Proving Theorem Help

• May 16th 2013, 08:05 PM
gfbrd
Proving Theorem Help
Hey I need help with this problem

Let R be an equivalence relation on S. For any 2 members x and y of S, define x http://w2.syronex.com/jmr/tex/img/simeq.latex.gif y provided that x and y belong to the same member of S/R. Then x http://w2.syronex.com/jmr/tex/img/simeq.latex.gif y if and only if xRy

I am hoping some one can guide me step by step to figure out this proof since I do not know how to even start this thanks.
• May 17th 2013, 12:13 AM
emakarov
Re: Proving Theorem Help
As far as I know, the notation S/R is not universally accepted when S is a just set and not, for example, a group or a ring. Nevertheless, my guess is that you don't know where to start because you don't know the definition of S/R used in your source.
• May 17th 2013, 03:29 AM
Plato
Re: Proving Theorem Help
Quote:

Originally Posted by gfbrd
Let R be an equivalence relation on S. For any 2 members x and y of S, define x http://w2.syronex.com/jmr/tex/img/simeq.latex.gif y provided that x and y belong to the same member of S/R. Then x http://w2.syronex.com/jmr/tex/img/simeq.latex.gif y if and only if xRy.

As far as I know, if we start with equivalence relation, $\displaystyle \mathcal{R}$, on $\displaystyle S$ then $\displaystyle S/\mathcal{R}$ is the collection of equivalence classes. It is a standard exercise to show that collection partitions the set $\displaystyle S$ and any partition determines an equivalence relation. That should be a theorem in your textbook.
• May 17th 2013, 09:10 AM
gfbrd
Re: Proving Theorem Help
yea I didn't understood the definition before but after understanding it now, I figured it out.
Thanks a lot for your help