set theory question from munkres topology Munkres topology book set theory chapter 1 question 10c
Let $\mathbb R$ denote the set of real numbers. for each of the following subsets of $\mathbb R\times\mathbb R$ determine whether it is equal to Cartesian product of two subsets of $\mathbb R$.
Now here $\{(x,y)| y\gt x\}$ is given as not equal to Cartesian product of two subsets of $\mathbb R$ but we can construct a product like that below:
$\{(x,y)| x \in \mathbb R, y\in B\text{ such that } B=\{y| y\gt x\}\}$
Still $x$ and $y$ are subsets of $\mathbb R$
Can anyone explain me why its not a true Cartesian product definition

If $(a_1,b_1),(a_2,b_2)\in A\times B$ then $a_1,a_2\in A$ and $b_1,b_2\in B$.  But then $(a_1,b_2)$ and $(a_2,b_1)$ are also in $A\times B$ since the latter is the set of all ordered pairs such that $a\in A$ and $b\in B$.  Apply this to the set $S=\{(x,y)\mid y\gt x\}$.  The pairs $(1,2)$ and $(3,4)$ are in $S$.  If $S$ is a Cartesian product then $(3,2)$ must be in $S$.  But this contradicts the definition of $S$.
