#### Answer

a. Statement: ∃ a program P such that ∀ questions Q posed to P, P gives the correct answer to Q.
b. Negation: ∀ programs P, there is a question Q that can be posed to P such that P does not give the correct answer to Q.

#### Work Step by Step

Recall the negation of a for all statement:
~($\forall x$ in D, P(x)) $\equiv \exists x$ in D such that ~P(x).
Recall the negation of an exists statement:
~($\exists x$ in D, P(x)) $\equiv \forall x$ in D such that ~P(x).
To negate a multiply quantified statement, apply the laws in stages moving left to right along the sentence.