Proving the Existence of Infinite Functions from One Domain to Another

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Discussion Overview

The discussion revolves around the existence of infinite functions from one set to another, specifically focusing on the sets of real numbers. Participants explore how to demonstrate that there are infinitely many functions between two sets, considering definitions and representations of functions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions how to prove the existence of many different functions from set A to set B, particularly when both sets are real numbers.
  • Another participant explains that a function can be represented as a set of ordered pairs, suggesting a pathway to explore the infinite nature of functions.
  • A follow-up post reiterates the idea that each subset of the Cartesian product can correspond to a function, implying that infinite subsets lead to infinite functions.
  • In response, a participant clarifies that not every subset corresponds to a function, providing an example of a non-function and emphasizing the need for specific forms of subsets to define valid functions.
  • This participant also notes that the number of functions is unbounded when considering nontrivial intervals of real numbers.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between subsets of sets and the existence of functions, indicating that the discussion remains unresolved regarding the criteria for defining functions and the implications for infinite functions.

Contextual Notes

There are limitations in the discussion regarding the definitions of functions and the nature of subsets, which may affect the conclusions drawn about the existence of infinite functions.

C0nfused
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Hi everybody,
I guess most of you know how a function is defined from a set A to a set B. How do we prove that many different functions exist (usually, if sets A and B are for example R) from A to B? Of course we can come up with many different functions, real ones for example, but is there any other way of proving that the set of real functions for example has infinite elements?
Thanks
 
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If A and B are sets, then by the definition of "function", a function from A into B can be represented as a set of ordered pairs (elements of the Cartesian product AxB), and thus as a subset of AxB. I'm sure you can see where to go from there. :)
 
hypermorphism said:
If A and B are sets, then by the definition of "function", a function from A into B can be represented as a set of ordered pairs (elements of the Cartesian product AxB), and thus as a subset of AxB. I'm sure you can see where to go from there. :)
So for each subset one function exists? For example RxR has infinite subsets so infinite different real functions can be defined?
 
C0nfused said:
So for each subset one function exists?
Not necessarily. The set {(a,b), (a,c)} is not a representation of any function, for example. Rather, you're looking for subsets of the form {a}xB for all a in the domain of f, where the image of f is the set consisting of exactly one element from each such set. Since each of those sets is infinite for your nontrivial real intervals, the amount of such functions is unbounded for even the trivial domain of one real number.
 

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