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?
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?
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 $\displaystyle g\circ f$.