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Math Help - Relations, part deux

  1. #1
    Member oldguynewstudent's Avatar
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    Relations, part deux

    I am completely lost on the terminology on this one.

    Let \rho be an equivalence relation on a non-empty set X.
    Let a \epsilon X. Show that [a ]_\rho \not= \emptyset

    Here [a ]_\rho := {b \epsilon X | a \rhob}

    Is this supposed to be b=a? What is this set X?

    The only thing I get is that X is non-empty so it must contain at least one element b. Then the equivalence relation has to be non-empty by definition of equivalence.
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    Quote Originally Posted by oldguynewstudent View Post
    I am completely lost on the terminology on this one.

    Let \rho be an equivalence relation on a non-empty set X.
    Let a \epsilon X. Show that [a ]_\rho \not= \emptyset

    Here [a ]_\rho := {b \epsilon X | a \rhob}

    Is this supposed to be b=a? What is this set X?

    The only thing I get is that X is non-empty so it must contain at least one element b. Then the equivalence relation has to be non-empty by definition of equivalence.
    You are correct that we assume that X\ne\emptyset.
    But to prove this we know that \rho is reflexive.
    So \left( {\forall x \in X} \right)\left[ {(x,x) \in \rho } \right] thus \left( {\forall x \in X} \right)\left[ {x \in [x]_\rho  } \right]
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  3. #3
    Member oldguynewstudent's Avatar
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    Thanks

    Quote Originally Posted by Plato View Post
    You are correct that we assume that X\ne\emptyset.
    But to prove this we know that \rho is reflexive.
    So \left( {\forall x \in X} \right)\left[ {(x,x) \in \rho } \right] thus \left( {\forall x \in X} \right)\left[ {x \in [x]_\rho } \right]
    I just found out that we skipped four sections in the textbook. I had not heard of an equivalence relation before and did not know that it is reflexive, symmetric and transitive. I also did not understand that [a] \rho was an equivalence class. So I have some more reading to do. I see the answer you supplied is extremely helpful.

    Wish I had you for my professor instead of my current one!

    Merry ChrisKwanuka.
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