The discussion revolves around the infinite pigeonhole principle and its implications for constructing one-to-one functions between sets of differing cardinalities. Participants explore the concept of cardinality, the definition of "length," and the application of these ideas to specific mappings, particularly involving the positive rationals and natural numbers.
Participants express differing views on the interpretation of "length" and its relevance to cardinality. There is no consensus on the application of the pigeonhole principle to one-to-one functions, and multiple competing perspectives remain regarding the examples provided.
Some participants' statements rely on specific definitions of cardinality and length, which may not be universally accepted. The discussion includes unresolved questions about the applicability of the pigeonhole principle to one-to-one functions and the nature of the mappings discussed.
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.