Proof Involving Rational Numbers

Question

I asked this same question last night and got some answers but still can't make sense of this, normally I'd move on but since I know how to do everything else for the test I'm going to try to get this down. Anyway .....

Stuck on a tutorial question trying to study for a test. The question is :

Consider the following statement: "Between any two different rational numbers, there are at least two different rational numbers." (a) Write this statement as a logical expression. The universe is all numbers. Use Q to denote the set of rational numbers.

(b) Prove or disprove this statement.

Thanks, proofs are what I'm having the hardest time with. There are actually other parts to the question but I know how to do those, can someone tell me what they'd consider the full answer?? Our prof gives us little to no examples so I have nothing to go on, plus I learn best from looking at example

Answer 1

You can just use the fact that rationals numbers are dense. So given $a$ and $b$, there is a rational number in $(a,b)$ and there is a rational number in both of $(a,x)$ and $(x,b)$, where $x$ is any number in $(a,b)$.

Answer 2

For the proof you can just give some explicit construction that works. E.g., note that if $x < y$, then $x < x + \frac{1}{3}(y - x) < x + \frac{2}{3}(y - x) < y$. Then argue that if $x, y \in\mathbb{Q}$, then the two new numbers are rationals, too.