Basic Equivalence Class Discrete Math

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?

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

• mod 3 = divides evenly? Nov 2, 2012 at 2:35
• @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 2, 2012 at 2:37
• Sorry, was looking at something else. So notation wise, how would you denote one of those equivalent classes? Nov 2, 2012 at 2:41
• @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 2, 2012 at 2:42
• Nevermind i'll figure it out from here, Thanks for your help. Nov 2, 2012 at 2:46

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 don't understand in terms of notations, it says equivalence class is defined [x] = { yEA | yRx } Where is the "y" from? Nov 2, 2012 at 2:15
• @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 2, 2012 at 2:17
• 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 ) Nov 2, 2012 at 2:18
• @Aaron: Are you familiar with set-builder notation?
– wj32
Nov 2, 2012 at 2:18
• Still can't understand why am I so dumb and yes I am familiar with the notation just cannot get the logic behind it Nov 2, 2012 at 2:19