Equivalence Class Question

**swarley** · February 17th 2010, 02:11 PM

Hey all, I was wondering if anyone could shed some light on this question.

Let X= {1,2,3,4,5} and Y={3,4}

b) What is the equivalence class of {1,2}?

I understand equivalence relations but can't seem to grasp the concept of equivalence classes. :|

Any explainations would be appreciated. Thanks in advance!

**Plato** · February 17th 2010, 02:18 PM

Originally Posted by swarley

Hey all, I was wondering if anyone could shed some light on this question.
Let X= {1,2,3,4,5} and Y={3,4}
b) What is the equivalence class of {1,2}?
I understand equivalence relations but can't seem to grasp the concept of equivalence classes. :|Any explainations would be appreciated.

That bit of a question makes no sense whatsoever.
Please post the entire question with its exact wording.

**swarley** · February 17th 2010, 02:22 PM

That bit of a question makes no sense whatsoever.
Please post the entire question with its exact wording.

Apologies.

Let X = {1,2,3,4,5} and Y = {3,4}.
Define a relation R on the power set P(X) of X by A R B iff A U Y = B U Y.

a) Prove that R is an equivalence relation.
b) What is the equivalence class of {1,2}?

**Plato** · February 17th 2010, 02:42 PM

Originally Posted by swarley

Apologies.

Let X = {1,2,3,4,5} and Y = {3,4}.
Define a relation R on the power set P(X) of X by A R B iff A U Y = B U Y.

a) Prove that R is an equivalence relation.
b) What is the equivalence class of {1,2}?

Is this the answer to part b): $\left\{ {\{ 1,2\} ,\{ 1,2,3\} ,\{ 1,2,4\} ,\{ 1,2,3,4\} } \right\}$
Can you explain why or why not?

**swarley** · February 17th 2010, 02:49 PM

Originally Posted by Plato

Is this the answer to part b): $\left\{ {\{ 1,2\} ,\{ 1,2,3\} ,\{ 1,2,4\} ,\{ 1,2,3,4\} } \right\}$
Can you explain why or why not?

Yes, this is the answer but I don't understand why.
For a start, why leave out 5?

**wgunther** · February 17th 2010, 02:53 PM

Originally Posted by swarley

Apologies.

Let X = {1,2,3,4,5} and Y = {3,4}.
Define a relation R on the power set P(X) of X by A R B iff A U Y = B U Y.

a) Prove that R is an equivalence relation.
b) What is the equivalence class of {1,2}?

Firstly, $a\cup Y = a\cup Y$ so $aRa$ for all $a\in\wp(X)$ .

Suppose $aRb$ and $bRc$ . Then $a\cup Y=b\cup Y=c\cup Y$ . So, by symmetry and transitivity of usual set equality, $bRa$ and $aRc$ So it's an equivalence relation.

For b, what is the equivalence class of $A=\{\,1,2\,\}$ is asking what things are R-related to A. If B is R-related to A, then $A\cup Y=B\cup Y$ . Then clearly, as 1 and 2 aren't members of Y, we can conclude that $\{\,1,2\,\}\subseteq B$ and furthermore, 3 or 4 could be on B (that wouldn't effect anything since they would be in their after unioned with Y anyway). So it's equivalence class is
$\{\,\{\,1,2\,\},\{\,1,2,3\,\},\{\,1,2,4\,\},\{\,1, 2,3,4\,\}\,\}$

**Plato** · February 17th 2010, 02:56 PM

Originally Posted by swarley

Yes, this is the answer but I don't understand why.
For a start, why leave out 5?

There is a simple answer.
$\{ 1,2\} \cup \{ 3,4\} \ne \{ 1,2,5\} \cup \{ 3,4\}$

**swarley** · February 17th 2010, 02:58 PM

Originally Posted by wgunther

Firstly, $a\cup Y = a\cup Y$ so $aRa$ for all $a\in\wp(X)$ .

Suppose $aRb$ and $bRc$ . Then $a\cup Y=b\cup Y=c\cup Y$ . So, by symmetry and transitivity of usual set equality, $bRa$ and $aRc$ So it's an equivalence relation.

For b, what is the equivalence class of $A=\{\,1,2\,\}$ is asking what things are R-related to A. If B is R-related to A, then $A\cup Y=B\cup Y$ . Then clearly, as 1 and 2 aren't members of Y, we can conclude that $\{\,1,2\,\}\subseteq B$ and furthermore, 3 or 4 could be on B (that wouldn't effect anything since they would be in their after unioned with Y anyway). So it's equivalence class is
$\{\,\{\,1,2\,\},\{\,1,2,3\,\},\{\,1,2,4\,\},\{\,1, 2,3,4\,\}\,\}$

Oh, I see!
Thanks very much.

Plato, thank you also.

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