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Rational Numbers Set Countable. I guess i'm interpreting the word countable different than the way the author/other mathematicians interpret it. If t were countable then r would be the union of two countable sets.
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The set of all computer programs in a given programming language (de ned as a nite sequence of \legal Cantor using the diagonal argument proved that the set [0,1] is not countable. Of course if the set is finite, you can easily count its elements.
You can make an infinitely long list of all rational numbers without leaving out one of them.
The set of positive rational numbers is countably infinite. The set \(\mathbb{q}\) of rational numbers is countably infinite. Prove that the set of rational numbers is countable by setting up a function that assigns to a rational number p/q with gcd(p,q) = 1 the base 11 number formed from the decimal representation of p followed by the base 11 digit a, which corresponds to the decimal number 10, followed by the decimal representation of q. See below for a possible approach.
