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.
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.
Hello, Discrete!
You must have been given a defintion of a rational number.$\displaystyle \text{Prove: If }x\text{ is rational and }x \neq 0\text{, then }\frac{1}{x}\text{ is rational.}$
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 . . .
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
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.
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
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
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