# The Union and Intersection of the Power Set

In the question, it's asked to find:$$\bigcup_{X=\mathscr{P}(\mathbb N)}{X}$$ and $$\bigcap_{X=\mathscr{P}(\mathbb N)}{X}$$Does this mean find the union and intersection of the power set of all natural numbers? Wouldn't the result be $$\mathbb N$$ for both cases since $$\mathbb N$$ is common for both sets. Please help, thanks!

• Isn't Null a part of the power set and wouldn't the intersection of null with any other set be the null set – Kitter Catter Oct 5 '19 at 3:38
• @KitterCatter I thought about that, but wouldn't all natural numbers be repeated for the intersection, including the null set? – Random Student Oct 5 '19 at 3:50
• The intersection consists of exactly those natural numbers that are in all subsets of $\mathbb{N}$. The union consists of exactly those natural numbers that are in at least one subset of $\mathbb{N}$. Is each natural numbers in at least one subset? If so, you are correct about the union; if not, then you are incorrect. Is each natural number in all subsets of $\mathbb{N}$? If so, you are correct about the intersection. If not, then yhou are incorrect. – Arturo Magidin Oct 5 '19 at 3:56
• Do you mean $X \in \mathscr{P}(\Bbb{N})$ instead of $X = \mathscr{P}(\Bbb{N})$? – J.-E. Pin Oct 5 '19 at 6:01

\begin{align*} \bigcup_{x \in \mathcal{P}(\mathbb{N})} x &= \bigcup_{x \subseteq \mathbb{N}} x \\ &= \mathbb{N} \cup \bigcup_{x \subseteq \mathbb{N}} x \\ &= \mathbb{N} \end{align*}
\begin{align*} \bigcap_{x \in \mathcal{P}(\mathbb{N})} x &= \bigcap_{x \subseteq \mathbb{N}} x \\ &= \varnothing \cap \bigcap_{x \subseteq \mathbb{N}} x \\ &= \varnothing \end{align*}