I am asked to proof this conjecture:

If x is rational and x is not equal to 0, then 1/x is rational. How do I proof this? Should I use direct or contraposition proof? I just learned about this stuff and I don't know where to start at.

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- Jun 24th 2007, 12:25 PMDiscretedirect proof or contraposition?
I am asked to proof this conjecture:

If x is rational and x is not equal to 0, then 1/x is rational. How do I proof this? Should I use direct or contraposition proof? I just learned about this stuff and I don't know where to start at. - Jun 24th 2007, 12:40 PMPlato
The answer depends upon the axiom set you are basing the proof upon.

What are you given about rational numbers? - Jun 24th 2007, 12:52 PMDiscrete
- Jun 24th 2007, 01:02 PMPlato
- Jun 24th 2007, 01:07 PMSoroban
Hello, Discrete!

Quote:

$\displaystyle \text{Prove: If }x\text{ is rational and }x \neq 0\text{, then }\frac{1}{x}\text{ is rational.}$

**must**have been given a defintion of a rational number.

Maybe somthing like:

. . $\displaystyle x$ is a rational number if it is of the form $\displaystyle \frac{a}{b}$ where $\displaystyle a$ and $\displaystyle b$ are integers and $\displaystyle b \neq 0$.

That's a good place to start . . .

- Jun 24th 2007, 01:12 PMDiscrete
yes soroban that's what I was actually given.

$\displaystyle x$ is a rational number if it is of the form $\displaystyle \frac{a}{b}$ where $\displaystyle a$ and $\displaystyle b$ are integers and $\displaystyle b \neq 0$.

but then I don't know the next step after that. - Jun 24th 2007, 01:15 PMJhevon
Let $\displaystyle x$ be a rational number, then $\displaystyle x = \frac {a}{b}$ for $\displaystyle a,b \in \mathbb {Z}, b \neq 0$. Since $\displaystyle x \neq 0$ it means that $\displaystyle a \neq 0$

So $\displaystyle \frac {1}{x} = \frac {1}{ \frac {a}{b}} = \frac {b}{a}$

Since $\displaystyle a,b \in \mathbb {Z}$ and $\displaystyle a \neq 0$, the fraction above represents a rational number

QED - Jun 24th 2007, 01:20 PMPlato
- Jun 24th 2007, 01:43 PMIlaggoodly
not true, his argument is based off of the given assumption that x is a rational number, rewriting 1/x as b/a is nothing more than perforrming simple algebra, and then reusing the given definition of rationals

- Jun 24th 2007, 01:52 PMThePerfectHacker
I think, Plato is asking how do you define 1/x to be.

The more elegant way to define a rational number is by using a "field of quotients of an integral domain"

In that case we can define Q=F(Z) where F(Z) represents the field of quotients of Z (integral domain). In that case it is valid to flip the fraction. - Jun 24th 2007, 01:53 PMPlato
- Jun 24th 2007, 02:00 PMIlaggoodly
yes as a matter of fact i have taken a logic and set theory course, and im wel aware that you can make this problem MUCH more complicated than it is likely intended to be, i believe the way it has been shown by soroban and jhevon would satisfy the poster... in which case it was my mistake in saying that you were not correct, my apologies

- Jun 24th 2007, 02:06 PMPlato
- Jun 24th 2007, 02:11 PMIlaggoodly
true enough, however considering some of his other posts...

S.O.S. Mathematics CyberBoard :: View topic - discrete math basic's

it would seem as if he/she is taking his or her first course in introduction to advanced mathematics...anyway, whatever - Jun 24th 2007, 02:25 PMJhevon
Yes, Plato is right. It is up to the poster to tell us how rigorous a proof is needed. My proof is not all that rigorous, it is simple algebra which assumes the reader knows the axioms behind it. If a more rigorous proof is needed, my post should only serve as an outline of what to do