Results 1 to 4 of 4

Math Help - Equivalence relation and Equivalence classes?

  1. #1
    Member Jason Bourne's Avatar
    Joined
    Nov 2007
    Posts
    132

    Equivalence relation and Equivalence classes?

    I'm not sure on the following question:

    (a) Define a relation R on Z by

         aRb   if and only if 3|(a+2b)

    Show that R is an equivalence relation, and describe the equivalence classes.

    ---------------------------------------------------------------------------

    I'm particularly interested on how you prove the transativity part of the equivalence relation and how to describe the equivalence classes. Can anyone help?

    (also any more explanation on Equivalence relations and classes would be much appreciated as I could certainly do with understanding it better.)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,957
    Thanks
    1780
    Awards
    1
    Quote Originally Posted by Jason Bourne View Post
    (a) Define a relation R on Z by
         aRb   if and only if 3|(a+2b)
    I'm particularly interested on how you prove the transativity part of the equivalence relation
    aRb\,\& \,bRc \Rightarrow \quad a + 2b = 3k\,\& \,b + 2c = 3j

    So add together: \begin{gathered}  a + 3b + 2c = 3k + 3j \hfill \\<br />
  a + 2c = 3k + 3j - 3b \hfill \\ \end{gathered}
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member Jason Bourne's Avatar
    Joined
    Nov 2007
    Posts
    132
    thanks for that, anything on Equivalence classes?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Apr 2005
    Posts
    16,396
    Thanks
    1846
    I would find symmetry most interesting because the rule defining equivalence does "look" symmetric! You must prove that if aRb, then bRa which means show that if a+ 2b is a multiple of 3, then so is b+ 2a. That is true but how did you show it?

    An equivalence class consists of all those things that are equivalent to one another. Here two integers, a and b, are equivalent if and only if a= 2b is divisible by 3. Now just start looking at integers: If b= 0, then in order to be equivalent, a must satisfy "a+ 0= a is divisible by 3" so one equivalence class is just the multiples of 3. If b= 1, then we must have a+ 2 divisible by 3 which is the same as saying a is a multiple of 3 plus 1: 1, 4, 7, -2, -5, etc. If b= 2, then we must have a+ 4 divisible by 3: 2, 5, -1, -4, etc. Those are the only three equivalence classes: every integer is in one of those.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Equivalence classes for an particular relation question
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: April 20th 2011, 03:38 PM
  2. Equivalence Relation and Classes
    Posted in the Discrete Math Forum
    Replies: 4
    Last Post: May 4th 2010, 09:07 AM
  3. equivalence relation and equivalence classes
    Posted in the Discrete Math Forum
    Replies: 6
    Last Post: January 7th 2010, 07:36 PM
  4. [SOLVED] equivalence relation/classes question.
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: July 28th 2009, 03:44 PM
  5. Equivalence Classes for a Cartesian Plane Relation
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: March 5th 2009, 11:31 PM

Search Tags


/mathhelpforum @mathhelpforum