Answer
a) There exists a value between 1 and 5 such that P(x) is true, this thus equivalent with $ P(1) \lor P(2) \lor P(3) \lor P(4) \lor P(5)$.
b) For every value between 1 and 5 we need P(x) is true, this thus equivalent with $ P(1) \land P(2) \land P(3) \land P(4) \land P(5)$.
c) This statement is negation of the statement in a, and is thus equivalent with $ \neg (P(1) \lor P(2) \lor P(3) \lor P(4) \lor P(5))$.
d) This statement is negation of the statement in b, and is thus equivalent with $ \neg (P(1) \land P(2) \land P(3) \land P(4) \land P(5))$.
e) This statement is equivalent with
$( P(1) \land P(2) \land P(4) \land P(5)) \lor (\neg P(1) \lor \neg P(2) \lor \neg P(3) \lor \neg P(4) \lor \neg P(5))$, because the first statement means that P(x) is true when x is not equal to 3 and the second statement means that there has to be a value of x where P(x) is false and thus $\neg P(x) $ is true.
Work Step by Step
See Answer.
