Originally Posted by

**ThePerfectHacker** This is the fundamental theorem of equivalence classes. It is used all the time in advanced mathematics.

It is a theorem that *partitions* a non-empty set. Hence it takes all the elements of the set and puts them into *disjoint* sets.

For example consider:

Z={0,1,-1,2,-2,3,-3,...}

And define x~y if and only if x=y(mod 2).

So pick any number, say 0:

Now find all y so that: 0 = y (mod 2).

Those are: 0 (itself), 2, -2, 4, -4, ....

Now pick any number which **is not** much those just listed above. Because if you do then the ones that are equivalent to it is going to be the same class of numbers.

Say you pick 1 (for it is not in that set)/

Now find all y so that: 1 = y(mod 2).

Those are: 1(itself), -1,3,-3,.....

As you can see we have completely depleted all the elements of the set. And divided the set into two disjoint sets:

{0,2,-2,...}

{1,-1,3,-3,...}