prove or disprove: If $c > 0$ and $a < b < c $ , then $a \not \equiv b$ mod $c$

Question

I hope this hasn't been asked already, I tried finding it but couldn't.

I claim it is FALSE:

Let $c > 0$ and $a < b < c$, take $c = 1$ (anything is congruent mod 1) but since $a,b$ are integers, we can let $b = 0$ and $a = -1$ and this works?

Thanks in advance. I am a noob to modular arithmetic. Sorry for being naive.

Answer 1

Yes, your example works. Good example of a problem where it is important to know where your variables are drawn from. For instance, if the problem specified $a,b,c\in \mathbb N$ then it would be true. In fact, had they not specified $c>0$ I'd have assumed that they meant for the three to be natural numbers. But the problem should not make us guess. If you read the problem as $a,b\in \mathbb Z$ and $c\in \mathbb N$ (which I think is the ordinary reading of the claim) then your example is correct.