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I read through the textbook definition of the equivalence class, but still cannot clearly understand what an equivalence class is.

Does anyone have a good example with a definition that can hit me home?

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2 Answers 2

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An equivalence class is just a set of things that are all "equal" to each other. Consider the set $$S=\{0,1,2,3,4,5\}.$$ There are many equivalence relations we could define on this set. One would be $xRy \Leftrightarrow x=y$, in which case the equivalence classes are: $$[0]=\{0\} \\ [1]=\{1\} \\ \vdots \\ [5]=\{5\}$$ We could also define $xRy$ if and only if $x \equiv y \pmod{3}$, in which case our equivalence classes are: $$[0]=[3]=\{0,3\} \\ [1]=[4]=\{1,4\} \\ [2]=[5]=\{2,5\}$$

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  • $\begingroup$ mod 3 = divides evenly? $\endgroup$
    – Aaron
    Nov 2, 2012 at 2:35
  • $\begingroup$ @Aaron: $x \equiv y \pmod{3}$ means $x-y$ is divisible by 3. I'm not sure what you mean by "= divides evenly". $\endgroup$
    – wj32
    Nov 2, 2012 at 2:37
  • $\begingroup$ Sorry, was looking at something else. So notation wise, how would you denote one of those equivalent classes? $\endgroup$
    – Aaron
    Nov 2, 2012 at 2:41
  • $\begingroup$ @Aaron: There are various notations that people use. You've already used the notation $[x]$. Some people like $[[x]]$. Or are you referring to something else? $\endgroup$
    – wj32
    Nov 2, 2012 at 2:42
  • $\begingroup$ Nevermind i'll figure it out from here, Thanks for your help. $\endgroup$
    – Aaron
    Nov 2, 2012 at 2:46
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maybe this example i found can help: If X is the set of all cars, and ~ is the equivalence relation "has the same color as", then one particular equivalence class consists of all green cars. X/~ could be naturally identified with the set of all car colors.

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    $\begingroup$ Still don't understand in terms of notations, it says equivalence class is defined [x] = { yEA | yRx } Where is the "y" from? $\endgroup$
    – Aaron
    Nov 2, 2012 at 2:15
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    $\begingroup$ @Aaron: It means the set of all things $y$ ("$y \in A$") such that $y$ is equivalent to $x$ ("$yRx$"). Here, if $x$ is a red car then $[x]$ is the set of all cars ("$y \in \mbox{Cars}$") which have the same color as $x$ ("$yRx$"). That is, the set of all red cars. $\endgroup$
    – wj32
    Nov 2, 2012 at 2:17
  • $\begingroup$ this is a general method for defining things. translated we have : the equivalience class [x] is defined such that ( y is an abstract element of a set A which has some relation R to x ) $\endgroup$ Nov 2, 2012 at 2:18
  • $\begingroup$ @Aaron: Are you familiar with set-builder notation? $\endgroup$
    – wj32
    Nov 2, 2012 at 2:18
  • $\begingroup$ Still can't understand why am I so dumb and yes I am familiar with the notation just cannot get the logic behind it $\endgroup$
    – Aaron
    Nov 2, 2012 at 2:19

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