How Many Types of Integrals Exist and Can They Be Classified?

[roger]

why can't functions be one to many ? from reals to reals ?

 

[TD]

Because we define a function otherwise, it uniquely associates elements from the domain with an element of the image.
They can be one-to-one or more generally, many-to-one but never one-to-many.

 

[roger]

so its a convention ?

how about the inverse function of a many to one function ?

 

[TD]

A function is only invertible iff it is a bijection (although you can generalize this for many-to-one functions such as sin(x) etc, by limiting the range to obtain the 'principle value').

 

[roger]

whats the differnce between bijection and surjection ?

what do you mean by generalise ?

 

[TD]

If f:A->B is a surjection, every element of B is the image of an element of A.
A bijection is one-to-one.

With generalize I meant that strictly, only bijections are invertible. We can however define inverse functions for many-to-one's such as sin(x), although we have to realize that this is no longer a many-to-one but a one-to-many then, which can be 'solved' in a way by using the principle value.

 

[roger]

by image do you mean range of function ?

but looking it up on mathworld, it shows two diagrams both of which are identical except there are a few points which are outside the range

Please could you make the differences clearer ?

 

[matt grime]

Properly a function is single valued, that is part of its definition. Some times we like to relax this to allow many valued things, usually for convenience, thus given f: X --> Y we may use the symbol f^{-1}(U) and call it the inverse image of U a *SUBSET* of Y and is the (possibly empty) set of elements in X mapped to U, this is called an abuse of notation, and it is thought of as acting on SUBSETS of the image.The range of a function is the set Y in the above, the image is the suibset of Y that f maps onto.

Occasionally one to many things are called correspondences instead of functions, we can make them functions by thinking of them as mapping subsets of X to subsets of Y instead of points to points.

 

[shmoe]

I just want to point out a small terminology annoyance.

If f is a function from X to Y, f:X-->Y, then

the domain is the set X
the codomain is the set Y
the image is the set of elements y in Y that have at least one x in X where f(x)=y

These three are pretty much universal (I've never seen them defined otherwise). On the other hand, range is sometimes used to refer to what I've called image above and sometimes to what I've called the codomain, so beware of how the writer has defined it.

 

[matt grime]

Very true, i think i tend to use whatever definition the course I'm teaching requires me to use. image and codomain certainly are universal.

 

[TD]

Right, thanks - it's a bit confusing to me since I'm not used to the Englisch terminilogy (in Dutch, we have domain, codomain and image, but nothing for 'range' afaik).

 

[hypermorphism]

As an addition to the list, preimage is generally used to talk about g^-1(B) if the set B is in the image of the domain of g. Ie., the preimage of B under g. This is the same notation matt was referring to.
I find the term preimage more geometrically pleasing than inverse, especially when buried in layers of differential geometry.

 

[roger]

How many different types of integrals are there ?
by type, I mean the way they are to be evaluated OR integrals which are non existent

and can they be classified ?