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Mathematics
OpenStudy (anonymous):

Is it possible to list the set of rational numbers between 0 and 1 by roster?

myininaya (myininaya):

\[\left\{ \frac{p}{q}| p \in \mathbb{Z}^+ \& q \in \mathbb{Z}^+ \& 0<p<q \right\} \]

OpenStudy (anonymous):

What?

myininaya (myininaya):

this set includes all the rationals numbers between 0 and 1 but not 0 or 1

myininaya (myininaya):

did you want to include 0 and 1 also?

OpenStudy (anonymous):

Wouldn't I have to list all the fractions between 0 and 1?

myininaya (myininaya):

yes thats what i did

OpenStudy (anonymous):

Through your first equation?

myininaya (myininaya):

equation?

OpenStudy (anonymous):

The first thing you posted

myininaya (myininaya):

i didn't post any equations i did post the set of rational numbers between 0 and 1 not including 0 and 1

myininaya (myininaya):

a rational number is in an integer over an integer with bottom integer not equal to zero, right?

OpenStudy (anonymous):

Yes

myininaya (myininaya):

so i said p/q and i wanted both p and q to be positive because you said btw 0 and 1

myininaya (myininaya):

we want 0<p/q<1

OpenStudy (anonymous):

Okay

myininaya (myininaya):

\[0 \cdot q <\frac{p}{q} \cdot q<1 \cdot q\]

myininaya (myininaya):

\[0<p<q\]

myininaya (myininaya):

we want the denominator to be bigger than the numerator right?

OpenStudy (anonymous):

yes

myininaya (myininaya):

that is what is said by 0<p<q

myininaya (myininaya):

0<numerator<denominator

OpenStudy (anonymous):

okay

OpenStudy (anonymous):

you can't list them by writing them out one at a time but you can make a list that counts them.

myininaya (myininaya):

yes there is infinity many rational numbers btw 0 and 1 no way to write them one by one so that is why i used a lot symbols to write the set

OpenStudy (anonymous):

@satellite73, how do you do that?

OpenStudy (anonymous):

What satellite73 is referring to is the fact that the rationals and those sets with the same cardinality are referred to as "countably infinite". For any set that is countably infinite, it is possible to setup a bijection between the natural numbers (the counting numbers) and the countably infinite set

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