(February 22, 2018 at 3:57 am)Jehanne Wrote: It's very counter intuitive that one could be able to "count" the set of rational numbers (those numbers that are quotient of two integers, say a/b), because such is an infinite set, but here's the proof:
Quote:A set is countable if you can count its elements. Of course if the set is finite, you can easily count its elements. If the set is infinite, being countable means that you are able to put the elements of the set in order just like natural numbers are in order. Yet in other words, it means you are able to put the elements of the set into a "standing line" where each one has a "waiting number", but the "line" is allowed to continue to infinity.
In mathematical terms, a set is countable either if it s finite, or it is infinite and you can find a one-to-one correspondence between the elements of the set and the set of natural numbers. Notice, the infinite case is the same as giving the elements of the set a waiting number in an infinite line.
https://www.homeschoolmath.net/teaching/...ntable.php
![[Image: rationals-countable.gif]](https://www.homeschoolmath.net/teaching/images/rationals-countable.gif)
While this is a function that is onto the *positive* rational numbers, it is NOT a one-to-one function. So for example, 4/1=8/2=12/3. In fact each postitive rational number is hit infinitely often by this counting method.
Since the definition requires a function that is BOTH onto and one-to-one, this doesn't give the result (at least not directly).
There *is* a result that takes an onto function from the natural numbers to an infinite set and gives a one-to-one onto function. But the argument needs to be made for this.
