Results 1 to 7 of 7

Math Help - Pointwise sum and product

  1. #1
    Junior Member
    Joined
    Apr 2007
    Posts
    32

    Pointwise sum and product

    If A and B are sets where B has the folowing binary operation defined on it.

    BxB -> B
    (b,b') -> b*b'

    Then given 2 maps, f,g contained in Map(A,B) we get another map f*g contained in Map(A,B) by setting

    Map(A,B)xMap(A,B) -> Map(A,B)
    (f,g) -> f*g

    Of particular importance is the case A=B=R (reals) and it is usual to refer to f and g as functions in this case. Then the addition and multiplication on R induce pointwise addition and multiplication on Map(R,R):

    f+g: R -> R
    a -> f(a) + g(a)

    f.g: R -> R
    a -> f(a)g(a)


    Can anyone explain whats going on here? I don't understand how maps can be contained in other maps nor the whole pointwise thing.

    PS does anyone happen to know if you can write greek characters on a keyboard using alt+ number codes as you can with other european characters?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,383
    Thanks
    1474
    Awards
    1
    Quote Originally Posted by Obstacle1 View Post
    If A and B are sets where B has the following binary operation defined on it.
    BxB -> B
    (b,b') -> b*b'

    Then given 2 maps, f,g contained in Map(A,B) we get another map f*g contained in Map(A,B) by setting Map(A,B)xMap(A,B) -> Map(A,B) (f,g) -> f*g

    Can anyone explain what’s going on here? I don't understand how maps can be contained in other maps nor the whole pointwise thing.
    PS does anyone happen to know if you can write greek characters on a keyboard using alt+ number codes as you can with other european characters?
    This appears to be a very interesting question. However, I am clueless about the notation. Is Map(A,B) the set of all mapping(functions) from A to B. If so the usual notations is B^A. What is the operation f*g? That doesn’t make any sense without a complete set of definitions( I do not have your textbook). Please give us more information.

    BTW. You should try to learn LaTeX.
    If you type [math]\lambda[/tex] we can see \lambda
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Apr 2007
    Posts
    32
    Quote Originally Posted by Plato View Post
    This appears to be a very interesting question. However, I am clueless about the notation. Is Map(A,B) the set of all mapping(functions) from A to B. If so the usual notations is B^A. What is the operation f*g? That doesn’t make any sense without a complete set of definitions( I do not have your textbook). Please give us more information.

    BTW. You should try to learn LaTeX.
    If you type \lambda we can see \lambda

    Yea, Map(A,B) is the set of all maps from A to B. Sorry, my textbook gave this as the standard notation and said that B^A is now seldom used. Although I won't be offended if you use it . I don't exactly understand what f*g is either, but I thought it was defined in some way by the initial binary operation given. i.e.
    \beta: BxB -> B where (b,b') -> b*b' (ordinary multiplication)
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,383
    Thanks
    1474
    Awards
    1
    Quote Originally Posted by Obstacle1 View Post
    I don't exactly understand what f*g is either, but I thought it was defined in some way by the initial binary operation given. i.e.
    \beta: BxB -> B where (b,b') -> b*b' (ordinary multiplication)
    Because each of f & g is a mapping of A to B, if we define f*g(x) as f(x)*g(x) where * is the binary operation on B, then f*g is a mapping from A to B also.

    But I still am not clear as to what the question is asking.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Apr 2007
    Posts
    32
    Quote Originally Posted by Plato View Post
    Because each of f & g is a mapping of A to B, if we define f*g(x) as f(x)*g(x) where * is the binary operation on B, then f*g is a mapping from A to B also.

    But I still am not clear as to what the question is asking.
    How do you deduce that?

    I guess i was asking 'what are pointwise addition and multiplication?'.

    So far i'm thinking along the lines of...because f and g are maps from A to B and B has a certain operation defined on it, you can also perform this operation on f and g...
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,383
    Thanks
    1474
    Awards
    1
    Quote Originally Posted by Obstacle1 View Post
    How do you deduce that?
    If each of f & g is a mapping of A to B and * is the binary operation on B, then \forall x \in A\,  \Rightarrow \; f(x) \in B \wedge  g(x) \in B. Therefore, f(x)*g(x) \in B is a well defined operation; thus defining \left( {f*g} \right)(x) = f(x)*g(x) creates a mapping f*g:A \mapsto B.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Junior Member
    Joined
    Apr 2007
    Posts
    32
    Quote Originally Posted by Plato View Post
    If each of f & g is a mapping of A to B and * is the binary operation on B, then \forall x \in A\, \Rightarrow \; f(x) \in B \wedge g(x) \in B. Therefore, f(x)*g(x) \in B is a well defined operation; thus defining \left( {f*g} \right)(x) = f(x)*g(x) creates a mapping f*g:A \mapsto B.

    ok. i see now, quite simple really.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Pointwise limit
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: November 23rd 2009, 12:50 AM
  2. Pointwise limit
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: November 22nd 2009, 06:37 PM
  3. Pointwise convergence.
    Posted in the Calculus Forum
    Replies: 2
    Last Post: May 17th 2008, 07:34 PM
  4. pointwise limit?
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 20th 2008, 06:15 PM
  5. Pointwise Convergence
    Posted in the Calculus Forum
    Replies: 3
    Last Post: October 21st 2007, 05:23 PM

Search Tags


/mathhelpforum @mathhelpforum