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I Quote: "let $X$ be any set, and let $R$ be the set of all those pairs $(x, y)$ in $X\times X$ for which $x = y$. The relation $R$ is just the relation of equality between elements of $X$; if $x$ and $y$ are in $X$, then $xRy$ means the same as $x = y$.

my question is about "for which $x = y$", what does it mean for two elements to be equal?

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It means $\forall a : a\in x \Leftrightarrow a\in y$. But on a more serious note, I think you are just a bit confused. The relation $R$ would be commonly written as $$R = \{ (x,x) : x\in X\}$$

For example, if $X= \{1,2,3\}$, then $R = \{(1,1),(2,2),(3,3)\}$.

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  • $\begingroup$ Also note that $(\forall \, a\,( a\in x \iff a \in y)) \rightarrow x = y$ is an axiom. $\endgroup$ – YoTengoUnLCD Jul 1 '16 at 15:47

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