MHB Is Adding Even Numbers to Fractions Enough to Prove Countable Infinity?

[JProgrammer]

I am trying to prove how this set is countably infinite:

q∈Q:q=a/b where a is even and b is odd

a needs to be even and b needs to be odd, so I thought this would prove that it would be countably infinite:

q = a/b + x/x, where x is any even number.

a always needs to be even and b always needs to be odd, so if they have the value of x added to them with x being any even number, they would always be positive or negative.

My question is: is this enough to prove that this set is countably infinite? If not, what do I need to do?

 

[I like Serena]

I am trying to prove how this set is countably infinite:

q∈Q:q=a/b where a is even and b is odd

a needs to be even and b needs to be odd, so I thought this would prove that it would be countably infinite:

q = a/b + x/x, where x is any even number.

a always needs to be even and b always needs to be odd, so if they have the value of x added to them with x being any even number, they would always be positive or negative.

My question is: is this enough to prove that this set is countably infinite? If not, what do I need to do?

Hey JProgrammer! ;)

How about proving that $\mathbb Q$ is countably infinite to begin with?
So any subset would be countably infinite as well.

Have you heard of the "swan-walk"?
That is, we run through all fractions in the following order:
$$\frac 11, \frac 21, \frac 12, \frac 13, \frac 22, \frac 31, \frac 41, \frac 32, \frac 23, \frac 14, \frac 15, ...$$
This is the swan walk. Its meaning will be apparent if we visualize it in X-Y coordinates.
It shows that we can iterate through all fractions in a countable fashion.
Put otherwise, we have a bijective function $\mathbb Z \to \mathbb Q$, proving that $\mathbb Q$ is countable... (Thinking)

 

[JProgrammer]

Hey JProgrammer! ;)

How about proving that $\mathbb Q$ is countably infinite to begin with?
So any subset would be countably infinite as well.

Have you heard of the "swan-walk"?
That is, we run through all fractions in the following order:
$$\frac 11, \frac 21, \frac 12, \frac 13, \frac 22, \frac 31, \frac 41, \frac 32, \frac 23, \frac 14, \frac 15, ...$$
This is the swan walk. Its meaning will be apparent if we visualize it in X-Y coordinates.
It shows that we can iterate through all fractions in a countable fashion.
Put otherwise, we have a bijective function $\mathbb Z \to \mathbb Q$, proving that $\mathbb Q$ is countable... (Thinking)

Thanks for your reply.

I have heard of this before, but I wasn't sure if it would work for this problem because a needs to be even and b needs to be odd. This would work for this problem?

Since this set has been proven to be countable, it needs to be proven to be infinite. How would I prove that?

 

[Evgeny.Makarov]

How about proving that $\mathbb Q$ is countably infinite to begin with?
So any subset would be countably infinite as well.

Well, this is an overstatement. (Smile)

Since this set has been proven to be countable, it needs to be proven to be infinite. How would I prove that?

You need to find infinite number of different elements. This is not hard.

 

[HallsofIvy]

"b odd" includes b= 1 so the set of all even integers is a subset.