I am studying partial ordered sets. I have a problem with the following example:
$ \text{Let X and Y be sets and } f \in Y^X. \text{The induced functions } f:P(X) \to P(Y)$ and $f^{-1}:P(Y)\to P(X) \text{ are increasing}$.
$P(X) \text{ and } P(Y)$ are power sets of X and Y.
I tried to find a counterexample: $P(X)=\text{{{a},{b},{c},{a,b},{a,c},{b,c},{a,b,c}} and Y={{d},{e},{d,e}}}$. If I assign {a} to {d,e} and {a,b,c} to {d} then I have {a}$\subseteq${a,b,c} and {d,e}$\nsubseteq${d}. What am I doing wrong?