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Math Help - Relations

  1. #1
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    Relations

    Question:

    The relation R on {1,2,3,....} where aRb means a|b.

    Is it reflexive, is it symmetric, is it antisymmetric, is it transitive?
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by oldguy View Post
    Question:

    The relation R on {1,2,3,....} where aRb means a|b.

    Is it reflexive,
    obviously, for all a \in \mathbb{N} we have a|a, thus the relation is reflexive

    is it symmetric
    for a,b \in \mathbb{N}, if a|b does it mean that b|a?

    is it antisymmetric
    for a,b \in \mathbb{N}, if a|b and b|a does it mean that a = b?

    is it transitive?
    for a,b,c \in \mathbb{N}. if a|b and b|c, does it mean that a|c?
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  3. #3
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    Hello, oldguy!

    The relation R on \{1,2,3, \cdots\} where a\text{R}b means a|b.

    Is it reflexive? .Symmetric? .Antisymmetric? .Transitive?

    For any natural numnber a,\;a \div a \:=\:1
    . . That is: . a|a
    Hence, \text{R} is reflexive.

    If a divides b, it does not follow that b divides a.
    Hence, it is not symmetric.

    If a|b and b|a, then a = b.
    Hence, \text{R} is antisymmetric.

    If a|b and b|c, then a|c. . **
    Hence, \text{r} is transitive.


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

    ** Proof of transitivity

    If a|b, then: . b \:=\:ma for some integer m.

    If b|c, then: . c \:=\:nb for some integer n.

    Then: . c \:=\:nb \:=\:n(ma) \:=\:(mn)a

    Therefore: . a|c.

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