1. Not finding help here? Sign up for a free 30min 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!

Equivalence relation and equivalence class

  1. Feb 2, 2012 #1
    i have two relations given to me which are both defined on the integers Z by

    relation 1: x~y if 3x^2 -y^2 is divisibale by 2

    and relation 2: x~y if 3x^2 -y^2 ≥0

    I have used three properties to figure out that relation 1 is eqivalence relation as it stands for all three properties i.e. reflexive, symmetric and transitive where as relation 2 is not equivalence as its not symmetric

    If this is correct- which I think it is. I have no idea what to do with second part
    which is:
    I have for relation 1: x~y if 3x^2 -y^2 is divisibale by 2 ( which is equivaleance), Show that :
    [3]={2k+1:k # Z} # means belongs to


    i would appriciate detail explaination and perhaps similar examples or show me how to do this.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Feb 2, 2012 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    [3] is the set of all numbers that are equivalent to 3 under this equivalence relation.

    Since x~ y if and only if [itex]3x^2- y^2[/itex] is divisible by 2, y is equivalent to 3 if and only if [itex]3(3^2)- y^2[/itex] is divisible by 2. That is, if and only if [itex]27- y^2= 2n[/itex] for some integer n. That, in turn, gives [itex]y^2= 27- 2n[/itex]. Now, n=1, that is 25 so y= 5 is in that set. [itex]27- 2n= y^2[/itex] is the same as 2n= 27- y^2. 27- y^2 is even if and only if [itex]y^2[/itex] is odd- if and only if y is odd. For example, if y= 7, [itex]27- 49= -22= 2(-11)[/itex] so [itex]3(3^2)- 7^2= -22[/itex] is divisible by 2. Every odd integer is equivalent to 3. Since equivalence classes "partion" the entire set, we then need to look at even numbers if y is equivalent to 3, then [itex]3(4)- y^2= 2n[/itex] so [itex]y^2= 12- 2n[/itex]. What integers, y, satisfy that?
     
  4. Feb 2, 2012 #3
    I understand this but my definition is [3] = {2k+1:k €Z} so what do I do with +1
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Equivalence relation and equivalence class
  1. Equivalence relation (Replies: 4)

  2. Equivalence Classes (Replies: 5)

  3. Equivalence relations (Replies: 1)

  4. Equivalence Relations (Replies: 11)

  5. Equivalence Relations (Replies: 17)

Loading...