Rational Numbers Set Countable. I guess i'm interpreting the word countable different than the way the author/other mathematicians interpret it. In order to show that the set of all positive rational numbers, q>0 ={r s sr;s ∈n} is a countable set, we will arrange the rational numbers into a particular order.
Being countable, the set of rational numbers is a null set, that is, almost all real numbers are irrational, in the sense of lebesgue measure. 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. It is well known that the set for rational numbers is countable.
Note that r = a∪ t and a is countable.
Any subset of a countable set is countable. We call a set a countable set if it is equivalent with the set {1, 2, 3, …} of the natural numbers. In other words, we can create an infinite list which contains every real number. As another aside, it was a bit irritating to have to worry about the lowest terms there.