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I If symmetric then transitive relation

  1. May 15, 2017 #1
    Isn't, if we have xRy and yRx then xRx will also make transitive? Because if I am right {(x,x),(y,y)} on set {x,y} is symmetric and transitive.

    Isn't the above similar to, if xRy and yRz then xRz is transitive relation?

    Thanks.
     
  2. jcsd
  3. May 15, 2017 #2

    TeethWhitener

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    ##R = \{(x,x),(y,y),(z,z),(x,y),(y,x),(y,z),(z,y)\}##
    is an example of a relation that's symmetric and reflexive without being transitive, because
    ##xRy \leftrightarrow yRx##
    and
    ##yRz \leftrightarrow zRy##
    but we also have ##xRy \wedge yRz## without ##xRz##.
     
  4. May 15, 2017 #3

    FactChecker

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    Similar but not nearly as strong. xRy and yRx => xRx is a statement about a much smaller set of x and y than the transitive property requires.
     
  5. May 15, 2017 #4
    Does this mean transitive relation require atleast 3 distinct element of a set e.g {x,y,z}.

    Also as I mentioned, is {(x,x),(y,y)} on set {x,y} reflexive along with symmetric and transitive.
     
  6. May 15, 2017 #5

    FactChecker

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    (xRy and yRx) => xRx only makes a statement about the x & y where both xRy and yRx. There might easily be none of those, so it might say nothing.

    PS. Even if the relation R is transitive, there may be no x & y where (xRy and yRx). An example is the order relation '>'. It's not possible for (x > y & y > x), even though '>' is a transitive relation.
     
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