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Math Help - properties of relations

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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:
    \forall a,b,c
    1) reflexive does(a,a) \in R?
    2) symmetric, if (a,b) \in R, is (b,a) \in R?
    3) transitive, if (a,b) \in R, is (b,a) \in R?
    4) irreflexive, does (a,a) \notin R?
    5) antisymmetric, if (a,b) \in R, and (b,a) \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 a = \pm b.But since a,b \in \mathbb{Z}^+, a=b.
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