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For my homework I have to determine whether $\{\{\emptyset\}\} \subseteq \mathcal P(\{\emptyset,\{\emptyset\}\})$ is true or false.

I believe the answer would be true because $\{\emptyset\} \in \mathcal P(\{\emptyset,\{\emptyset\}\})$ , but I am not sure.

I have been reading previous questions but am still confused about this problem. I know that $\{\{\emptyset\}\} \in \mathcal P(\{\emptyset,\{\emptyset\}\})$, but have been unable to determine the relationship between the two. Any help would be greatly appreciated

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  • $\begingroup$ As would say that as at set, $\{\{\emptyset\}\}$ is both an element and a subset of the set $\mathcal{P}$. $\endgroup$ – anderstood Jun 14 '16 at 22:12
  • $\begingroup$ Do you see why $\{\{a\}\}\subseteq \mathcal{P}(\{a,b\})$? Now, replace $a$ by $\emptyset$ and $b$ by $\{\emptyset\}$... $\endgroup$ – JMoravitz Jun 14 '16 at 22:14
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It's best to work a problem like this in stages:

First, recall the

Definition of subset. $A\subseteq B$ if every element of $A$ is also an element of $B$.

OK, so let's look at this specific case: we want to know if $\{\{\emptyset\}\}\subseteq \mathcal{P}(\{\emptyset, \{\emptyset\}\})$. This is true if and only if all the elements of $\{\{\emptyset\}\}$ - namely, $\{\emptyset\}$ - are also elements of $\mathcal{P}(\{\emptyset, \{\emptyset\}\})$.

So we want to know: Is $\{\emptyset\}$ in $\mathcal{P}(\{\emptyset, \{\emptyset\}\})$? Now we need the

Definition of powerset. $A\in\mathcal{P}(B)$ iff $A\subseteq B$.

So we want to know: Is $\{\emptyset\}$ a subset of $\{\emptyset, \{\emptyset\}\}$?

Going back to the definition of subset, do you see how to answer this question?

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  • $\begingroup$ Aaahh yes, I see why it is true now.. Thank you $\endgroup$ – Turtle Jun 14 '16 at 22:46

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