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Prove Relationship between Equivalence Relations and Equivalence Classes

  1. Oct 13, 2011 #1
    I'm not sure if I did these 2 questions correctly, so would someone please check my work for any missing ideas or errors?

    Question 1:
    1. The problem statement, all variables and given/known data
    Prove:
    For every x belongs to X, TR∩S(x) = TR(x) ∩ TS(x)

    2. Relevant equations

    3. The attempt at a solution
    TR(x) = {x belongs to X such that <x,y> belongs to R}
    TS(x) = {x belongs to X such that <x,y> belongs to S}
    TR∩S(x) = {x belongs to X such that <x,y> belongs to R∩S}

    <x,y> belongs to R∩S if <x,y> belongs to R and also belongs to S, which satisfy the definition above for TR(x) and TS(x)

    Question 2:

    1. The problem statement, all variables and given/known data
    R and S are equivalence relations over X
    Prove R ∩ S is also an equivalence relation over X

    2. Relevant equations


    3. The attempt at a solution
    Since R and S are equivalence relations over X, then for x in X, R and S satisfy properties:
    Reflexive:
    <x,x> belongs to R
    <x,x> belongs to S
    Symmetry:
    <x,y> and <y,x> belong to R
    <x,y> and <y,x> belong to S
    Transitivity:
    <x,y> belongs to R, <y,z> belongs to R; then <x,z> belongs to R
    <x,y> belongs to S, <y,z> belongs to R, then <x,z> belongs to S

    If R∩S is equivalence relation, then it must satisfy:
    1/ <x,x> belongs to R∩S, meaning <x,x> belongs to R and also belongs to S
    2/ <x,y> and <y,x> belong to R∩S, meaning <x,y> belongs to R and also belongs to S.
    3/ <x,y> belongs to R∩S and <y,z> belongs to R∩S, then <x,z> belongs to R∩S, meaning <x,z> belongs to R and also belongs to S
    All of these are satisfied by hypotheses.
    So R∩S is equivalence relation over X.
     
  2. jcsd
  3. Oct 13, 2011 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    First, be carful with your wording:
    should be "For all x in X, <x,x> belongs to R and <x,x> belongs to S"
    And you really should say "therefore <x, x> belongs to [itex]R\cap S[/itex]".

    would normally be interpreted as "for all x and y in X, <x, y> and <y, x> belong to R", etc. but that is not what you want to say. IF <x, y> is in R, then <y, x> if in R.

    and the same for "transitive": IF <x, y> is in R AND <y, z> is in R, then <x, z> is in R.

    And you cannot just say "all of these are satisfied by hypotheses". You must show exactly why each of reflexive, symmetric, and transitive is satisfied for [itex]R\cap S[/itex].
     
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