How is the set of rational numbers countably infinite?
1,288
Because you can construct a mapping of the rationals into the integers.
One easy way is to map the positive rational $\dfrac{a}{b}$ to the integer $2^a 3^b$.
This shows that there are at least as many integers as there are positive rationals.
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DilllyBar
Updated on October 28, 2022Comments
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DilllyBar about 1 year
How is $\mathbb{Q}$ countably infinite?
The definition says all elements of the set must have a one-to-one relation to the natural numbers. I do not understand this.
How do the elements in $\mathbb{Q}$ have a one-to-one relation with natural numbers?
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copper.hat almost 7 yearsDo you see that $\mathbb{N}$ , $\mathbb{Z}$ and $\mathbb{Z}^2$ all have the same cardinality?
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Daniel Xiang almost 7 yearsyou've already asked this question and I've posted a solution. Also I'm sure that if you googled "rational numbers countable proof" it would be the first link to appear.
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Masacroso almost 7 yearsJust for the record see that.
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