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Thread: properties of relations

  1. #1
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    properties of relations

    Which properties does each of the following relations satisfy among reflexive, symmetric, transitive, irreflexive, and antisymmetric.

    R = {(a,b) l a squared = b squared} over the real numbers

    R = {(x,y) l x divides y} over the positive integers

    I thought the first one might be irreflexive, but I am not sure, and I don't know on the second one.
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  2. #2
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    Quote Originally Posted by patches View Post
    Which properties does each of the following relations satisfy among reflexive, symmetric, transitive, irreflexive, and antisymmetric.

    R = {(a,b) l a squared = b squared} over the real numbers

    R = {(x,y) l x divides y} over the positive integers

    I thought the first one might be irreflexive, but I am not sure, and I don't know on the second one.
    You are essentially asking yourself:
    $\displaystyle \forall $ a,b,c
    1) reflexive does(a,a) $\displaystyle \in R$?
    2) symmetric, if (a,b) $\displaystyle \in R$, is (b,a) $\displaystyle \in R$?
    3) transitive, if (a,b) $\displaystyle \in R$, is (b,a) $\displaystyle \in R$?
    4) irreflexive, does (a,a) $\displaystyle \notin R$?
    5) antisymmetric, if (a,b) $\displaystyle \in R$, and (b,a) $\displaystyle \in R$ then is a=b?


    A)
    R = {(a,b) l a squared = b squared} over the real numbers
    "Equal" is a reflexive, symmetric, transitive relation
    Try all the 5 conditions out for R = {(x,y) l x = y}

    B)
    R = {(x,y) l x divides y} over the positive integers
    Note: I will use a|b for "a divides b"

    Start checking:
    1) a|a
    2) Not symmetric , See 5)
    3)if a|b and b|c then b = au and c = bv. Thus c = bv = (au)v = a(uv) and therefore a|c.
    4) Since 1) holds, 4) cant
    5) if a|b and b|a then $\displaystyle a = \pm b$.But since a,b $\displaystyle \in \mathbb{Z}^+$, a=b.
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