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: Equivalence relation on the Cartesian plane

  1. Aug 6, 2010 #1
    1. The problem statement, all variables and given/known data
    A relation p is defined on R^2 (fancy R, as in Reals) by (a,b)p (c,d) if a+d=b+c
    Show that p is an equivalence relation.

    b) Consider R^2 to be the Cartesian Plane. Describe p's equivalence classes geometrically. (Consider which points will be in the particular equivalence classes by taking an arbitrary point in the same equivalence class as (x,y). )

    I have done part a. In part b I only got as far as drawing the Cartesian axes and a table of values. I'll show you below. I think that I have not made enough of a start for you to be able to give me a clue, but I thought you might be able to point me in the direction of a book that would cover this. I am finding my course really hard because I am studying by distance education and we don't have a textbook. When I browse through the library I am not finding anything that quite fits my course. Direction to an online resource would be particulary good or a text book that you think would be readily available at my university library.

    2. Relevant equations
    To be an equivalence relation, p must be reflexive symmetric and transitive. I have shown all that.


    3. The attempt at a solution

    I drew up a list of values
    a b c d
    1 2 1 2
    4 3 3 2
    1 2 2 3
    -1 0 1 2


    I did about 30 so that I had a really good idea of what was happening.
    What I figured out:
    Points on line y=x+1 map to (1,2)
    Points on line y=x+2 map to (1,3)
    Points on line y=x+3 map to (1,4)
    etcetera

    I discovered that I couldn't draw it on the Cartesian axes. Am I meant to be able to?

    I don't even understand if "decribe p's eqivalence classes geometrically" means I am meant to draw or use words.

    I have been puzzling over this one for about a month now.
    Any clues you can give me to point me in the right direction will be greatly appreciated I assure you.
    Many thanks is anticipation.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Aug 7, 2010 #2

    HallsofIvy

    User Avatar
    Science Advisor

    (a, b) p (c, d) if and only if a+ d= b+ c. Since you are talking about the Cartesian plane, it might help to use (x, y) and (x', y') instead. x+ y= x'+ y' is the same as y- y'= x- x' or (y- y')/(x- x')= 1. That gives exactly the result you talk about. An equivalence class is a straight line with slope 1.


    By the way, this same equivalence relation can be used to define the integers given on the positive integers (counting numbers). If x> y and (x, y)p (x', y') then it must be true that x'> y' and, in fact, x'- y'= x- y. So we can think of the equivalence class [(x, y)] as defining "x- y". For x> y, of course, that is just a positive integer but if x= y and (x, y) p (x', y') then x'= y' and we can think of the equivalence class as defining "0". Similarly, if x< y and (x', y')p (x, y), x'< y' and we can think of the equivalence class as defining the negative integer x- y= -(y- x).
     
  4. Aug 7, 2010 #3
    Thank you very much for your help. Much appreciated!
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook