Thread: 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?

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 . 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

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 . 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 we can see

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 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.
: BxB -> B where (b,b') -> b*b' (ordinary multiplication)

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.
: 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.

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?

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...

Originally Posted by Obstacle1

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 . Therefore, is a well defined operation; thus defining creates a mapping .

Originally Posted by Plato

If each of f & g is a mapping of A to B and * is the binary operation on B, then . Therefore, is a well defined operation; thus defining creates a mapping .

ok. i see now, quite simple really.