Fiber (mathematics)

In mathematics, the archetype is a term related functions. The archetype of a set under a function is the set of elements which are represented by elements in. An element from the definition of quantity is exactly then the preimage of when in lies. This is the archetype of a subset of the target set of a subset of the set of definitions.

Definition

Is a function, and a subset of the. Then we denote the set

As the inverse image of M under f

A prototype is therefore a value of the so-called inverse image function that maps the archetype as an element of the power set of the set of definitions each element of the power set of the target amount.

The archetype of a singleton to write as

And it is called the inverse image of b under f This amount but need not be a singleton ( so it can be empty or contain more than one element ).

The inverse image of an element is sometimes also called the fiber of this element, in particular in the context of fiber bundles.

Examples

For the function ( an integer ) with the following applies:

Properties

Injectivity, surjectivity, bijectivity

• Under a bijective function, the preimage of each element ( exactly ) is a singleton. The mapping which assigns to each element of the (one, that is uniquely determined) element of its archetype, ie inverse function of. They are called (even - as the prototype function) with. This can easily lead to misunderstandings if you do not detail writes for the inverse function (which they will then be clearly distinguished from the prototype function).
• Under an injective function is the inverse image of an element at most a singleton (that is always a singleton or empty).
• Under a surjective function is the inverse image of an element at least a singleton (that is always non-empty ).

Set operations and properties

It is a function, and and are subsets of. Then:

It refers to the complement of the respective base amount.

The statements about union and intersection can be generalized as two treatments on any family of subsets.

Image and preimage

It is a function of a subset and a subset of the. Then:

• Is injective, then the equality holds.
• Is surjective, then the equality holds ( is sufficiently already that that is a subset of the image of is ).