# Thread: Creating a bijection to show equal cardinality

1. ## Creating a bijection to show equal cardinality

Hello,

I have to create a bijection between two sets $\displaystyle Q$- a set of rational numbers, and $\displaystyle Q^{+}$- a set of positive rational numbers, to show that these sets have equal cardinality.

Can anyone give me a hint?

2. ## Re: Creating a bijection to show equal cardinality

Originally Posted by MachinePL1993
I have to create a bijection between two sets $\displaystyle Q$- a set of rational numbers, and $\displaystyle Q^{+}$- a set of positive rational numbers, to show that these sets have equal cardinality.

Both in this question and the other you posted, it seems to me as if someone is asking you "to reinvent the wheel". You already have the Cantor-Bernstein' theorem.

Think about the mapping $\displaystyle \Phi:\mathbb{Z}\times\mathbb{Z}\to\mathbb{Q}^+$, $\displaystyle (m,n)\mapsto 2^n\cdot 3^m$.
Is that an injection?

In this new problem, you know that subsets of countable sets are countable.
There us a bijection $\displaystyle f:\mathbb{Q} \leftrightarrow \mathbb{Z}^ +$ as well as a bijection $\displaystyle g:\mathbb{Z}^ + \leftrightarrow \mathbb{Q}^ +$

Consider $\displaystyle g\circ f$.