1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: R is symmetric iff R = R^-1

  1. Mar 8, 2010 #1
    1. The problem statement, all variables and given/known data
    In Velleman's "How to Prove it", he gives a proof that "R is symmetric iff R = R-1, which I find to be confusing when he is proving that [tex]R^{-1}\subseteq{R}[/tex]:

    Now suppose [tex](x,y)\in R^{-1}[/tex]. Then [tex](y,x)\in R[/tex], so since R is symmetric, [tex](x,y)\in R[/tex]. Thus, [tex]R^{-1}\subseteq R[/tex] so R=R-1

    It seems to me that he is saying that since [tex]xRy\rightarrow yRx[/tex] and yRx, xRy, which makes no sense.

    Basically my question is this: how this part of his proof could be correct?
    Last edited: Mar 8, 2010
  2. jcsd
  3. Mar 8, 2010 #2


    User Avatar
    Science Advisor

    It would help if you would tell us what "R" is! A relation?

    If R is a relation, then it is a set of ordered pairs. [itex]R^{-1}[/itex] is defined as the set of pairs [itex]\{(x, y)| (y, x)\in R\}[/itex].

    What he is saying is that if (x,y) is in R-1, then (y, x) is in R. Since R is symmetric, (x, y) is in R and so [itex]R^{-1}\subset R[/itex].
  4. Mar 8, 2010 #3
    Ah, okay. I guess I was stuck thinking that (x,y) was in R and didn't consider that R being symmetric could mean that if yRx then xRy.

    And yes, R was a relation haha.

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook