A Surjective Function for All [R]

[Atran]

Hi, I wonder if there are any surjective functions whose range covers all real numbers.
Note that I'm not implying countability, since |R| is uncountable.

For instance,
Sum[rⁿ/n!, {n, 0, m}] : {r, n, m} ∈ R

Thanks for help...

 

[mathwonk]

as asked, one answer is the identity function from R to itself.

 

[Robert1986]

as asked, one answer is the identity function from R to itself.

And the inverse of mathwonk's function :)

The function f(r) = r + e is surjective. Since it is surjective for each e in R, there are an infinite number of such functions; indeed uncountably infinite number of such funcitons.

 

[Goongyae]

f(x)=a+bx+cx^2+dx^3+...+ex^(2n+1) with e nonzero is surjective on R

Obviously f(x) will approach minus or plus infinity as x approaches minus or plus infinity, or vice versa. Only the last term matters when it comes to this outcome. Since the function is continuous, it ranges from -inf to +inf and hence is surjective.

But this kind of sum might not be surjective if there are a infinite number of terms, for example consider sin(x).

Other surjective functions:
f(x)=x^3+sin(x)
f(x)=1/x, taken to be 0 at x = 0.
The Gamma function
The Riemann Zeta function

 

[Atran]

as asked, one answer is the identity function from R to itself.

How can a number be defined mathematically?

And the inverse of mathwonk's function :)

The function f(r) = r + e is surjective. Since it is surjective for each e in R, there are an infinite number of such functions; indeed uncountably infinite number of such funcitons.

If for instance, 2n+1 is an expression always having an odd output iff n∈Z; is there a analogous expression for all real outputs?

 

[Atran]

f(x)=a+bx+cx^2+dx^3+...+ex^(2n+1) with e nonzero is surjective on R

Are you saying, f(x)= a + bx + cx² + dx³ + ... + [coefficient]_n-1x^n-1 + [coefficient]_nxⁿ?

But this kind of sum might not be surjective if there are a infinite number of terms, for example consider sin(x).

I don't clearly understand you.

Other surjective functions:
f(x)=x^3+sin(x)
f(x)=1/x, taken to be 0 at x = 0.
The Gamma function
The Riemann Zeta function

Thanks, I'll check.

 

[Goongyae]

>Are you saying, f(x)= a + bx + cx2 + dx3 + ... + [coefficient]n-1xn-1 + [coefficient]nxn?

Yes except the last term needs a nonzero coefficient and odd power. An even power would not produce a surjective function. For example, x^2 is not surjective since it can't produce a negative number from a real number.

>I don't clearly understand you.

sin(x)=x-x^3/6+x^5/120-x^7/5040+... is essentially an infinite polynomial. It is not surjective even though every term has an odd power of x.

 

[Robert1986]

How can a number be defined mathematically?

If for instance, 2n+1 is an expression always having an odd output iff n∈Z; is there a analogous expression for all real outputs?

Well, f(n)=2n+1 is not onto Z. Are you asking for a function from R to R such that f(r) is always an odd number? I don't understand what you mean by analogous in this instance.

 

[Atran]

Well, f(n)=2n+1 is not onto Z. Are you asking for a function from R to R such that f(r) is always an odd number? I don't understand what you mean by analogous in this instance.

A surjective function from X to R; I don't know what X should be.

 

[Robert1986]

A surjective function from X to R; I don't know what X should be.

Oh ok, gotcha. Well, there can't be a surjective function from Z to R or from Q to R because both Q and Z are both countable. But, as we have shown, there are plenty of functions from R onto R. In fact, I showed that functions of the form: f(r)=r+e, e in R, is a function from R onto itself. Indeed, there are uncountably many such functions. So, if you want something other than a surjective function from R to R, let X = {f(r) + e : e in R} and define the function g: X -> R by g(x)=g(f(r)+e)=e. That is, you map the set of functions (X) onto the set of reals (R).

Another onto function from X -> R, where X isn't R, might be something involving the Cantor set. Also, you could map from the complex plane to R and find an onto function that way.


Does any of this help? Do you mind if I ask what your motivation is?

 

[Atran]

So, if you want something other than a surjective function from R to R, let X = {f(r) + e : e in R} and define the function g: X -> R by g(x)=g(f(r)+e)=e. That is, you map the set of functions (X) onto the set of reals (R).

Why is this necessarily not a function from R to R?

Another onto function from X -> R, where X isn't R, might be something involving the Cantor set. Also, you could map from the complex plane to R and find an onto function that way.

I don't really know what the Cantor set is; I'll check it later.
Can you write a function from C to R?

Does any of this help? Do you mind if I ask what your motivation is?

I'm trying to "solve" the continuum hypothesis, in spite of me being in high-school.
Do you know any webpage showing the proofs why the hypothesis is undecidable in ZFC?
Is there a mathematical definition of number?

 

[Robert1986]

Why is this necessarily not a function from R to R?

I defined X to be a set of functions, then defined a function from X to R. So, by definition, it is a function from X to R, not R to R.

I don't really know what the Cantor set is; I'll check it later.
Can you write a function from C to R?

The Cantor set is an uncountably infinite subset of the reals. It has a lot of completely un-intuitive properties. It is constructed by removing intervals from the unit interval. As for a function, I'm sure I could, but it would be better for you to do so (and I don't feel like thinking up a good one .

I'm trying to "solve" the continuum hypothesis, in spite of me being in high-school.
Do you know any webpage showing the proofs why the hypothesis is undecidable in ZFC?
Is there a mathematical definition of number?

I don't know much about finding proofs w.r.t. CH and ZFC as this sort of thing has never interested me.

As for a mathematical definition of number, I suggest you make your way to some University Library and get a book on Real Analysis (there are also several published by Dover in the Math section of Borders.) There, the natural numbers are created, then the integers, then the rationals and finally the reals. Additionally, and this is the route I prefer (again, because this sort of Foundations stuff doesn't interest me) the reals can be defined in the terminology of Fields, which is a concept of Modern Algebra.

If you have any plans to do University Math, I would certainly suggest getting familiar with Modern Algebra (also called Abstract Algebra) and you will have a leg up on (probably) every other student.