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Thread: Symmetric relation. Cannot understand.

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    Super Member maxpancho's Avatar
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    Symmetric relation. Cannot understand.



    I would guess the relation R would be symmetric if A was equal to {b,c,d}, but not {b,c,d,e} as in the textbook?


    But let me try to reason through it.

    Let's say we consider eRb. Then if the relation is symmetric, it must follow that bRe. But since there's no (e,b) in R (i.e. ~eRb), the statement $ \forall x,y \in A, xRy \implies yRx $ is vacuously true for x=e, y=b, and so with any other pair involving e. Hence R is symmetric. Is that correct?

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    Re: Symmetric relation. Cannot understand.

    Quote Originally Posted by maxpancho View Post

    I would guess the relation R would be symmetric if A was equal to {b,c,d}, but not {b,c,d,e} as in the textbook?
    But let me try to reason through it.
    Let's say we consider eRb. Then if the relation is symmetric, it must follow that bRe. But since there's no (e,b) in R (i.e. ~eRb), the statement $ \forall x,y \in A, xRy \implies yRx $ is vacuously true for x=e, y=b, and so with any other pair involving e. Hence R is symmetric. Is that correct?
    Here are some quick tests.
    Refelcxive means that \DeltaČ_{\mathscr{R}}\subseteq\mathscr{R}
    Symmetric means that \mathscr{R}=\mathscr{R}^{-1}
    Transitive means that \mathscr{R}\circ\mathscr{R}\subseteq\mathscr{R}
    Last edited by Plato; Feb 7th 2017 at 09:05 AM.
    Thanks from topsquark
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  3. #3
    Super Member maxpancho's Avatar
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    Re: Symmetric relation. Cannot understand.

    Thanks. Although I don't understand the notation quite that well yet (in regard to analysis) and more concerned with the particular paragraph for now.

    But okay if no one's going to correct me, then I will assume I'm right.
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