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Can someone please verify this?

Show that if $B \subseteq C$, then $\mathcal{P}(B) \subseteq \mathcal{P}(C)$

let $x \in \mathcal{P}(B)$.

Then, $x \subseteq B$

This implies that $$\forall a \in x, a \in B$$

This further implies that $$\forall a \in x, a \in C$$

Therefore, $$x \subseteq C$$

This implies that $$x \in \mathcal{P}(C)$$

Joffysloffy
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user154185
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  • Looks good to me – Mathmo123 Jun 27 '14 at 10:46
  • Yes, it is correct. Basically, your proof is the statement $(X \in \mathcal{P}(B) \Leftrightarrow X \subseteq B \Rightarrow X \subseteq C \Leftrightarrow X \in \mathcal{P}(C)) \Rightarrow \mathcal{P}(B) \subseteq \mathcal{P}(C)$ elaborated at the lower level. More specifically, all you need to prove is the transitivity of the $\subseteq$ relation. – M. Vinay Jun 27 '14 at 11:01

1 Answers1

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The proof is indeed correct! :)

Belgi
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