Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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:

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?

share|cite|improve this question
To get the curly braces you have to use \{ and \}; plain { and } vanish. – Brian M. Scott Nov 7 '13 at 13:13
up vote 0 down vote accepted

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

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

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.