Answer
a)$P(-5) \lor P(-3) \lor P(-1) \lor P(1) \lor P(3) \lor P(5)$
b)$P(-5) \land P(-3) \land P(-1) \land P(1) \land P(3) \land P(5)$
c) $P(-5) \land P(-3) \land P(-1) \land P(3) \land P(5)$
d) $P(1) \lor P(3) \lor P(5)$
e) $(\neg P(-5) \lor \neg P(-3) \lor \neg P(-1) \lor \neg P(1) \lor \neg P(3) \lor \neg P(5)) \land (P(-1) \land P(-3) \land P(-5))$
Work Step by Step
a) $\exists xP(x)$ means that there exists a value of x for which P(x) is true, thus P(-5) is true or P(-3) is true or P(-1) is true or P(1) is true or P(3) is true or P(5) is true. Using above interpretation of symbols, we can rewrite the proposition as :
$$P(-5) \lor P(-3) \lor P(-1) \lor P(1) \lor P(3) \lor P(5)$$
b)$\forall x$P(x) means that for all possible values of x: P(x) is true, thus P(-5) is true and P(-3) is true and P(-1) is true and P(1) is true and P(3) is true and P(5) is true. Using above interpretation of symbols, we can rewrite the proposition as:
$$P(-5) \land P(-3) \land P(-1) \land P(1) \land P(3) \land P(5)$$
c) $\forall x((x\ne 1) \rightarrow P(x))$ means that P(x) is true when x is not equal to 1, thus P(-5) is true and P(-3) is true and P(-1) is true and P(3) is true and P(5) is true. Using above interpretation of symbols, we can rewrite the proposition as:
$$P(-5) \land P(-3) \land P(-1) \land P(3) \land P(5)$$
d) $\exists x((x\geq 0)\land P(x))$ means that for one of the positive value of x: P(x) is true, thus P(1) is true or P(3) is true or P(5) is true. Using above interpretation of symbols, we can rewrite the proposition as:
$$P(1) \lor P(3) \lor P(5)$$
e) $\exists x \neg P(x)$ means that there exists a value of x for which $\neg P(x)$ is true, thus $\neg P(-5)$ is true or $\neg P(-3)$ is true or $\neg P(-1)$ is true or $\neg P(1)$ is true or $\neg P(3)$ is true or $\neg P(5)$ is true. Using above interpretation of symbols, we can rewrite the proposition as :
$$\neg P(-5) \lor \neg P(-3) \lor \neg P(-1) \lor \neg P(1) \lor \neg P(3) \lor \neg P(5)$$
$\forall x((x\lt 0)\rightarrow P(x))$ means that for all of the negative value of x: P(x) is true, thus P(-1) is true and P(-3) is true and P(-5) is true. Using above interpretation of symbols, we can rewrite the proposition as:
$$P(-1) \land P(-3) \land P(-5)$$
The given statement $\exists x \neg P(x) \land \forall x((x\lt 0)\rightarrow P(x))$ is the conjunction of $\exists x \neg P(x)$ and $\forall x((x\lt 0)\rightarrow P(x))$, thus the answer is
$$(\neg P(-5) \lor \neg P(-3) \lor \neg P(-1) \lor \neg P(1) \lor \neg P(3) \lor \neg P(5)) \land (P(-1) \land P(-3) \land P(-5))$$
