So i've been stuck on this for a while now and im just not sure what to do. I'm giving this definition of a product set:

A relation R on a set U is said to be a product set if there are subsets A,B of U such that R = A × B.

How do I use that to prove the following theorem? Let U be a set and let R a relation on U. If there exists four elements a,b,c,d of U such that (a,b) ∈ R, (c,d) ∈ R and (a,d) is not in R, then R is not a product set.

Suppose the relation is a product set , but then as , then

we'd get that , by definition of cartesian product, in contradiction with

the given

Check carefully the above and then try, with the new understanding, to answer the following two questions

Tonio
Also, is the set {(x, y) ∈ R2 : xy not equal 0} a product set, is the set {(x, y) ∈ R2 : xy not equal 1} a product set? and how would I prove them? I'm completely blank.