Floor and Ceiling Function: Injective, Surjective

[jsyc9301]

I was wondering if someone can verify if my assumptions are correct about this problem:

For each real number x, there exists a unique integer n, such that
Floor(x)= n where n <= x < n +1
and
Ceiling(x)= n where n -1 < x <=n

Essentially, I am trying to determine if 1)are these functions equivalent functions,2)are these functions one-to-one functions, and 3) are these functions onto functions.

What I have come up with so far:

1. Yes, they are equivalent functions because: -Floor(-x)=Ceiling(x) * Not to sure about this though
2. No, they are not one-to-one functions because each unit interval is mapped to the same integer.
3. No, they are not onto functions because the range consists of the integers, so the functions are not onto the reals.

Thanks again everyone. If you think I am mistaken for any of these, please feel free to point out where my logic is flawed

 

[daon]

I'm not sure what you mean by "equivalent functions" but the relationship between them is:

\(\displaystyle \lfloor{x}\rfloor = \lceil{x}\rceil, \,\, x \in Z\)

\(\displaystyle \lfloor{x}\rfloor = \lceil{x}\rceil-1, \,\, x \not\in Z\)

edit: Yours seems fine, too.

If the function is defined as mapping R->R, then yes, it is not 1-1 or onto. If it is defined as from R->Z, it is onto.

edited to make things more spacey :mrgreen:

 

[jsyc9301]

From reading my textbook, it says that floor and ceiling functions are defined from R->Z? So, then it would be an onto function, but not one-to-one?

 

[pka]

From reading my textbook, it says that floor and ceiling functions are defined from R->Z? So, then it would be an onto function, but not one-to-one?

Correct!

 

[jsyc9301]

Thank you very much everyone. Now I get the joy of spending the next couple hours proving or providing a counterexample

 

[darlenehill19]

Let me try that;
will get back to you after my results;
As an apparent read it seems pretty simple and proven; :shock:
but allow me to share my results
At this point i can't tell much; :roll:
My cousin checked it and according to him it is CORRECT!
BUT to do is believe in MATHS!
____________________
Life is a bubble flooring ....

 

[lookagain]

I'm not sure what you mean by "equivalent functions" but the relationship between them is:

\(\displaystyle \lfloor{x}\rfloor = \lceil{x}\rceil, \,\, x \in Z\)

\(\displaystyle \lfloor{x}\rfloor = \lceil{x}\rceil-1, \,\, x \not\in Z\)

Put these lines spaced apart. As it is, someone can't tell what the symbols are.

 

[daon]

Put these lines spaced apart. As it is, someone can't tell what the symbols are.

I'm someone, and I can tell :wink:. Seriously though, it looks fine to me.

 

[lookagain]

Put these lines spaced apart. As it is, someone can't tell what the symbols are.

I'm someone, and I can tell :wink:. Seriously though, it looks fine to me.

Your being able to tell is irrelevant as the readers are the ones who need to tell what they are.
Seriously, it didn't look "fine" at all; it was a mess/mesh of symbols. They weren't even
separate from each other. When they are crammed so that they are meshed together,
then they are too close together and it has to be fixed. Prevention is worth a pound of cure.
To reasonably explain it, *someone* would have submitted it without previewing it first.

Yours was as unreadable as this:

\(\displaystyle \sum_{x=2}^nx_n\)
\(\displaystyle \sum_{x=1}^nx\)


\(\displaystyle x_n = 2 ?\)

\(\displaystyle x = n^2 ?\)


\(\displaystyle The \ lower \ limit \ of \ the \ upper \ summation \ is \ "x = 2" \ and \ the \ upper\ limit\)

\(\displaystyle \ of \ the \ lower \ summation \ is \ "n" \ below \ it \ ?\)


That is why they are spaced apart, not to make them "spacey."


_/\ _\/_/\ _\/_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _\/_/\ _\/ _/\ _


\(\displaystyle \text{My} \\) :wink: \(\displaystyle \ \ \times \infty \ > \ \text{your single}\) :wink:.


\(\displaystyle (\) + \(\displaystyle )/2\ = \\) :| \(\displaystyle \ . \ . \ . \ . \ . \ . The \ average \ of \ happy \ and \ sad \ is \ neutral.\)


\(\displaystyle 4\) :!: \(\displaystyle \ \ \ = \ 4 \times 3\) :!: \(\displaystyle \ =\ 4 \times 3 \times 2\) :!: \(\displaystyle \ = \ 4 \times 3 \times 2 \times 1\) :!:

 

[daon]

Okay, I'll admit the summation looks like it could be confusing. I didn't think it looked bad when I posted that, maybe because I posted it and knew what I was attempting to communicate, but now i know it can be an issue to others.