Okay, so I'm stuck on a question and I'm not sure how to solve it, so here it is: In the following questions, $B_n = \mathcal{P}(\{1, ... , n\})$ is ordered by containment, the set $\{0,1\}$ is ordered by the relation $0 \leq 1$, and $\{0,1\}^n$ is ordered using the product order.
a) Draw Hasse diagram of $\{0,1\}^3$
b) Give an order isomorphism from $B_3$ to $\{0,1\}^3$
c) Prove that $B_n$ is isomorphic to its dual. Your proof must include a clear explicit definition of the order isomorphism you use to prove this.
d) Prove that $\{0,1\}^n$ is isomorphic to $B_n$. Your proof must include a clear and explicit definition of the order isomorphism you use to prove this.
Okay so I got a) and b), I started on c) but what's throwing me off is that $B_n$ is infinite (meaning I have to show that its isomorphic to its dual for all sets (or elements?) of numbers.
I started c) by "Let $R$ be an order relation on $S$. The dual order is $R^{-1}$. In other words if $a R b$ then $b R a$."