I wish to prove De Morgan's Laws. I saw the Wikipedia page of De Morgan's Law proof, which seems to confuse me.
Say $x\in (A\cap B)^{\complement}\implies x\not\in (A\cap B)$ which makes perfect sense to me, but in the very next line, they say that this implies $x\not\in A \lor x\not \in B$ which is not very intuitive because if we think about it even in terms of Venn Diagrams (which is not formal, but still), for an element that fails to be in the intersection of $2$ sets, it's not at all necessary for it to not be a member of those $2$ sets.
Why is this implication meaningful?
Best Answer
It may be easier to note $x\not\in(A\cap B)^\complement\iff x\in A\cap B\iff x\in A\land x\in B$, so taking the contrapositive $x\in(A\cap B)^\complement\iff x\not\in A\lor x\not\in B$.