(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Given:

R is an equivalence relation over a nonempty set X

Prove:

dom(R) = X

and range(R) = X

2. Relevant equations

3. The attempt at a solution

I have the following thoughts:

About the given:

Since R is an equivalence relation over X by hypothesis, R satisfies:

Reflexivity: <x,x> belongs to R

Symmetry: <x,y> belongs to R, and <y,x> belongs to R

Transitivity: <x,y> belongs to R, <y,z> belongs to R, and <x,z> belongs to R

with x, y, z E X

About the conclusion:

Base on definition of domain and range of a relation R over a set X, I have:

dom(R) = {x E X : there exists y belongs to Y such that <x,y> E R}

range(R) = {y E Y : there exists x belongs to X such that <x,y> E R}

What I'm confused is that I don't know how to connect my ideas together. The properties that R satisfies is with x, y, and z E X. And R is a subset of X x X. There is no Y whatsoever. So what should I do (or say) next to come to the conclusion?

Thank you for your help.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Equivalence Relation, prove dom(R) = range(R) = X

**Physics Forums | Science Articles, Homework Help, Discussion**