Question

Prove that the set \( \mathbb{Q^+} \) of positive rationals is denumerable

Answer 1

Can someone tell me what is wrong with the following proof

Answer 2

Consider the positive rationals in the array in Figure 1. Order this set by listing all the rationals in the first row, then the second row and so forth. Omitting the fractions that are not in the lowest terms, we have an ordering of \( \mathbb{Q^+} \) in which every poitive rational appears. Therefore \( \mathbb{Q^+} \) is denumerable

Answer 3

There are a few things wrong in my opinion. But what stands out the most?

Answer 4

I am no professional, but isn't an ordering of a set different from a one-to-one correspondence?

In that respect, I do not have a problem with the proof. (Remember, the union of two countable sets is countable!)

Answer 5

@KingGeorge ?

Answer 6

There's two (minor) things that bug me, and these are mostly notational I think.

First, you don't have vertical dots at the bottom signifying you have fractions with a numerator larger than 7. So \(\frac{8}{9}\) doesn't appear anywhere.

Second, you shouldn't have arrows going from 2/1 to 3/1 and 4/1 to 5/1 and 6/1 to 7/1.

Answer 7

wait a sec hmmmm

Answer 8

Whoops! I didn't even look at the picture

Answer 9

okk well This wasnt the exact picture but this what i found when i googled for it. There are no arrow going across the first row and there shld be dots continuing at the bottom too

Answer 10

I am just not sure if you cld prove that teh set is denumerable from a diagram

Answer 11

Those reasons are exactly the ones that I gave, so that's good I guess.

By the simple fact that you could make the diagram, and put those arrows there, it is denumerable. If you tried doing it with a set that's infinite, and not countable, it wouldn't work.

Answer 12

Gonna drive you crazy with one more question. But if u dont have the time just tell me soo ok?

Answer 13

I've still got some time. I'm not on a super tight schedule.

Answer 14

if its denumerable then it must have a bijection is that correct?

Answer 15

correct.

Answer 16

so wldnt u have to prove that the set is denumerable if it has a bijection

Answer 17

And those arrows give you the bijection.

Answer 18

okkkkkkkkkkkkkkk gotcha. Thanks KG :DDDDDDDD
I totally drive you crazy

Answer 19

No worries :P

Answer 20

Thanks @Herp_Derp