# Proof Involving Rational Numbers

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

• This is marked as a duplicate of a question, but that question is marked as a duplicate of this one. To break the circuit, I vote to reopen this one. – Jonas Meyer Mar 15 '15 at 4:20

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.

• Very nicely (and simply) argued! – Namaste Feb 22 '14 at 17:58
• For a I have this: (a,b∈Q∧a<b)→(∃p,q∈Q|a<p<q<b) would you say that's correct?? – JUBE Feb 22 '14 at 18:01
• Yes, looks correct to me. – Arno Feb 22 '14 at 18:28
• Can you explain where this comes from: x+1/3(y−x) – JUBE Feb 22 '14 at 22:41
• @JoseMartinez What do you mean with "come from"? This is just the rational number one third on the way from x to y. – Arno Feb 22 '14 at 23:45

"Thanks, proofs are what I'm having the hardest time with."

I am creating some autocorrected step-by-step tutorials on proving theorems, including those at the pre-calculus level, at the following location: http://www.public-domain-materials.com/folder-student-exercise-tasks-for-mathematics-language-arts-etc---autocorrected.html

The tutorials include, for example, proving the infinitude of primes, the irrationality of the square root of 2, and deriving the Quadratic Formula.

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 $(a,b)$.

• I would expect that on the level where such a question is asked, the statement "$(\mathbb{Q}, <)$ is dense" is not yet available as a basic fact. – Arno Feb 22 '14 at 17:46