Question

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

Answer 1

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

Answer 2

What?

Answer 3

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

Answer 4

did you want to include 0 and 1 also?

Answer 5

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

Answer 6

yes thats what i did

Answer 7

Through your first equation?

Answer 8

equation?

Answer 9

The first thing you posted

Answer 10

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

Answer 11

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

Answer 12

Yes

Answer 13

so i said p/q

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

Answer 14

we want
0<p/q<1

Answer 15

Okay

Answer 16

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

Answer 17

\[0<p<q\]

Answer 18

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

Answer 19

yes

Answer 20

that is what is said by 0<p<q

Answer 21

0<numerator<denominator

Answer 22

okay

Answer 23

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

Answer 24

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

Answer 25

@satellite73, how do you do that?

Answer 26

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