How to prove something is an equivalence class?

Question

I don't understand equivalence class and representative function: http://www.cdhmhome.com/uic/math215/S.pdf , I'm looking at examples 37, 1-7 and have already determined which ones are equivalence relations. But I don't quite understand what to do with an equivalence class. And I don't understand the idea before the representative (in Definition 40) and the idea that there can be many representatives relating them to example 37. Please, any help is greatly appreciated.

Answer 1

An equivalence class is just a set of all the things which are like each other in some way.
For example, let's say we're looking at the integers, $\mathbf Z$, but we're only interested in whether the integers we're looking at are even or odd.
Here, the equivalence relation is a~b iff $2|(b-a)$.
Then the equivalence classes would be
[$0$] = { $0, 2, -2, 4, -4, 6, -6, ...$ }
[$1$] = { $1, -1, 3, -3, 5, -5, 7, ...$ }

The representatives 0 and 1 are arbitrary representations of these equivalence classes which one might also call even integers and odd integers. One could easily have chosen [$2$] and [$1$]. Or [$-68$] and [$47$].

Or, as another example, we could look at $\mathbf R$ under the equivalence relation a~b iff sgn(a)=sgn(b).
Then the equivalence classes are [$-1$] = $\mathbf R^-$, [$0$]={$0$}, and [$1$]=$\mathbf R^+$.
Again, the choice of representatives $-1$, $0$, and $1$ is completely arbitrary.

Equivalence classes are used when you want to work with objects which behave similarly to each other in some way, such as how any two even numbers, when summed, will yield an even number, or any two positive numbers, when multiplied, will yield a positive number.