This question is related to another answered question, but I am struggling to relate the two (as a self-learner, I have no other avenues to get some feedback).
The problem which I am trying is the one shown in the below screenshot. The red marked lines are where I think there could be a problem.
On the first red mark, I am wondering whether distributivity of OR over AND is missing something.
If there is no problem in that line (the linked answer for the related problem had assumed that line to be correct), then the error is in the last line. I suppose the reasoning $x \in (A \cup B) \space and \space y \in (C \cup D) \iff (x,y) \in L $ cannot be taken to conclude that the two sets $L$ and $R$ are equal.
But if my above reasoning is correct, how to fix the proof to show that $R \subseteq L$ ? Would a simple statement that $ (x,y) \in L \implies R \subseteq L$ enough?
From
$x \in A$ or $x \in B$ and $y \in C$ or $y \in D$
you cannot reach the conclusion
$(x \in A$ and $y \in C)$ or else $(x \in B$ and $y \in D)$
so
$(x \in A$ and $y \in C)$ or else $(x \in B$ and $y \in D) \implies$ $x \in A$ or $x \in B$ and $y \in C$ or $y \in D$
but not the other way around, which explains both why the proof is bogus (the statements are not an joined by an iff) and why $R \subseteq L$