Set Theory Simple Proof

It has only just occurred to me, there is a simple set theory proof for a simple problem.

Typical Venn diagram

XX ≠ Ø, XY ≠ Ø 

XX ≠ XY

XX⊄XY, XY⊄XX

Importantly

XX ≠ XY', XY ≠ XX'

Then 

XX ∪ XY ≠ XX, XX ∪ XY ≠ XY

If 

XX x a = XY, and a ≠ Ø, what is a?

XY x b = XX, and b ≠ Ø, what is b?

As above a and b cannot be a NOT operator.

Instead of finding the vector a and b, it would be easier to draw a new set.