If I am correct it is true that: 

(1) "$P(A)$ is included in $P(B)$" implies "$A$ is included in $B$". 

(2) "$P(A) = P(B)$" implies "$A = B$". 

Might I conclude from this that the power sets of two sets always have the same relations as these two sets have with one another? 

Are there **classical counterexamples** to this (hasty) generalization? 

I can think of this as a counterexample : 

The fact that $A$ and $B$ are disjoint does NOT imply that $P(A)$ and $P(B)$ are disjoint. 

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Attenpt to prove (1) using the theorem : "$\{ x \}$ belongs to $P(S)$" $\Longleftrightarrow$ "$x$ belongs to $S$.

Let's admit that : $P(A)$ is included in $P(B)$. 

Now, suppose (in view of refutation) that $A$ is not included in $B$. 

It means that there exists an x such that x belongs to $A$ but not to $B$. And consequently that there is an $x$ such that $\{ x \}$ belongs to $P(A)$ but not to $P(B)$. If this were true, there would be a set $S$ such that $S$ belongs to $P(A)$ but not to $P(B)$. This contradicts our hypothesis according to which $P(A)$ is included in $P(B)$. 

Conclusion: "$P(A)$ is included in $P(B)$" implies "$A$ is included in $B$".