Results 1 to 9 of 9

Math Help - Equivalence relations X x X

  1. #1
    Junior Member
    Joined
    Oct 2011
    Posts
    47

    Equivalence relations X x X

    Let X ={1,2, ...,10} Define a relation R on X x X by (a,b) R (c,d) if a+d=b+c
    How can I show that R is an equivalence relation on X x X?
    Thank you!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Siron's Avatar
    Joined
    Jul 2011
    From
    Norway
    Posts
    1,250
    Thanks
    20

    Re: Equivalence relations X x X

    What's the definition of an equivalence relation R on X? (or even better: for which conditions is a relation an equivalence relation?)
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Oct 2011
    Posts
    47

    Re: Equivalence relations X x X

    A relation that is reflexive, symmetric, and transitive on a set is called an equivalence relation.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Siron's Avatar
    Joined
    Jul 2011
    From
    Norway
    Posts
    1,250
    Thanks
    20

    Re: Equivalence relations X x X

    Quote Originally Posted by mathproblems View Post
    A relation that is reflexive, symmetric, and transitive on a set is called an equivalence relation.
    Indeed! So use them to prove that R is an equivalence relation on X.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Oct 2011
    Posts
    47

    Re: Equivalence relations X x X

    I am not sure how to start, and how to use a+d=b+c
    Do I plug numbers between 1 to 10 in there?

    Let (a,b) R (c,d) be symmetric
    therefore R { (a,b) , (c,d) ,(b,a), (d,c)}

    Let (a,b) R (c,d) be reflexive
    therefore R {(a,b), (c,d)}

    Let (a,b) R (c,d) be transitive
    therefore R {(a,b) (b,c) (c,d)}

    So the Equivalence relation of (a,b) R (c,d) will be {(a,b) , (c,d) ,(b,a), (d,c) (b,c)}

    where shall I use a+d= b+c?
    Thank you.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1573
    Awards
    1

    Re: Equivalence relations X x X

    Quote Originally Posted by mathproblems View Post
    I am not sure how to start, and how to use a+d=b+c
    Do I plug numbers between 1 to 10 in there?
    Let (a,b) R (c,d) be symmetric
    therefore R { (a,b) , (c,d) ,(b,a), (d,c)}
    Let (a,b) R (c,d) be reflexive
    therefore R {(a,b), (c,d)}
    Let (a,b) R (c,d) be transitive
    therefore R {(a,b) (b,c) (c,d)}
    So the Equivalence relation of (a,b) R (c,d) will be {(a,b) , (c,d) ,(b,a), (d,c) (b,c)}
    where shall I use a+d= b+c?
    Once again, that sort of example has absolutely nothing to do with proof.
    Do you understand what constitutes a proof?

    I will get you started.
    We know that a+b=b+a therefore (a,b)\mathbf{R}(a,b).
    Therefore, \mathbf{R} is reflexive.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Junior Member
    Joined
    Oct 2011
    Posts
    47

    Re: Equivalence relations X x X

    could you give me a hint on the numbers, how do I use them a+d=b+c? Thank you.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1573
    Awards
    1

    Re: Equivalence relations X x X

    Quote Originally Posted by mathproblems View Post
    could you give me a hint on the numbers, how do I use them a+d=b+c? Thank you.
    Numbers have nothing to do with proof.
    Did you understand the proof the \mathbf{R} is reflexive?

    Now suppose (a,b)\mathbf{R}(c,d).
    To prove symmetry we must show that (c,d)\mathbf{R}(a,b).

    We know that (a,b)\mathbf{R}(c,d) means that a+d=b+c.
    But we can rearrange that to get c+b=d+a.
    That means (c,d)\mathbf{R}(a,b).
    Thus it is symmetric.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Junior Member
    Joined
    Oct 2011
    Posts
    47

    Re: Equivalence relations X x X

    thank you. i understood.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Equivalence relations
    Posted in the Discrete Math Forum
    Replies: 11
    Last Post: October 16th 2011, 02:53 PM
  2. Replies: 1
    Last Post: September 19th 2011, 01:09 PM
  3. Equivalence Relations
    Posted in the Discrete Math Forum
    Replies: 7
    Last Post: November 8th 2010, 10:55 AM
  4. Replies: 10
    Last Post: January 14th 2010, 12:28 PM
  5. equivalence relations
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: January 12th 2010, 07:17 PM

Search Tags


/mathhelpforum @mathhelpforum