Question

Show that this is a countably infinite{2q:q∈Q+}

Answer 1

SOLUTION

Given the question in the question tab, the following are the solution steps to solve the question

STEP 1: Define a countably infinite set

A set is countably infinite if its elements can be put in one-to-one correspondence with the set of natural numbers. In other words, one can count off all elements in the set in such a way that, even though the counting will take forever, you will get to any particular element in a finite amount of time.

STEP 2: Write the given set

$\mleft\lbrace2q\colon q\in Q+\mright\rbrace$

This set is read as a set of values 2q such that q is an element of positive rational numbers

STEP 3: Show the proof

$\begin{gathered} \text{Let }Q_{\pm}=\mleft\lbrace q\in Q\colon q>0\mright\rbrace \\ \text{For ev}ery\text{ q}\in Q_+,\text{ there exists at least one pair (m,n)}\in N\times N \\ \text{such that q}=\frac{m}{n} \end{gathered}$

Therefore, we can find an injection:

$i\colon Q_+\Rightarrow N\times N$

By Cartesian Product of Natural Numbers with Itself is Countable, N×N is countable.

Hence Q+ is countable, by Domain of Injection to Countable Set is Countable.

Note that the function defined by is a bijection. Therefore the composition is a bijection. So the given set is countably infinite.