Thread: Equivalence relation and Functions

The problem is:
"Suppose that A is a nonempty set and R is an equivalence relation on A. Show that there is a function f with A as its domain such that (x,y) are elements of R if and only if f(x)=f(y)"

I don't understand how to connect a relation with a function

thank you

Originally Posted by mremwo

The problem is:
"Suppose that A is a nonempty set and R is an equivalence relation on A. Show that there is a function f with A as its domain such that (x,y) are elements of R if and only if f(x)=f(y)"

I don't understand how to connect a relation with a function

thank you

A function is actually a relation, but that's just not how many people think about it. Consider that a function is really just it's graph . This is the relation.

Try finding a function (the power set of ) such that two elements are in the same equivalence class under if and only if they have the same image under . Can you think of a natural choice for the image under of an element from ?

Originally Posted by topspin1617

Try finding a function (the power set of ) such that two elements are in the same equivalence class under if and only if they have the same image under . Can you think of a natural choice for the image under of an element from ?

What does this have to do with anything? Unless you really meant .

No. The function given by does exactly what the question asks.

Originally Posted by topspin1617

No. The function given by does exactly what the question asks.

Touche sir, touche. I misread the question.