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Math Help - Equivalence Relation

  1. #1
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    Equivalence Relation

    Hi, i'm studying some basic group theory at the moment, and i'm not sure whether or not i've got the jist of these things yet?

    Question: Consider the set {1,2,3}, how many equivalence relations are there on this set? Justify your answer.

    I've found what i think are 5:

    {(1,1),(2,2),(3,3)}
    {(1,1),(1,2),(2,1),(2,2),(3,3)}
    {(1,1),(1,3),(3,1),(2,2),(3,3)}
    {(1,1),(2,3),(3,2),(2,2),(3,3)}
    {(1,1),(1,2),(2,1),(1,3),(3,1),(2,3),(3,2),(2,2),( 3,3)}

    as these are the only ones that satisfy the transitivity, reflexive and symmetric properties. Is this correct?

    Also,
    *Given a partition {A_i : i belongs to I} of S describe, without proof, the unique equivalence relation on S whose equivalence classes form this partition.*
    I'm not sure how to go about this. Any help would be greatly appreciated.

    Thanks.
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  2. #2
    Super Member
    Joined
    Oct 2007
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    London / Cambridge
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    Quote Originally Posted by DeFacto View Post
    Hi, i'm studying some basic group theory at the moment, and i'm not sure whether or not i've got the jist of these things yet?

    Question: Consider the set {1,2,3}, how many equivalence relations are there on this set? Justify your answer.

    I've found what i think are 5:

    {(1,1),(2,2),(3,3)}
    {(1,1),(1,2),(2,1),(2,2),(3,3)}
    {(1,1),(1,3),(3,1),(2,2),(3,3)}
    {(1,1),(2,3),(3,2),(2,2),(3,3)}
    {(1,1),(1,2),(2,1),(1,3),(3,1),(2,3),(3,2),(2,2),( 3,3)}

    as these are the only ones that satisfy the transitivity, reflexive and symmetric properties. Is this correct?

    Also,
    *Given a partition {A_i : i belongs to I} of S describe, without proof, the unique equivalence relation on S whose equivalence classes form this partition.*
    I'm not sure how to go about this. Any help would be greatly appreciated.

    Thanks.
    yes you got all the relations there. not sure exactly what you mean in the second part, but given partition on a set the relation "x is related to y, if they belong to the same partition" is an equivalence relation.

    Bobak
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