# Equivalence classes

• Nov 13th 2011, 11:34 AM
Lowoctave
Equivalence classes
Hello there,

I have troubles with finding equivalence classes for my homework.
Question goes like this:
Let A={1,2,3,4} and B = {2,4}. P(A) is power set. Relation R defined on P(A) by
X R Y, X, Y elements of P(A) if X ∩ B = Y ∩ B. Now, i have to show that R is equivalence relation on P(A), which i did. and i have to find equivalence classes for R.
Now, im stuck here.. im not too sure how start it. I was wondering if you could guide me please.

EDIT: SOLVED>
• Nov 13th 2011, 11:59 AM
FernandoRevilla
Re: Equivalence classes
Perhaps the following outline can help you. Given $X\in P(A)$ we have four cases:

$(i)\;X\cap B=\emptyset\quad (ii)\;X\cap B=\{2\}\quad (iii)\;X\cap B=\{4\}\quad (iv)\;X\cap B=B\quad$

So , $[\;\emptyset\;]=\{\;\emptyset,\{1\},\{3\},\{1,3\}\;\}$ , etc
• Nov 13th 2011, 12:57 PM
Lowoctave
Re: Equivalence classes
I have only one question sir,

how did you find [∅] = {∅, {1}, {3}. {1,3}}? I have troubles understanding that..
• Nov 13th 2011, 01:10 PM
FernandoRevilla
Re: Equivalence classes
Quote:

Originally Posted by Lowoctave
how did you find [∅] = {∅, {1}, {3}. {1,3}}? I have troubles understanding that..

We have found all elements $Y\in P(A)$ such that $\emptyset\cap B=Y\cap B$ that is , all elements $Y\in P(A)$ such that $\emptyset\; R\; Y$ .
• Nov 13th 2011, 02:19 PM
Lowoctave
Re: Equivalence classes
Alrighty! makes sense! thank you very much sir:)