5. (9 points) Recall that, fixed a set $U$ (which we call the universe of discourse), we have certain operations on subsets of $U$ so that, for all $A, B, C \subseteq U$ the following equivalences and equalities hold.
$A \setminus B \subseteq C \iff A \subseteq B \cup C$,
$C \subseteq A \to B \iff C \cap A \subseteq B$,
$A \subseteq B \iff B^c \subseteq A^c$,
$A^\prime = A$,
$(A \cup B)^\prime = A^\prime \cap B^\prime$,
$(A \cap B)^\prime = A^\prime \cup B^\prime$.
You can answer just one of the following parts, not both. You can support your answer with drawings of Venn diagrams, but you need to give an argument according to the specifications for full credit.
2
(a) Prove that for any given sets $A, B \subseteq U$, we have that $B \setminus A^\prime = (A \to B)^\prime$ using only the above equations and equivalences. (Hint: Notice that two sets $X, Y \subseteq U$ are equal if and only if, for every $C \subseteq U$, we have $X \subseteq C \iff Y \subseteq C$.)
(b) Prove that for any given sets $A, B \subseteq U$, we have that $B \setminus A^\prime = (A \to B)^\prime$ using the definitions of the operations $(\)^\prime$, $\setminus$, and $\to$ in terms of the elements of $U$, $A$, and $B$.
