simplify:$$
(A\cup B)\cap (B\cup C\cup D)\cap(B\cup C\cup D')
$$
the end result should only have one $\cup$ and one $\cap$ symbol.
a bit confused on how to start. I think we can use the distributive property here. Take $x=A\cup B$ and the other terms as $y$ and $z$, respectively.
however, i'm not sure which distributive property since the equation has 2 intersect symbols and the properties have union and intersect symbol (I hope that made sense).
Best Answer
As you said, you can use the distributive law of union over intersection: $$ \begin{align} (A \color{red}{\cup B})\cap (\color{red}{B \cup} C\cup D)\cap(\color{red}{B \cup} C\cup D') &= \color{red}{B \cup} \left(A \cap (\color{green}{C \cup} D) \cap (\color{green}{C \cup} D')\right) \\ &= B \cup \left( A \cap \left(\color{green}{C \cup} (D \cap D') \right) \right) \\ &= B \cup (A \cap C) \end{align} $$ Note that the last equality is because $D \cap D' = \varnothing$ for any set $D$.