Can you tell me more about bijective functions, please?
Also, the fact that Zero leads to infinity, can be seen with Obama and our national debt...
Cheers!
So the upshot is that a 1-1+onto or bijective function has a unique inverse.
Thus this is a way of extending the ordinary notion of counting elements to infinite sets. When we count finite collections, we are putting them into 1-1 correspondence with a subset of the integers. To extend that notion to infinite sets, two sets have the same cardinality or "size" if there is a bijection between them.
The cardinality of the even integers is the same as the cardinality of the integers. Why? Here is a bijection f(N) = 2N.
Every non-empty open subinterval of the real line, no matter how small, has the same cardinality as the whole real line. Why? Here is a bijection: f(x) = arctan(αx); with "α" some suitable scaling factor that maps the arbitrary interval into (-π/2, π/2).
To prove the reals do not have the same cardinality as the integers, produce an enumeration of the reals, then show there is always a real number it doesn't contain. That's Cantor's Diagonalization Theorem.
Here's another way, more abstract but actually less difficult. Define the powerset of a set to be the set of all subsets of a set. So the powerset of {1, 2} is the set {{1}, {2}, {1, 2}, {}} [It's called the powerset because if a finite set has "S" elements, the set of all its subsets has 2S elements.]
Show that there is no bijection between any set and its powerset. Cantor did this already. It's the so-called "who shaves the barber" proof. Then show that there is a bijection between the reals and the powerset of the integers. Since there's a bijection between the reals and the powerset of the integers, there can't be one between the reals and the integers themselves.
In this extended sense [that there is no bijection] there are "more" reals than there are integers.