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

In summary, R^R can refer to either the set of functions from R to R or the set of R-fold tuples of real numbers. Both interpretations are commonly used in mathematics.
  • #1
karnten07
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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?


Homework Equations





The Attempt at a Solution

 
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  • #2
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.
 
  • #3
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?
 
  • #4
"solutions" is not the right word, but I get what you mean.
 
  • #5
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.
 
  • #6
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.
 

1. What is the definition of Set R^R?

Set R^R is a mathematical concept that refers to the set of all functions from the real numbers to the real numbers. In other words, it includes all possible combinations of inputs and outputs using real numbers.

2. How is Set R^R used in function analysis?

Set R^R is a fundamental tool in function analysis as it allows for the study and manipulation of functions with real number inputs and outputs. It enables mathematicians to analyze the behavior and properties of functions, such as continuity, differentiability, and integrability.

3. What are some common examples of functions in Set R^R?

Some common examples of functions in Set R^R include polynomial functions, exponential functions, logarithmic functions, and trigonometric functions. These functions can have various forms and expressions, but they all fall under the category of functions with real number inputs and outputs.

4. How is Set R^R related to other mathematical concepts?

Set R^R is closely related to other mathematical concepts such as calculus, analysis, and topology. It serves as the foundation for many advanced mathematical theories and is essential in the study of mathematical functions and their properties.

5. What is the significance of understanding Set R^R?

Understanding Set R^R is crucial for any mathematician or scientist working with real-world data that can be represented by functions. It allows for a deeper understanding of how functions behave and how they can be manipulated to solve complex problems. Set R^R is also essential in many fields such as physics, engineering, and economics, where mathematical functions are used to model real-world phenomena.

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