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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?

saeed
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Aaron
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2 Answers2

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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: $$[0]=[3]=\{0,3\} \\ [1]=[4]=\{1,4\} \\ [2]=[5]=\{2,5\}$$

wj32
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  • mod 3 = divides evenly? – Aaron Nov 02 '12 at 02:35
  • @Aaron: $x \equiv y \pmod{3}$ means $x-y$ is divisible by 3. I'm not sure what you mean by "= divides evenly". – wj32 Nov 02 '12 at 02:37
  • Sorry, was looking at something else. So notation wise, how would you denote one of those equivalent classes? – Aaron Nov 02 '12 at 02:41
  • @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? – wj32 Nov 02 '12 at 02:42
  • Nevermind i'll figure it out from here, Thanks for your help. – Aaron Nov 02 '12 at 02:46
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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.

still_learning
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    Still don't understand in terms of notations, it says equivalence class is defined [x] = { yEA | yRx } Where is the "y" from? – Aaron Nov 02 '12 at 02:15
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    @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. – wj32 Nov 02 '12 at 02:17
  • 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 ) – still_learning Nov 02 '12 at 02:18
  • @Aaron: Are you familiar with set-builder notation? – wj32 Nov 02 '12 at 02:18
  • Still can't understand why am I so dumb and yes I am familiar with the notation just cannot get the logic behind it – Aaron Nov 02 '12 at 02:19
  • expertise comes with practice, try and get in the habit of translating set notation and understanding what each part of a notation means and then obtaining an overall idea for what the set represents. abstraction is by no means easy. – still_learning Nov 02 '12 at 02:22
  • To update, I know what an equivalence relation is. – Aaron Nov 02 '12 at 02:22
  • @Aaron: Did you read and understand my first comment? You need to be more specific with your difficulties. – wj32 Nov 02 '12 at 02:23
  • Instead of an arbitrary example, do you have one with defined numbers? – Aaron Nov 02 '12 at 02:26
  • The set all even integers (like the set of all odd integers) is an equivalence class on $\mathbb{N}$. Note also that the union of all (in this case both) equivalence classes on a set IS the set, and the intersection of any two equivalence classes on a set is empty, i.e. the equivalence classes of a relation on a set form a partition of the given set. – amWhy Nov 02 '12 at 02:31
  • @Aaron: See my answer. – wj32 Nov 02 '12 at 02:31