0
$\begingroup$

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!

$\endgroup$
  • 1
    $\begingroup$ Isn't Null a part of the power set and wouldn't the intersection of null with any other set be the null set $\endgroup$ – Kitter Catter Oct 5 '19 at 3:38
  • $\begingroup$ @KitterCatter I thought about that, but wouldn't all natural numbers be repeated for the intersection, including the null set? $\endgroup$ – Random Student Oct 5 '19 at 3:50
  • 1
    $\begingroup$ 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. $\endgroup$ – Arturo Magidin Oct 5 '19 at 3:56
  • 3
    $\begingroup$ Do you mean $X \in \mathscr{P}(\Bbb{N})$ instead of $X = \mathscr{P}(\Bbb{N})$? $\endgroup$ – J.-E. Pin Oct 5 '19 at 6:01
1
$\begingroup$

\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*}

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.