Understanding Set R^R: Definition and Use in Function Analysis

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Homework Help Overview

The discussion revolves around the set R^R, particularly its definition and application in function analysis. Participants explore whether R^R is used to define functions and how it relates to real solutions.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Some participants attempt to clarify the notation R^R, noting its common interpretation as the set of functions from R to R and its relation to Cartesian products. Others question the terminology used, particularly the use of "solutions" in this context.

Discussion Status

The discussion is ongoing, with participants providing insights into the meaning of R^R and its implications for functions. There is a recognition of the need for precise language when discussing mathematical concepts, particularly regarding the mapping of functions to real numbers.

Contextual Notes

Participants are navigating the nuances of mathematical terminology and its implications for understanding functions, with some uncertainty about the correct usage of terms like "solutions."

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Homework Statement


What is the set R^R? Is it used to define functions and show that the function produces real solutions?


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The Attempt at a Solution

 
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X^Y usually denotes the set of functions from Y to X.

But I've also seen R^R denote an R-fold Cartesian product of R.
 
quasar987 said:
X^Y usually denotes the set of functions from Y to X.

But I've also seen R^R denote an R-fold Cartesian product of R.

Okay, in this question after it writes R^R it says "of all functions from R to R", so i think they just mean that solutions are real?
 
"solutions" is not the right word, but I get what you mean.
 
quasar987 said:
"solutions" is not the right word, but I get what you mean.

Oh sorry, i mean the function maps to a real number.
 
Formally, R^R is indeed the set of all R-fold tuples of real numbers. That is, an element of R^R is sort of a vector, containing in each slot a real number, and having as many slots as there are real numbers.
In everyday mathematical usage we call such an object a function from R to R.
 

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