My real analysis class is using Terence Tao's "Analysis 1" this semester, which unfortunately doesn't have solutions available anywhere. Part of the current homework is exercise 3.4.11:
"Let $X$ be a set, let $I$ be a non-empty set, and for all $\alpha \in I$ let $A_\alpha$ be a subset of $X$. Show that
$$X – \bigcup_{\alpha \in I} A_\alpha = \bigcap_{\alpha \in I} (X – A_\alpha)$$
and
$$X – \bigcap_{\alpha \in I} A_\alpha = \bigcup_{\alpha \in I} (X – A_\alpha).$$"
The book suggests comparing them with de Morgan's laws, and I see the similarities, but it also says that one cannot derive these identities from de Morgan's laws since $I$ could be infinite.
I'm not sure where to go with this. Since we're comparing sets, I know that the place to start is "Let $x \in X – \bigcup_{\alpha \in I} A_\alpha$." I also think from this we can deduce that $x \in X$ and $x \notin \bigcup_{\alpha \in I} A_\alpha$, but I'm not sure where to go from there.
Best Answer
De Morgan's laws are not necessary. We have \begin{align} x\in X\setminus\bigcup_{\alpha\in I} A_\alpha &\iff x\in X, x\notin A_\alpha \text{ for all $\alpha\in I$}\\ &\iff x\in X\setminus A_\alpha \text{ for all $\alpha\in I$}\\ &\iff x\in \bigcap_{\alpha \in I}(X\setminus A_\alpha), \end{align} and similarly for the other statement.
Also, De Morgan's laws hold for sets of arbitrary cardinality.