The five$-$variable propositional formula

P ::$=($A$$B$$C $$D$$E $)($A $$B$$C $$E $)$

is in Disjunctive Normal Form with two $$AND$-$of$-$literal$$ clauses.

$($a$)(5$ points$)$ Find a full Disjunctive Normal Form$1$ that is equivalent to P $,$ and explain your reasoning. Hint: Can you narrow in on the important parts of the truth table without writing all of it$?$ Alternatively, can you avoid the truth table altogether?

$($b$)(5$ points$)$ Let C be a full Conjunctive Normal Form that is equivalent to P$.$ Assume that C has been simplified so that none of its $$OR$-$of$-$literals$$ clauses are equivalent to each other. How many clauses are there in C$?$ Please don$$t try to write out any of these clauses. Briefly explain your answer.The five$-$variable propositional formula

$P$::$=(A^{{?}^{?}}\frac{B^{{?}^{?}}}{b}ar(C{)}^{{?}^{?}}\frac{D^{{?}^{?}}}{b}ar(E))vv(\frac{?}{\text{b}\text{a}\text{r}}(A{)}^{{?}^{?}}\frac{B^{{?}^{?}}}{b}ar\frac{(C{)}^{{?}^{?}}}{b}ar(E))$

is in Disjunctive Normal Form with two "AND$-$of$-$literal" clauses.

$($a$)(5$ points$)$ Find a full Disjunctive Normal Form ${?}^1$ that is equivalent to $P,$ and explain your

reasoning. Hint: Can you narrow in on the important parts of the truth table without

writing all of it$?$ Alternatively, can you avoid the truth table altogether?

$($b$)(5$ points$)$ Let $C$ be a full Conjunctive Normal Form that is equivalent to $P.$ Assume

that $C$ has been simplified so that none of its $"$OR$-$of$-$literals" clauses are equivalent to

each other. How many clauses are there in $C?$ Please don't try to write out any of these

clauses. Briefly explain your answer.