Describe a partition of Ropf2 into 5 sets What equivalence r

Describe a partition of Ropf^2 into 5 sets. What equivalence relation is associated with this partition? If you assume that the 5 sets are, in fact, a partition of Ropf^2, do you need to prove that your proposed relation is an equivalence relation on Ropf^2? Why or why not?

Solution

First of all let me tell you that you can consider any particular 5-partition and then construct an equivalence relation. Here for symmetry reasons we take following partition:

A := {(x, y)| x > 0 and y > 0}

B := {(x, y)| x > 0 and y < 0}

C := {(x, y)| x < 0 and y > 0}

D := {(x, y)| x < 0 and y < 0}

E := {(x, y) | xy = 0}

Now consider a relation R such that (x1, y1) R (x2, y2) iff both points belongs to same set (A or B or C or D or E). And this is why this is an equivalence relation. Consider reflexivity symmetry and transitivity . It is obvious because the sets are exactly the equivalance classes.