I have a proposed solution, but am not sure if I am correct.
I am looking to simplify a set. With only using $A, B, B', A'$ expressions. $A, B$ are the subsets of Universe.
\begin{align} (A' \cap \emptyset') \cap (A' \cup B') &= A' \cap (A' \cup B') \end{align}
I am not certain on how to simplify or statements. Any tips on where I am wrong, and examples are greatly appreciated.
If you keep doing distribution, you'll just end up going around in circles:
$$A' \cap (A' \cup B') =$$
$$ (A ' \cap A') \cup (A' \cap B') = A' \cup (A' \cap B') = (A ' \cup A') \cap (A' \cup B') = $$
$$A' \cap (A' \cup B')$$
Instead, you can do:
$$A' \cap (A' \cup B') = (A' \cup \emptyset) \cap (A' \cup B' ) = A' \cup (\emptyset \cap B') = A' \cup \emptyset = A'$$