\,Name\land q\,\epsilon\,City,\quad p\,\,lives\,\,in\,\,city\,\,q\}$$ suppose we get, $$R_1=\{<x,a>,<y,b>\}$$ $$R_2=\{<x,a>,<y,b>\}$$ That is, x was born in the same city which he/she lives in.
Can we say,$$R_1=R_2$$

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  • $\begingroup$ Your definition of equality of relations is wrong. It only accomplishes that both relations are subsets of the same product, not actually equal. $\endgroup$
    – Christoph
    Commented Dec 26, 2020 at 10:57
  • $\begingroup$ Thanks, @Christoph I totally skipped that one condition. I have edited the question. $\endgroup$ Commented Dec 26, 2020 at 11:58
  • $\begingroup$ Your condition on tuples when defining equality of relations is not what you want to say. It implies that $R_1=R_2$, but also that this relation contains only a single tuple. $\endgroup$ Commented Dec 26, 2020 at 15:49
  • $\begingroup$ @MaliceVidrine I have edited the condition. I hope I have removed any sort of ambiguity. $\endgroup$ Commented Dec 26, 2020 at 18:43
  • $\begingroup$ Also let me clear my doubt further. I was confusing equality of two relations with them being same. I think my concepts are pretty much clear now. $\endgroup$ Commented Dec 26, 2020 at 18:57

1 Answer 1

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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.

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