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Math Help - equivalence relations

  1. #1
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    equivalence relations

    say we have a set W. We also have a non-empty set Z such that each element of Z is an equivalence relation on W.

    Must show that the nZ is an ER in W. (where n is intersection)
    how would this be proven, like i know that u must prove that it is reflexive, transitive and symmetric but how exactly would u start it?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by g rad23 View Post
    say we have a set W. We also have a non-empty set Z such that each element of Z is an equivalence relation on W.

    Must show that the nZ is an ER in W. (where n is intersection)
    how would this be proven, like i know that u must prove that it is reflexive, transitive and symmetric but how exactly would u start it?
    I am sorry, I don't understand what you mean.
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  3. #3
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    my question is
    let W be a set.
    Let Z be a non empty set such that each element of Z is an equivalence relation on W

    i must show that the nZ is an equivalence relation on W. (where nZ is the intersection of Z)

    how would i go about proving this is true?

    i understand that i have to prove that the relation is reflexive, anti-symmetric and transitive, but how would i do that
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by g rad23 View Post
    my question is
    let W be a set.
    Let Z be a non empty set such that each element of Z is an equivalence relation on W

    i must show that the nZ is an equivalence relation on W. (where nZ is the intersection of Z)

    how would i go about proving this is true?

    i understand that i have to prove that the relation is reflexive, anti-symmetric and transitive, but how would i do that
    I'm sorry. I don't mean to sound ignortant, but I was unsure of notation. Do you mean nz=\bigcap_{x\in z}x?
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  5. #5
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    dont worrry about it and yes i did mean the intersection as u stated
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  6. #6
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by g rad23 View Post
    dont worrry about it and yes i did mean the intersection as u stated
    Maybe some other member can help you better than I, but I don't really undertsand what this question could possibly mean? What is the intersection of two equivalence relations? Unless, you mean the intersection of all the partitions induced by the relations?
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  7. #7
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    well im not sure myself this is the question i was given for my hmwk. well im assuming its intersection it looks exactly like the symbol for intersection but has no limit.
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  8. #8
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by g rad23 View Post
    well im not sure myself this is the question i was given for my hmwk. well im assuming its intersection it looks exactly like the symbol for intersection but has no limit.
    Ok, now that I think about it I believe that I understand what this is saying. if \sim is an equivalence relation on E then \sim is charcterized by R=\left\{(a,b)\in E\times E:a\sim b\right\}. Maybe that if we consider the different relations on E to be charchertized by R_1,R_2,\cdots (not necessarily countable..I just wrote it that way for clarity) then perhaps they mean to show that R_1\cap R_2\cdots=R is an the characterization of an equivalence relation. Does that sound right? Another member may swoop in an answer this btw.
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  9. #9
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    umm it might be right im not sure i thought in order to prove it was an equivalence relation on something one would have to prove that the relation is reflexive, transitive and symmetric
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  10. #10
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    The gist of the problem is to prove the following. Let W be a set and let R_1 and R_2 be relations (i.e., subsets of W\times W) on W. Moreover, suppose that R_1 and R_2 are equivalence relations. Show that R_1\cap R_2 is an equivalence relation a well.

    (This solves the problem when the set Z of equivalence relations has size 2, or, in fact, any finite size. To be strict, one has to prove this also for infinite Z, but the proof is essentially the same.)

    This should be easy to show by definition.
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  11. #11
    Senior Member Shanks's Avatar
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    Yeah, i agree with emakarov!
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