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Math Help - Equivalence Class Question

  1. #1
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    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!
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  2. #2
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    Quote Originally Posted by swarley View Post
    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.
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  3. #3
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    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}?
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  4. #4
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    Quote Originally Posted by swarley View Post
    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?
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  5. #5
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    Quote Originally Posted by Plato View Post
    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?
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  6. #6
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    Quote Originally Posted by swarley View Post
    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\,\}\,\}
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  7. #7
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    Quote Originally Posted by swarley View Post
    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\}
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  8. #8
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    Quote Originally Posted by wgunther View Post
    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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