How to prove answer to solution of a material conditional statement for any x

Question

I recently enrolled for a class on discrete mathematics and this problem came up:

Assume x is a natural number (0,1,2,3,4...) and let P(x) denote "x divided by 3 gives no remainder". Determine the truth value of the following proposition: $$\forall x (P(x) \rightarrow \lnot(P(x+2))$$

Using regular reasoning I can tell that this must be true for any x (or so I think), since any number that is divisible by 3 can't be divisble by itself + 2 since then it would not be a multiple of 3 anymore. In addition, if x is not divisible by 3 aka P(x) is False, then the conditional statement automatically evalutes to true due to the laws of Material Conditional. Though I feel like I got the answer right I'm not sure where to start when coming up with a proof. I've tried setting it up as an equation but I'm a bit puzzled on how to do that for any x. How would I go about determining the truth value of a proposition like this and then prove my answer?

Answer 1

You're right that the statement is true.

Let $x$ be arbitrary. If $P(x)$ is false, we're done instantly. Otherwise, $P(x)$ is true, so $P(x+3)$ is also true, so $P(x+2)$ must be false (because subtracting $1$ from a multiple of $3$ yields a non-multiple of $3$).

Depending on how pedantic you need to be:

Lemma: subtracting 1 from a multiple of 3 yields a non-multiple of 3. More precisely, if $n$ is a multiple of $3$, then $n-1$ is not a multiple of $3$.

Proof: by contradiction. Suppose $n$ and $n-1$ are both multiples of $3$. Then their difference is $1$; but the difference of two multiples of $3$ is a multiple of $3$, so we have that $1$ is a multiple of $3$, which is patently rubbish (it's less than $3$).