Suppose that R1 and R2 are equivalence relations on a set A. Let P1 and P2 be the partitions that correspond to R1 and R2, respectively. Show that R1 is a subset of R2 if and only if P1 is a refinement of P2.
We must show these two implications:
If R1 is a subset of R2 then P1 is a refinement of P2 AND
If P1 is a refinement of P2 then R1 is a subset of R2.
1)I am not sure how to do this one...
2) Assume P1 is a refinement of P2. Then every set in P1 is a subset of one of the sets in P2 (lets call it p). since P1 is a partition of R1 then the union of all sets in P1 = R1 and because all sets in P1 are in p then all elements of R1 are in P2 and since P2 is a subset of R2, R1 is a subset of R2.
The second one sounds too informal, I would really appreciate some help (especially with the first implication) THANKS!!!!!!!!!