Determine whether $x-1>0$ implies $x=0$ over a given domain

Question

I'm taking Mathematics for my degree, mostly all the time I don't know how to answer mathematic question(not because I don't know the answer) and how to start to answer the question.

For Example :(logic) consider the statement $P(x) :x-1>0$ , $Q(x) :x=0$ . where $P(x)$ and $Q(x)$ over domain A. $A=\{-1,0,1\}$ Determine whether $P(x)\implies Q(x)$ is true. So how do I start answering the question?

Answer 1

You could try some algebra or brute force it by listing the table of all possible arguments and values and checking if those form a valid implication or comparison.

$$ \begin{array}{|c|c|c|c|c|} \hline x & P(x) & Q(x) & P(x)\, R \, Q(x) & \text{valid} \\ \hline \vdots & \vdots & \vdots & \vdots & \vdots \\ \hline \end{array} $$

Answer 2

One thing that may help is to decide whether you need a universal proof or a counterexample. For instance, in order to prove that $x - 1 > 0$ implies $x > -1$ where the domain of $x$ is all integers, you would need to use facts of algebra and rules of logic from some previously-developed sets of facts and rules in order to form a logical sequence of statements. (I cannot tell you what facts and rules would be appropriate to use in your exercises, because it depends on context; for a given homework question, it would be all the facts and rules presented in the class so far, plus any others you may have been told you can assume.)

If $x$ is restricted to just the set $\{-1, 0, 1\}$ you might build a logical sequence of statements to prove the fact just as you would for all integers; but for this domain you have also have the ability to simply test the statement for every possible value of $x$ and show that it is true for each and every value.

On the other hand, if you are asked whether a statement is true, and the statement is false, all you need to do is to come up with one counterexample. That is, if you can deduce (or even just guess) a value of $x$ that, when plugged into the two parts of the implication, results in a false implication, then all you need to do is to name that value of $x$ and evaluate all parts of the implication assuming $x$ has that value. Your answer might start like this: "Let $x = \ldots$."

So the first step in writing your answer is to decide whether the statement is true or false. Which is it?