# Math Help - Pointwise sum and product

1. ## 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?

2. Originally Posted by Obstacle1
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 &#091;math]\lambda[/tex] we can see $\lambda$

3. Originally Posted by Plato
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)

4. Originally Posted by Obstacle1
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.

5. Originally Posted by Plato
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?

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 $f*g:A \mapsto B$.
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 $f*g:A \mapsto B$.