This is to check my work on a problem from Discrete Math and Its Applications. Here is the problem.
My question is on part d. I would say that c does not follow from a and b because it is true that all professors are not ignorant and that ignorance implies being vain. However in this case, the proposition "all ignorant people are vain" does not say anything about non ignorance peoples and their vanity. (non ignorant peoples can still possibly be vain and that would would not violate proposition b(All ignorant people are vain)). Therefore you cannot imply that non ignorant peoples(the professors) are not vain. Is my logic correct here or am I missing something?
C does not follow from a and b.
We know that no professors are ignorant and that all ignorant people are vain. We cannot conclude from this knowledge that no professors are vain. We can only conclude than no professors are ignorant and therefore none are vain due to their ignorance.
All you can say for certain is that professors are not ignorant and that you have eliminated one of a possible multitude of ways a professor could 'arrive at' vanity.
That is, it is still entirely possible for these non-ignorant professors to be vain.