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Thread: Smallest equivalence relation

  1. #1
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    Smallest equivalence relation

    Find the smallest equivalence relation R on M = {1; 2; 3; 4; 5} which contains the subset Ro = {(1; 1); (1; 2); (2; 4); (3; 5)} and give its equivalence classes.


    How would I go about doing this ?

    Any help would be good

    cheers
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  2. #2
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    Re: Smallest equivalence relation

    To make it reflexive, it needs (2,2), (3,3), (4,4), (5,5)
    To make it transitive: (1,2) and (2,4) needs (1,4).
    To make it symmetric, it needs (2,1), (4,2), (4,1), (5,3)

    Equivalence classes: [1]=[2]=[4]={1,2,4}, [3]=[5]={3,5}
    Thanks from bee77 and HallsofIvy
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    Re: Smallest equivalence relation

    Thanks slip eternal ...when it says smallest equivalence ration are we just basically breaking it down to the smallest amount of sets possible?
    Thanks
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    Re: Smallest equivalence relation

    Quote Originally Posted by bee77 View Post
    Thanks slip eternal ...when it says smallest equivalence ration are we just basically breaking it down to the smallest amount of sets possible?
    Thanks
    When it says smallest, it means with the minimum number of pairs.
    Thanks from bee77
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    Re: Smallest equivalence relation

    Ah Thanks SlipEternal
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  6. #6
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    Re: Smallest equivalence relation

    Quote Originally Posted by bee77 View Post
    Find the smallest equivalence relation R on M = {1; 2; 3; 4; 5} which contains the subset Ro = {(1; 1); (1; 2); (2; 4); (3; 5)} and give its equivalence classes.
    How would I go about doing this ?
    Quote Originally Posted by SlipEternal View Post
    Equivalence classes: [1]=[2]=[4]={1,2,4}, [3]=[5]={3,5}
    Just a note on this process.
    SlipEternal gave you the real key to your question "How would I go about doing this ?"
    Once the Equivalence classes are identified the your answer comes:
    $\mathscr{R}=[\{1,2,4\} \times\{1,2,4\}]\cup[\{3,5\}\times\{3,5\}]~.$

    As point of interest, there is a one-to-one relationship between partitions of a set and equivalence relations on that set.
    Thanks from bee77
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    Re: Smallest equivalence relation

    Thanks for the help...not sure if i suck or my lecturer does ..but you guys explain it better,cheers
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