Equivalence Classes in PxP for (1,2)

[mamma_mia66]

Homework Statement

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)

Homework Equations

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) ...

 

[steelphantom]

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.

 

[mamma_mia66]

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)

 

[Dick]

Looks fine to me.