Answer
a. $\forall x (P(x)\rightarrow\neg Q(x))$, if a person is a professor than he is not ignorant.
b. $\forall x (Q(x)\rightarrow R(x))$, if a person is ignorant then he is vain.
c. $\forall x (P(x)\rightarrow\neg R(x))$, same as part a.
d. The part c does not follow from part a and b because part b does not rule out that person who is not ignorant can be vain, so a professor who is not ignorant can still be vain.
Work Step by Step
Same as above.
