Prove equal cardinality between two sets?

Question

I'm preparing for a discrete math course in September and I'm trying to study on my own this summer. I've run into a bit of trouble with a practice problem I found online and can't really figure it out on my own.

Let A = set of all integers with 2 as a factor

Let B = set of all integers with 3 as a factor

Prove |A| = |B|

I understand that I have to prove a 1-1 correspondence by finding a function that maps A to B and proving bijectivity. But, I'm not sure how to find this function?

Answer 1

Hint: write simply

$$A=2\Bbb Z:=\{...-4,-2,0,2,4,...,2m,...\}\;,\;\;B=3\Bbb Z=\{...-3,0,3,6,...,3m,...\}$$

Can you guess now a nice bijection between the above sets?

Answer 2

Another idea :

Take the function $\Phi$ defined by $\Phi(2^n 3^m q) = 2^m 3^n q$ (where $q$ is not dividible by $2$ or $3$) and show that it's a bijection $\Bbb N^* \to \Bbb N^*$

Then it's easy to see that the set of multiple of $2$ is in bijection with the set of multiples of $3$

Answer 3

Define $f:A\rightarrow B$ by $f(n)=\frac{3}{2}n$. This is well defined and injective. Its inverse is also clearly injective. So this defines a bijection.