(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that if [tex]A_{1}[/tex], [tex]A_{2}[/tex],... are countable sets, so is [tex]A_{1}[/tex][tex]\cup[/tex] [tex]A_{2}[/tex][tex]\cup[/tex]....

2. Relevant equations

3. The attempt at a solution

Part one of the question is okay, I would like to believe I can handle that but, part B, I am not so sure.

My solution is as follows ( using the matrix in the attached picture)

Line up [tex]A_{1}[/tex], [tex]A_{2}[/tex],[tex]A_{3}[/tex] ... in the "X-axis" ( it's more of an axis of positive integers) and then the line up the elements of of set along the vertical columns associated with each set. Then use the function is part A of the picture to define a bijection.

[tex]\varphi[/tex] : N XN [tex]\rightarrow[/tex] N by [tex]\varphi[/tex](i,j) = j + [tex]\frac{k(k+1)}{2}[/tex] where k= i+j.

Of course, I use the fact that the set of positive integers and natural numbers are of the same cardinality.

What do you guys think ? Is this okay ? Would this suffice?

I'm looking for another way of showing this same result, but one that is less "forced". Forced ,in the sense that, I was basicallly given the answer in part A of the question in the attached picture.

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# Homework Help: Show that the union of countable sets is countable

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