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: Proving an equivalence relation

  1. Jan 24, 2010 #1
    1. The problem statement, all variables and given/known data
    On the set of integers, define the relation R by: aRb if ab>=0.

    Is R an equivalence relation?

    2. Relevant equations



    3. The attempt at a solution

    R is an equivalence relation if it satisfies:

    1) R is reflexive
    Show that for all a∈Z, aRa.

    Let a∈Z. Then if a is a negative integer, aa>=0. If a is a positive integer, aa>=0. And if a = 0, aa>=0.
    Hence aRa

    I feel like it is too simple.. lacking something??

    2) R is symmetric
    Show that for all a∈Z, aRb --> bRa

    Let a∈Z, b∈Z such that aRb. By the definition of R, ab>=0.
    This is not symmetric. Take a = -1, b = 2.
    Then we have ab = -2 which is not >= 0.

    3) R is transitive
    if aRb and bRc implies aRc for all a,b,c ∈ Z

    Let a, b, c ∈ Z s/t aRb, bRc --> aRc

    Now I think this one is true.. but I'm not sure. But since aRb, and bRc, then you would always have ab or bc >=0 yea? so that means aRc must be true..
    How would I prove it properly if it is correct?

    Any help is appreciated! :) Thanks.
     
  2. jcsd
  3. Jan 24, 2010 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    1) is too simplistic because you didn't take account of the case a<0. 2) That (-1)R(2) is false doesn't prove anything. You want to prove if ab>=0 THEN ba>=0. For 3) suppose one of a,b and c is 0. Can you show with a simple example that it's not true?
     
  4. Jan 24, 2010 #3
    1) I had that if it was a negative integer, which is the case of a<0 no?

    The problem was I was doing them backwards, assuming the relation first rather than the property.

    2)
    Let a, b ∈ Z s/t ab>=0 --> a>=0, b>=0, so ab>=0 then ba>=0

    So I think this is actually symmetric. Im not sure though, because I used the inequality to solve for a and b, but am I allowed to do that??

    3) Let a, b, c ∈ Z s/t ab>=0 and bc>=0
    So I should prove ac>=0
    Let a = 1, b = 0, c = -1
    Then we have ab>=0, bc>=0, but ac<0
    so its false.

    Thanks for the help!
     
    Last edited: Jan 24, 2010
  5. Jan 24, 2010 #4

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Oh yeah, I see you did cover the negative case for 1). Sorry, somehow I didn't see that. But then for 2) ab>=0 doesn't imply a>=0 and b>=0. But it certainly does imply ba>=0. And yes, 3) is false. Just as you say.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook