1. ## Equivalence Class Question

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!

2. 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.

3. 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}?

4. 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): $\displaystyle \left\{ {\{ 1,2\} ,\{ 1,2,3\} ,\{ 1,2,4\} ,\{ 1,2,3,4\} } \right\}$
Can you explain why or why not?

5. Originally Posted by Plato
Is this the answer to part b): $\displaystyle \left\{ {\{ 1,2\} ,\{ 1,2,3\} ,\{ 1,2,4\} ,\{ 1,2,3,4\} } \right\}$
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?

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

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

For b, what is the equivalence class of $\displaystyle A=\{\,1,2\,\}$ is asking what things are R-related to A. If B is R-related to A, then $\displaystyle A\cup Y=B\cup Y$. Then clearly, as 1 and 2 aren't members of Y, we can conclude that $\displaystyle \{\,1,2\,\}\subseteq B$ 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
$\displaystyle \{\,\{\,1,2\,\},\{\,1,2,3\,\},\{\,1,2,4\,\},\{\,1, 2,3,4\,\}\,\}$

7. Originally Posted by swarley
Yes, this is the answer but I don't understand why.
For a start, why leave out 5?
$\displaystyle \{ 1,2\} \cup \{ 3,4\} \ne \{ 1,2,5\} \cup \{ 3,4\}$

8. Originally Posted by wgunther
Firstly, $\displaystyle a\cup Y = a\cup Y$ so $\displaystyle aRa$ for all $\displaystyle a\in\wp(X)$.

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

For b, what is the equivalence class of $\displaystyle A=\{\,1,2\,\}$ is asking what things are R-related to A. If B is R-related to A, then $\displaystyle A\cup Y=B\cup Y$. Then clearly, as 1 and 2 aren't members of Y, we can conclude that $\displaystyle \{\,1,2\,\}\subseteq B$ 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
$\displaystyle \{\,\{\,1,2\,\},\{\,1,2,3\,\},\{\,1,2,4\,\},\{\,1, 2,3,4\,\}\,\}$
Oh, I see!
Thanks very much.

Plato, thank you also.