(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.

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

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