00:03
In this question, it gave the setting that a is some set that is not empty, and we have r as equivalent relation on a.
00:13
We are asked to show that there exists a function, called it f, with a as a domain, such that for any other pair x, y in the relation r, x y will be in relation r even only if the value of the function f x equals to f y this may look a bit confusing as first but essentially x and y are element in a and we want the function be such that this this two things happen whenever the other happen so so if x y is in the relation, then the function map them to the same thing.
01:11
Or if the function map them to the same thing, it must be in the relation.
01:15
Okay, so i will show one of the way to build this.
01:19
There are many ways to do this.
01:22
First, since a is not empty and has equivalent relation are, a will have a partition, call it a1 to an.
01:36
Where ai is equivalent class of r.
01:41
So each ai will be equivalent class of r.
01:44
We can do this.
01:47
You probably have seen some example or other questions that use this property.
01:55
So any equivalent relation can create a partition on its set.
02:08
And the partition is such that the union of everything made up the whole, but any pair of the part, oh, sorry, is equal, has intersection as empty set...