How to determine if a predicate statement is true or false while giving reasons?

I am currently studying discrete maths at university. However the text book doesnt seem to explain predicate statements in much detail (only one chapter on them).

In the assignment which i am currently working on (and is due tomorrow), it asks me to determine whether a set of statements is true or false, and give reasons. The statements are mostly like:

for all x in the integers Z, not S(x) implies P(x)
where P(x) = x >=0, and S(x) = x^2 - 1 = (x+1)(x-1)

However no where in the text book does it explain how to do this.

Note that we need to evaluate the truth of the statement $$\forall x \in \mathbb Z, ( \lnot S(x) \rightarrow P(x))$$
Now, for all $x\in \mathbb Z$ (and all $x \in \mathbb R)$, $$S(x) \equiv (x^2 - 1) = (x + 1)(x-1)$$ This is a tautology, true whatever the value of $x$.
That makes $\lnot S$ false. An implication is true whenever the antecedent is false, or when the consequent is true. Here, the antecedent is clearly false (it is never satisfied by any real number, let alone integer). Hence the implication, as a whole, is true.
The fact that $\lnot S(x)$ is false is sufficient justification to conclude that the implication is true. (If this seems confusing, jot down the truth table for $\rightarrow$ to refresh your memory. An implication is false if and only if the antecedent is true and the consequent is false.