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Thread: Properties of Relations

  1. #1
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    Properties of Relations

    Hello,

    I just have a question concerning properties of relations.
    I'm wondering, if we wanna check wheter a relation is symmetric or not, is it ok to consider identical pairs..??

    You know like.. R = {(a,b) | a^2 = b^2}.. R is defined on the integers.
    Now the way I'm thinking of this is like.. lets take a look at (1,2) & (2,1) obviously the square of 1 doesnt equal the square of 2 so its not symmetric. But can we instead, consider a pair like (1,1) or (2,2).. which in that case can make the relation symmetric...???

    Another question... can a relation be not symmetric and not anti-symmetric at the same time ???

    I hope somebody can clarify theses point.
    Thanks for your help.
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  2. #2
    MHF Contributor red_dog's Avatar
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    Let $\displaystyle R$ be a binary relation.
    $\displaystyle R$ is symmetric if $\displaystyle aRb\Rightarrow bRa, \ \forall a,b$.

    Now, about our relation:
    If $\displaystyle aRb\Rightarrow a^2=b^2\Rightarrow b^2=a^2\Rightarrow bRa$
    So, the relation is symmetric.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by MMM88 View Post
    Hello,

    I just have a question concerning properties of relations.
    I'm wondering, if we wanna check wheter a relation is symmetric or not, is it ok to consider identical pairs..??

    You know like.. R = {(a,b) | a^2 = b^2}.. R is defined on the integers.
    Now the way I'm thinking of this is like.. lets take a look at (1,2) & (2,1) obviously the square of 1 doesnt equal the square of 2 so its not symmetric. But can we instead, consider a pair like (1,1) or (2,2).. which in that case can make the relation symmetric...???

    You relation $\displaystyle R$ is symmetric because if $\displaystyle (a,b) \in R$, then so is $\displaystyle (b,a)$

    A specific pair can show that a relation is not symmetric, but cannot show
    that it is symmetric.

    Also a relation is not defined on all pairs. In this case $\displaystyle (a,b) \in R$ if and only
    if $\displaystyle a^2=b^2$, $\displaystyle R$ has nothing to say about any other pairs.

    RonL
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by MMM88 View Post

    Another question... can a relation be not symmetric and not anti-symmetric at the same time ???
    $\displaystyle R=((1,2),(2,1),(3,4))$

    is not anti-symmetric because $\displaystyle (1,2) \in R$ as is $\displaystyle (2,1)$, also it is not symmetric
    because $\displaystyle (3,4) \in R$ but $\displaystyle (4,3) \not\in R$

    RonL
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  5. #5
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    Awesome!!! I get it now.
    Thanks for the help guys.
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