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I'm studying about equivalence relations. My book has the following definition for an equivalence class:

If $R=(G,A,A)$ is a relation of equivalence over the set $A$, the equivalence class of $a$ is denoted as $[a]$ is the set

$$[a] = \{b \in A : a\mathbin{R}b\}$$

Recently, I found a document online about the topic: http://www2.uca.es/matematicas/Docencia/ESI/1710003/Apuntes/Leccion8.pdf

The document is in Spanish, but here is its definition for an equivalence class:

If $R$ is a relation of equivalence over a set $A$, for each $a \in A$, we'll call the equivalence of $a$ to the set formed by all elements in $A$ that are related to it. It will be denoted $[a]$, that is:

$$[a] = \{x \in A : x\mathbin{R}a\}$$

I am a bit confused now. It seems to me that both documents' definitions don't quite match.

This is just an idea, but maybe it doesn't matter, since the relation is supposed to be symmetric?

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  • $\begingroup$ To get the curly braces you have to use \{ and \}; plain { and } vanish. $\endgroup$ Commented Nov 7, 2013 at 13:13

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Your idea in the last line is correct: we talk about equivalence classes only when we have an equivalence relation, which by definition is symmetric. Thus, the two definitions are equivalent: for any $a,x\in A$, $a\mathbin{R}x$ if and only if $x\mathbin{R}a$, and therefore

$$\{x\in A:a\mathbin{R}x\}=\{x\in A:x\mathbin{R}a\}\;.$$

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