Results 1 to 11 of 11
Like Tree4Thanks
  • 1 Post By emakarov
  • 1 Post By emakarov
  • 1 Post By emakarov
  • 1 Post By emakarov

Math Help - Mapping an interval to R - Proving that its a group.

  1. #1
    Junior Member
    Joined
    Nov 2013
    From
    Germany
    Posts
    40
    Thanks
    1

    Mapping an interval to R - Proving that its a group.

    M := {f:[0,1] --> ℝ} ~+: M x M --> M

    ∀x∈[0,1]: (f~+g)(x) = f(x)+g(x)

    I need to show that (M, ~+) is a group and answer additionally if its an abelian group.

    The interval somehow just throws me off and I dont know how to start to proof anything.

    (f~+g)(x) = f(x)+g(x) looks like group homomorphism to me.

    A group has to be assoziative and it has an inverse/neutral element.

    Where do I go from here?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,517
    Thanks
    771

    Re: Mapping an interval to R - Proving that its a group.

    Hint: The inverse of f is the function that sends x ∈ [0, 1] to -f(x).
    Thanks from Cyganek
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Nov 2013
    From
    Germany
    Posts
    40
    Thanks
    1

    Re: Mapping an interval to R - Proving that its a group.

    f:ℝ --> [0,1]?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,517
    Thanks
    771

    Re: Mapping an interval to R - Proving that its a group.

    Could you state your question more clearly?

    I mean inverse in the sense of ~+, not the inverse function.
    Thanks from Cyganek
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Nov 2013
    From
    Germany
    Posts
    40
    Thanks
    1

    Re: Mapping an interval to R - Proving that its a group.

    I am so sorry if I created confusion, but I forgot to say that I mean + with a tilde above, simply to show that I am talking about linking elements in terms of groups.

    Something like this:
    ~
    +

    I really need to start using LaTex...
    Last edited by Cyganek; December 1st 2013 at 09:24 AM.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,517
    Thanks
    771

    Re: Mapping an interval to R - Proving that its a group.

    OK, so you have an operation \tilde{+} on functions. You need the inverse element for each f ∈ M and the neutral element in M. The inverse of f should also be in M and should produce the neutral element when combined with f using \tilde{+}.
    Thanks from Cyganek
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Junior Member
    Joined
    Nov 2013
    From
    Germany
    Posts
    40
    Thanks
    1

    Re: Mapping an interval to R - Proving that its a group.

    Considering that I am looking at the interval [0,1], its hard for me to come up with inverse elements. I would say it would be [0,-1] but I dont think I can go into this interval.

    And thank you for your patience by the way.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,517
    Thanks
    771

    Re: Mapping an interval to R - Proving that its a group.

    OK, it may be hard for someone to compute in his/her head that, for example, 123 m/s times 26 s equals 3198 m, but it is easy to see that we multiply speed (m/s) by time (s) and get distance (m). Similarly, let's forget about the definition of \tilde{+} and look at the dimension, or type, of objects we are dealing with.

    What are elements of M? They are functions from [0, 1] to reals. What does the operation on M do? It takes two such elements (functions) and returns another function. What is the type of the neutral element? It is also a function from [0, 1] to reals, just like all elements of M. (The neutral element of a group is first of all an element of that group; it has the same type as other elements. If a group consists of numbers, the neutral element is not a function.) What does the inverse operation do? It maps a function from [0, 1] to reals to another such function.

    Quote Originally Posted by Cyganek View Post
    Considering that I am looking at the interval [0,1], its hard for me to come up with inverse elements. I would say it would be [0,-1] but I dont think I can go into this interval.
    What is the type of [0,-1]? It is a set of real numbers. What should be the type of the inverse of an element of M? It should be another element of M, i.e., a function from [0, 1] to reals. A set of reals in not the same as a function from [0, 1] to reals.

    Quote Originally Posted by emakarov View Post
    Hint: The inverse of f is the function that sends x ∈ [0, 1] to -f(x).
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Junior Member
    Joined
    Nov 2013
    From
    Germany
    Posts
    40
    Thanks
    1

    Re: Mapping an interval to R - Proving that its a group.

    I feel ashamed that I still cant get on the right track... . Its just so abstract. From what I understand [0,0] is the neutral element, because it is an element of M. I hope this is a small step into the right direction.
    Follow Math Help Forum on Facebook and Google+

  10. #10
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,517
    Thanks
    771

    Re: Mapping an interval to R - Proving that its a group.

    M is a set of functions. Is [0, 0] a function?

    I agree that it is abstract and goes beyond operations we deal with every day. Specifically, \tilde{+} is an operation (i.e., a function) that takes elements of M as input. Now, elements of M are themselves functions. That is, \tilde{+} is a function that acts on functions, or a higher-order function. This may be hard to grasp initially because we usually deal with functions that act on concrete objects, like numbers.

    As an example, you may think about functions as instructions written on paper, e.g., take x and turn it into y. Then a higher-order function takes pieces of paper with instructions written on them and produces another piece of paper with an instruction. For example, if one instruction says, "Take $6 and buy (i.e., turn $6 into) a gallon of milk" and another says, "Take $3 and buy a loaf of wheat bread", you can combine them into "Take $9 and buy a gallon of milk and a loaf of wheat bread".
    Thanks from Cyganek
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Junior Member
    Joined
    Nov 2013
    From
    Germany
    Posts
    40
    Thanks
    1

    Re: Mapping an interval to R - Proving that its a group.

    [0,0] itself is not a function. I thought [0,0] --> ℝ is a function, but I realized right now that Im on he wrong way here. I just cant change the function that Im working with as its part of the assignment.
    Functions from [0,1] --> ℝ are elements from M, nothing else.

    I dont know what makes me so much trouble. I have to work with an interval here. Usually I just work with letters or a given set of numbers. The interval just confuses me.

    On the other hand your example helped me to understand this topic a bit more from a more common perspective.

    This instruction says:" Take one element from the interval [0,1] and map it to the real numbers."
    An the follow-up question is:"If you link together 2 elements from the interval [0,1] and map them together to the real numbers, are tehy the same as mapping to elements seperately and then linking them."


    Litte note: Its getting late here in Germany. I will answer again in about 8-9 hours. Thank you very much for your help and patience so far.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Question on group mapping
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: September 2nd 2011, 05:14 PM
  2. proving that an interval I = [a,b] has length |b-a|
    Posted in the Differential Geometry Forum
    Replies: 8
    Last Post: December 7th 2010, 12:37 PM
  3. Infinite set mapping to infinitesimal interval
    Posted in the Differential Geometry Forum
    Replies: 7
    Last Post: August 1st 2010, 12:17 AM
  4. Proving a point in an interval
    Posted in the Calculus Forum
    Replies: 10
    Last Post: November 24th 2008, 12:58 PM
  5. Group Homomorphism Mapping
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: April 13th 2008, 07:37 AM

Search Tags


/mathhelpforum @mathhelpforum