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irony of truth
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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.
The pigeonhole principle, also known as the "drawer principle", is a mathematical concept that states that if there are more pigeons than pigeonholes, then at least one pigeonhole must contain more than one pigeon.
The pigeonhole principle is used to prove various mathematical theorems and to solve problems in combinatorics and number theory. It is also commonly used in computer science and cryptography.
One example of the pigeonhole principle is the "birthday problem", which states that in a group of 23 people, there is a greater than 50% chance that two people will have the same birthday.
The pigeonhole principle is important because it provides a simple and powerful tool for solving problems and proving theorems in mathematics. It also has practical applications in various fields such as computer science and statistics.
Yes, there are several variations of the pigeonhole principle, such as the generalized pigeonhole principle, the strong pigeonhole principle, and the infinite pigeonhole principle. These variations have different conditions and can be applied to different types of problems.