Let sets $A$ and $B$ be given as follows: $$A := \{ (x,y) \in \mathbf{R}^2 | \ \ x < y \ \ \} $$ and $$B := \{ (x,y) \in \mathbf{R}^2 |\ \ x^2 + y^2 < 1 \ \ \}.$$ Can we express $A$ or $B$ as a Cartesian product of two subsets of $\mathbf{R}$? If we can, then how? If we can't, then why not?
What is the most general statement that we can make regarding an arbitrary given set of points in the plane?
We have $X \times Y = \{ (x,y) | x \in X, y \in Y \}$. In particular, if $(x_1,y_1), (x_2,y_2) \in X \times Y$, then $(x_1,y_2), (x_1,y_2) \in X \times Y$.
Neither $A$ nor $B$ can be expressed as a Cartesian product.
Take $A$ first. We have $(1,2),(-2,-1) \in A$, but $(1,-1) \notin A$. Hence $A$ cannot be expressed as a Cartesian product.
Similarly for $B$. Let $x \in (\frac{1}{\sqrt{2}},1)$. Then $(x,0),(0,x) \in B$, but $(x,x) \notin B$.