Showing final part of equivalence relation
Let S be the set of all ordered pairs (m,n) of positive integers. For (a1,a2) in S and (b1,b2) in S, define (a1,a2)~(b1,b2) if a1+b2=a2+b1.
I need to show we have an equivalence relation.
The first two parts I have already determined to be true, so i will omit my work on that.
For the third part I need to show if for all a,b,c in S if (a,b) is in ~ and (b,c) is in ~ then (a,c) is in ~.
We want to show:
I'm unsure of how to get there.
I tried using b1=a1-a2+b2 and b2=a2-a1+b1
That didn't really get me anywhere, though.