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Thread: equivalence relations

  1. #1
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    equivalence relations

    I am having a very hard time understanding how to tell whether relations are
    reflexive, symmetric, and transitive.

    reflexive x = y
    symmetric x = y and y = x
    transitive x = y and y = z

    I just dont know how to go about working the problems. Here are the problems.

    Determine which of the reflexive, symmetric, and transitive properties are satisified by the given relation R defined on se S, and stat whether R is an equivalence relation on S.

    S = {1,2,3,4,5,6,7,8} and x R y means that x - y = 0.

    S = {1,2,3,4,5,6,7,8} and x R y means that x > y

    If someone can help me work thru the first one I can probably figure out the second one.
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  2. #2
    MHF Contributor

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    For #1.
    Reflexive: is it true that x-x=0 for all x? If so, the relation is reflexive.
    Symmetric: is it true that if x-y=0 then y-x=0? If so, the relation is Symmetric.
    Transitive: is it true that if x-y=0 & y-z=0 then x=z=0? If so the relation is transitive.

    Do note that x-y=0 if and only if x=y.
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  3. #3
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    So would I be safe to assume that #1 is only reflexive?
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  4. #4
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    And number 2 is reflexive and transitive?
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  5. #5
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    No. #1 are all three.
    And
    #2 only transitive.

    $\displaystyle x-x=0$ thus it is reflexsive for all $\displaystyle x\in S$.

    If $\displaystyle x-y=0$ then $\displaystyle y-x=0$, thus it is symterric.

    If $\displaystyle x-y=0$ and $\displaystyle y-z=0$ then $\displaystyle x-y+y-z=x-z=0$, thus it is transitive.

    Similary with <
    It is not true that,
    $\displaystyle x<x$ for all $\displaystyle x\in S$

    It is not true that,
    If $\displaystyle x<y$ then $\displaystyle y<x$

    It is true that,
    $\displaystyle x<y$ and $\displaystyle y<z$ then $\displaystyle x<z$
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  6. #6
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    Thanks for the explanation.
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