aaaa202 said:
I don't know if you can say this, but my teacher said today that there is just as many numbers between 0 and 1 as 2 and ∞. He then said this could easily be seen by looking at the bijective function 1/x. Can anyone try to explain what he meant?
The idea to think about is what does it mean for two sets to have the same size. For finite sets we could just count each one and see if the number is the same but underlying this is a more fundamental idea.
That is: Two sets have the same size if there is a way to match the elements of one to the elements of the other so that both sets are used up and only one thing gets matched to one thing. If it is not possible to completely uses up one of the two sets with any matching, then we would say that that set is bigger.So for instance, the set {1,2} and the set {3.4} have the same size because the matching
1 -> 3 and 2->4 uses up both sets and only one number is matched to one other number,
the matching 1->4 ,2->3 also works.
We would not say that the matching
1-> 3, 2->3 means that {3,4} is bigger and that shows why the matching must pair only one number to one other number.
For infinities the idea is exactly the same. Two infinite sets have the same size if it is possible to match each to the other without repetitions. So for instance, the even integers have the same size as all of the integers because each integer can be matched to its double.
1 is matched to 2, 2 to 4, 3 to 6, and so on. This matching is "1 to 1", that is each integer is matched to a different even integer, and "onto" that is both sets are completely matched up.
he function 1/x maps the interval 1 to infinity to the interval 0 1 in a 1-1 and onto way. So the two intervals have the same size.
Interestingly, it is not possible to match the integers 1-1 and onto to the interval 0 to1. The interval can never be completely used up. So its is bigger. It is a bigger infinity.