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Math Help - Symmetric Relation question

  1. #1
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    Symmetric Relation question

    I know that a symmetric relation on a set X with for every a,b ∈X is noted by:

    aRb ⇒ bRa

    But I was wondering if this could be an alternative definition (please explain why it is not if this fails, thank you).

    A relation on set X is symmetric iff for every (x,y) ∈X, (x, y) ∈X ⇒(y, x) ∈X



    The reason I suspect this might be true is because the "relation" might be defined as an ordered pair. x is related to y by x and y matched as an ordered pair, (x, y).




    Thanks for your time.
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  2. #2
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    Quote Originally Posted by SunRiseAir View Post
    I know that a symmetric relation on a set X with for every a,b ∈X is noted by:
    A relation on set X is symmetric iff for every (x,y) ∈X, (x, y) ∈X ⇒(y, x) ∈X
    How does xRy and xRy imply yRx? Is there a typo here?

    Having an extra xRy doesn't added anything so I am confused on what is going.

    Since there is a Relation on X, \displaystyle (x,y)\in X\rightarrow xRy
    Last edited by dwsmith; November 17th 2010 at 05:17 PM.
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  3. #3
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    Quote Originally Posted by SunRiseAir View Post
    A relation on set X is symmetric iff for every (x,y) ∈X, (x, y) ∈X ⇒(y, x) ∈X
    You write (x,y)\in X. However, X is the set on which a relation (which one, by the way?) is considered, so there is no reason to assume that X contains pairs. To say that x and y are elements of X, one writes x\in X, y\in X or x,y\in X. Some pedants even write (x,y)\in X^2.
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  4. #4
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    Quote Originally Posted by dwsmith View Post
    How does xRy and xRy imply yRx? Is there a typo here?

    Having an extra xRy doesn't added anything so I am confused on what is going.

    Since there is a Relation on X, \displaystyle (x,y)\in X\rightarrow xRy
    What I mean is that for every x,y in X, xRy implies yRx.

    This is the notation of a symmetric relation according to wikipedia.

    http://en.wikipedia.org/wiki/Symmetric_relation

    But then maybe that's a bad source.

    Quote Originally Posted by emakarov View Post
    You write (x,y)\in X. However, X is the set on which a relation (which one, by the way?) is considered, so there is no reason to assume that X contains pairs. To say that x and y are elements of X, one writes x\in X, y\in X or x,y\in X. Some pedants even write (x,y)\in X^2.
    That's what I meant. I just forgot to right the square for X.

    Let's say we're talking about a set whose elements are ordered pairs.

    Could a symmetric relation in a set whose elements are ordered pairs be one that I described above?


    Thanks again.
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  5. #5
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    Symmetry: when aRb, then bRa. Written as ordered pairs: (a,b)\in R \ \mbox{and} \ (b,a)\in R.
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  6. #6
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    A relation on set X is symmetric iff for every (x,y) ∈X, (x, y) ∈X ⇒(y, x) ∈X
    This is like saying, "A subset of a given set X is called nonempty iff there exists an x in X". Since the left-hand side did not introduce a notation for the subset in question ("A subset Y of a given set X..."), the right-hand side cannot talk about this subset, and the whole definition is meaningless. Could you reformulate your definition and collect all relevant info in one post?
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