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

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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.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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