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True or False:

If $A_1=\{\emptyset\}$, then $A_1$ $∈$ the power set of $A$ for all sets $A$.

I'm confident this would be true if $A_1=\emptyset$, but does the question change since $A_1=\{\emptyset\}$?

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  • $\begingroup$ Did you mean "$A_1\subset$ the power set"? $\endgroup$ – user147263 Feb 9 '15 at 1:20
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A1 is included in all power sets. The notions of $\subset$ and $\in$ are different. A counterexample is $A_1\notin\mathcal P(\{1\})=\{\emptyset,\{1\}\}$. But it is true that $A_1\subset\mathcal P(\{1\})$.

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The power set of $\emptyset$ is $\{\emptyset\}$. And $\{\emptyset\}\notin\{\emptyset\}$ because the only element of this set is $\emptyset$, which is different from the non empty set $\{\emptyset\}$.

However, since $X\subset Y$ implies $P(X)\subset P(Y)$, we can surely say that $$ \{\emptyset\}=P(\emptyset)\subset P(A) $$ for every set $A$, since $\emptyset\subset A$.

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