Thread: 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.

Thank you in advance.

EDIT: SOLVED>

Perhaps the following outline can help you. Given $\displaystyle X\in P(A)$ we have four cases:

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

So , $\displaystyle [\;\emptyset\;]=\{\;\emptyset,\{1\},\{3\},\{1,3\}\;\}$ , etc

I have only one question sir,

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

Originally Posted by Lowoctave

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

We have found all elements $\displaystyle Y\in P(A)$ such that $\displaystyle \emptyset\cap B=Y\cap B$ that is , all elements $\displaystyle Y\in P(A)$ such that $\displaystyle \emptyset\; R\; Y$ .

Alrighty! makes sense! thank you very much sir