Results 1 to 4 of 4

Thread: Prove or disprove that the complement of a relation is symmetric

  1. #1
    Member
    Joined
    Jan 2010
    Posts
    232

    Prove or disprove that the complement of a relation is symmetric

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

    I only really know this much:
    $\displaystyle R\subset A\times A$ and $\displaystyle \overline{R}\subset A\times A-R$
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,577
    Thanks
    790
    In fact, $\displaystyle \bar{R}$ is not only a subset, but is equal to $\displaystyle (A\times A)-R$ (by definition).

    To get a better intuition, consider an example, such as $\displaystyle xRy$ iff $\displaystyle x-y$ is even ($\displaystyle x,y$ are integers), or $\displaystyle A\,R\,B$ iff $\displaystyle A\cap B\ne\emptyset$ ($\displaystyle A,B$ are sets). Take a pair $\displaystyle (x,y)$ from $\displaystyle \bar{R}$ and see if $\displaystyle (y,x)\in\bar{R}$. Do this several times and then try to have a different outcome for whether $\displaystyle (y,x)\in\bar{R}$.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Jan 2010
    Posts
    232
    Quote Originally Posted by emakarov View Post
    In fact, $\displaystyle \bar{R}$ is not only a subset, but is equal to $\displaystyle (A\times A)-R$ (by definition).

    To get a better intuition, consider an example, such as $\displaystyle xRy$ iff $\displaystyle x-y$ is even ($\displaystyle x,y$ are integers), or $\displaystyle A\,R\,B$ iff $\displaystyle A\cap B\ne\emptyset$ ($\displaystyle A,B$ are sets). Take a pair $\displaystyle (x,y)$ from $\displaystyle \bar{R}$ and see if $\displaystyle (y,x)\in\bar{R}$. Do this several times and then try to have a different outcome for whether $\displaystyle (y,x)\in\bar{R}$.
    I'm afraid that didn't really help me. Would it be possible for you to show me?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,736
    Thanks
    2811
    Awards
    1
    If $\displaystyle (x,y)\in \overline{R}$ then $\displaystyle (x,y)\notin R$.
    But $\displaystyle R$ is symmetric so $\displaystyle (y,x)\notin R$.
    That means that $\displaystyle (y,x)\in \overline{R}$. DONE.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Symmetric relation v.s. symmetric matrix
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Oct 14th 2010, 11:37 PM
  2. prove or disprove complement of probability
    Posted in the Advanced Statistics Forum
    Replies: 7
    Last Post: Sep 13th 2010, 05:38 PM
  3. Replies: 12
    Last Post: Jun 24th 2010, 12:00 PM
  4. Prove or disprove that R^4 is a transitive relation
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: Mar 16th 2010, 03:52 PM
  5. Replies: 4
    Last Post: Apr 12th 2009, 12:05 PM

Search tags for this page

Click on a term to search for related topics.

Search Tags


/mathhelpforum @mathhelpforum