Originally Posted by

**tn11631** 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 $\displaystyle \Longrightarrow R=A\times B\,,\,\,A,B\subset U$ , but then as $\displaystyle (a,b)\,,\,(c,d)\in R=A\times B$ , then

we'd get that $\displaystyle a,c\in A\,,\,\,b,d \in B\Longrightarrow (a,d)\in R$ , by definition of cartesian product, in contradiction with

the given $\displaystyle (a,d)\notin R$

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.