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

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?

• Can you explain what P and Q are? They seem like logical statements to me, but apparently they're functions? Try to carefully copy out what the question is asking. EDIT: Is it boolean algebra? – Kevin Long Oct 28 '15 at 2:57
• That edit changed the relation $R$ in $P R Q$ from implicatiom to greater or equal comparison. What was intended? – mvw Oct 28 '15 at 3:02
• "all the time I don't know how to answer mathematic question(not because I don't know the answer)". That seems contradictory. Do you mean that you usually know the final answer, but not how to explain it? In your specific example, what does "not because I don't know the answer" refers to? Do you mean that you guess it is false but don't know how to justify it? – Taladris Oct 28 '15 at 5:49
• If you already know whether $P(x)\implies Q(x)$ is true, and the difficulty is merely in how to justify your answer, then you already know something about this problem that you have not explained in the question. Namely, is the implication true or false, and what makes you think so (in whatever way you can describe that, not necessarily in a mathematical way)? – David K Oct 28 '15 at 13:35
• By the way, if you're having difficulty signing in as user284608, try the help pages, or look at questions such as meta.math.stackexchange.com/questions/12599/… so that you can edit your question appropriately if the answers you have received are not answering your real question. (Also so that you can accept an answer if it does answer your question.) – David K Oct 28 '15 at 13:39

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