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Math Help - Prove or disprove that the complement of a relation is symmetric

  1. #1
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    Prove or disprove that the complement of a relation is symmetric

    Consider a relation R on a set A. Suppose that you are told that R is symmetric. Prove or disprove that \overline{R} is also symmetric.

    I only really know this much:
    R\subset A\times A and \overline{R}\subset A\times A-R
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  2. #2
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    In fact, \bar{R} is not only a subset, but is equal to (A\times A)-R (by definition).

    To get a better intuition, consider an example, such as xRy iff x-y is even ( x,y are integers), or A\,R\,B iff A\cap B\ne\emptyset ( A,B are sets). Take a pair (x,y) from \bar{R} and see if (y,x)\in\bar{R}. Do this several times and then try to have a different outcome for whether (y,x)\in\bar{R}.
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  3. #3
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    Quote Originally Posted by emakarov View Post
    In fact, \bar{R} is not only a subset, but is equal to (A\times A)-R (by definition).

    To get a better intuition, consider an example, such as xRy iff x-y is even ( x,y are integers), or A\,R\,B iff A\cap B\ne\emptyset ( A,B are sets). Take a pair (x,y) from \bar{R} and see if (y,x)\in\bar{R}. Do this several times and then try to have a different outcome for whether (y,x)\in\bar{R}.
    I'm afraid that didn't really help me. Would it be possible for you to show me?
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  4. #4
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    If (x,y)\in \overline{R} then (x,y)\notin R.
    But R is symmetric so (y,x)\notin R.
    That means that (y,x)\in \overline{R}. DONE.
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