So I was given this problem by my friend, and apparently I got the answer wrong.
Problem:
Consider the set $S=\big\{1,3,2,5,7,4,9,11,6,13,17,8,19,21,10,\ldots\big\}$. Now, form a ratio $\#O:\#E$ such that $O$ is the set of all odd numbers and $E$ is the set of all even numbers.
What is the value of the ratio?
Notation: $\#X = $ the cardinality (cardinal number) of a set $X$.
My Answer:
Obviously, we can see a pattern. Let $o_n$ be the $n^\text{th}$ odd number and $e_n$ be the $n^\text{th}$ even number. Then, $$S=\big\{o_1,o_2,e_1,o_3,o_4,e_2,\ldots\big\}=\bigcup_{n\in O}^\infty\big\{\{o_n,o_{n+1}\}\cup \{e_{(n+1)/2}\}\big\}.$$ So now, we have three such elements to consider: $o_n,o_{n+1}$ and $e_{(n+1)/2}$. Two of those elements belong to the set $O$ and only one element belongs to the set $E$. Thus, $$\#O:\#E=2:1$$ which means that $2/3$ of the set is odd and $1/3$ of the set is even.
However, my friend said the answer was wrong! Actually, $\#O:\#E=1:1$. But I don't understand where I might have gone wrong.
Possible reason I was wrong:
If a set $X=\bigcup_{i=k}^n\big\{x_i\big\}$ then $n=\#X$. Thus, $O$ and $E$ have the same carnality (infinite), so $$\#O:\#E=\infty:\infty=1:1.$$ But are we allowed to reduce ratios like that? I mean, $\infty$ is not a number! My friend said that my reasoning for why the answer is wrong is incorrect, and I understand that... so how else is my answer incorrect? He gave me the following hint:
Hint:
$X=\{a,b,c\}=\{b,a,c\}$.
But how is this useful?
Thank you in advance.
The ratio of the cardinalities of the two sets is undefined as they are both infinite and "infinity divided by infinity" isn't allowed. You can however talk about the sequence and the relative density of the odds and evens within that sequence.
Let $O$ represent the set of odd natural numbers and let $E$ represent the set of even natural numbers.
Let $A_1=\{1\},A_2=\{1,3\},A_3=\{1,3,2\},A_4=\{1,3,2,5\},A_5=\{1,3,2,5,7\},\dots$
$\dots,A_{3n}=\{1,3,2,5,7,4,\dots,4n-3,4n-1,2n\},\dots$
We can then talk about $\lim\limits_{n\to\infty}\dfrac{|O\cap A_n|}{|A_n|}$ and $\lim\limits_{n\to\infty}\dfrac{|E\cap A_n|}{|A_n|}$ which will be $\dfrac{2}{3}$ and $\dfrac{1}{3}$ respectively.
In that sense, the "ratio" that you might have been thinking of is $2:1$ when treating this in terms of the relative density in which the odds versus evens appear.
Compare this to $[n]=\{1,2,3,\dots,n\}$ and how $\lim\limits_{n\to\infty}\dfrac{|O\cap [n]|}{|[n]|}=\dfrac{1}{2}$ and $\lim\limits_{n\to\infty}\dfrac{|E\cap [n]|}{|[n]|}=\dfrac{1}{2}$
Notice that $\lim\limits_{n\to\infty}A_n=\lim\limits_{n\to\infty}[n]=\Bbb N^+$.
This goes to show that depending on the order in which things are written, certain subsets can seem to "appear more frequently." Once we accurately define the ratios we are talking about we can see exactly what that means and why it happens.