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Equivalence Relations

  1. Nov 6, 2008 #1
    1. The problem statement, all variables and given/known data

    On set PxP, define (m,n)[tex]\approx[/tex](p,q) if m*q=p*n
    Show that [tex]\approx[/tex] is an equivalence relation on PxP and list three elements in equivalence class for (1,2)



    2. Relevant equations





    3. The attempt at a solution
    I will appreciate any help how to start this problem. I now I have to show R, S and T properties, but I am confused from the notation above m*q=p*n

    do I have to start with listing some pairs like
    (0,0) (0,1) (1,0) (0,2) (1,1) (2,0) (0,3) (1,2) (2,1) (3,0) ....
     
  2. jcsd
  3. Nov 6, 2008 #2
    For reflexive, show that (m, n) ~ (m, n) is true. For symmetric, show that if (m, n) ~ (p, q) then (p, q) ~ (m, n). Just check these explicitly to see if they work out.
     
  4. Nov 7, 2008 #3
    How is that?

    Reflexive: b/c m*n=m*n then (m,n)[tex]\approx[/tex](m,n)

    Symmetric: if (m,n)[tex]\approx[/tex](p,q)

    then m*q=p*n and p*n=m*q

    => (p,q)[tex]\approx[/tex](m,n)

    Transitive: if (m,n) [tex]\approx[/tex](p,q) and (p,q)[tex]\approx[/tex](r,s)

    then m*q=p*n and p*s=r*q

    m/n=p/q and p/q=r/s

    m/n=r/s

    m*s=r*n

    (m,n)[tex]\approx[/tex](r,s)

    the three elements:
    (1,2)= (3,6) (1,2)= (4,8) (1,2)= (5,10)
     
  5. Nov 7, 2008 #4

    Dick

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    Looks fine to me.
     
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