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For example: the relation given is $x\sim y$ if $f(x)=f(y)$.

What do you have to say when describing a equivalence class?

user3209698
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Young
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    "The equivalence classes are separated into those values of $x$ for which their output are the same." For example, with $f(x)=x^2$ you have $0$ is in an equivalence class of its own, $1$ and $-1$ share an equivalence class, $4$ and $-4$ share an equivalence class, ... in general for that specific $f$ there will be an equivalence class of $y$ for every non-negative real number $y$, which will be made up of two elements: $\sqrt{y}$ and $-\sqrt{y}$. In general, the equivalence classes are "the different groups" you can place things into so that they are grouped with what they are equivalent to. – JMoravitz Apr 12 '16 at 21:45
  • Can you describe the equivalence classes of all the function by a simple notation? – Young Apr 12 '16 at 21:47
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    There is notation for it, but it is no more meaningful than saying the sentence "the set of all equivalence classes (of $X$ for a relation $\sim$)." In mathematical notation, you could write it as $X/\sim$ – JMoravitz Apr 12 '16 at 21:49
  • Worth to say that, in JMoravitz example, it does not care the representant you choose to describe the class. $[2]=[-2] ={-2,2}$. – JnxF Apr 12 '16 at 21:49

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If $f:A\to B$ denotes a function and $\sim$ is a relation on $A$ prescribed by: $$x\sim y\iff f(x)=f(y)$$ then the equivalence class represented by $a\in A$ is the set: $$f^{-1}(\{f(a)\}):=\{x\in A\mid f(x)=f(a)\}$$ You can say that $[a]_{\sim}$ is actually the fiber of $f(a)\in B$.

drhab
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