Proof of duality law $E-\bigcap_i A_i = \bigcup_i (E-A_i)$

Question

The proof of the theorem :

given a set $E$ and any family of sets $\{A_i\}$; $\quad E-\bigcap_i A_i = \bigcup_i (E-A_i)$

has been left to the reader in Basic Concepts of Mathematics by Elias Zakon. I have attempted the proof but, as I am inexperienced in proof writing, I am not are if there are faults in my logic, or areas where I can be more succinct or clearer. My proof is:

$x \in E- \bigcap_i A_i \\x \in E \quad\text{but} \quad x\notin \bigcap_i A_i \\ \text{As $x\in E$ but $x$ is not in all the sets $A_i$, therefore $x$ is in some of the sets $A_i$} \\ x \in \bigcup_i (E-A_i) \quad\text{therefore $ \ E-\bigcap_i A_i = \bigcup_i (E-A_i)$}$

Answer 1

$x$ is not in all the sets $A_i$, therefore $x$ is in some of the sets $A_i$

This is not true. It could be that $x$ isn't in any of the $A_i$. The conclusion is supposed to be "Therefore there is at least one $k$ with $x\notin A_k$". Now, since $x\in E$ and $x\notin A_k$, we have $x\in E-A_k$. Finally, apply the definition of union.

Once you've done that, you've proven that $$ E-\bigcap_i A_i\subseteq \bigcup_i(E-A_i) $$ It remains to show the opposite inclusion. You do that basically the same way. Take an $x\in \bigcup_i(E-A_i)$, and reason that we must have $x\in E-\bigcap_i A_i$, or by being careful pointing out that each line in your argument is actually an equivalence, so the argument works both ways.

Answer 2

Besides the issues that Arthur pointed out, you should include some transition words to introduce each of your steps which indicate how they are logically related to each other. For instance, instead of just presenting your first two lines, you could write something like:

Suppose that $x \in E- \bigcap_i A_i$. Then $x\in E$ but $x\notin \bigcap_i A_i$.

The words "suppose that" are extremely important: they indicate that you are assuming $x$ is an arbitrary element of the set $E- \bigcap_i A_i$ (and will subsequently prove something about such an $x$). If you write just $x\in E- \bigcap_i A_i$, you are using a variable $x$ which you have never defined and whose role in your logic is unclear. The word "then" introducing the second sentence indicates that this is a consequence of the first sentence.