This question applies with the so called "infinite" pigeonhole principle. Why is it possible to construct a one-one function out of two sets where the codomain has a length smaller than the length of the domain?
You can also think it of as an infinite set broken into a finite number of subsets, then one of the subsets must be an infinite set.irony of truth said:This is what I understood so far...
I was able to find that in the infinite case, "If n holes contains an infinite number of points, then at least one of the holes contain an infinite number. In particular, if the holes are labeled or ordered from 1 to n, then there must be a first hole with infinitely many points in it."
Pardon?What confuses me is regarding whether this concept I have researched applies to a one-one function...
? Whats the problem ?I was thinking of cramming Q+, the positive rationals into N the natural numbers with space left over to do it infinitely more times. Let p/q always be a reduced fraction in Q+ and define the map Q+--->N;p/q--->(2^p)(3^q). I can know that
this is a 1-1 map by the fundamenntal theorem of arithmetic. (unique prime
factorization) and no number that has any other prime in it's decomposition
other than 2 or 3 is in the range.
Yes this is true, it follows from the schroeder-bernstein theorem. It goes to show how dense the set of (-1,1) can be.Also, I can construct a cartesian plane with all the values of y between -1 and 1, exclusively, ie., (-1,1) and all the values of x indefinitely. From here, I can assigned a one-one function in which every value in x corresponds to a unique value of y.