[Home]Injective, surjective and bijective functions

HomePage | Recent Changes | Preferences

Difference (from prior major revision) (no other diffs)

Added: 30a31,32
* A function f : X -> Y is bijective if and only if there exists a function g : Y -> X such that gof is the identity on X and fog is the identity on Y. In this case, g is uniquely determined by f and we call g the inverse function of f and write f -1 = g.
* The bijective functions X -> X, together with functional composition, form a mathematical group, the symmetric group of X denoted by S(X).

Three important kinds of mathematical functions deserve special names: a function f : X -> Y is called

Motivation

These kinds of functions are generally useful for showing the relationships between different kinds of mathematical objects. Often, an injective relationship will correspond to a subset-like relationship, and the existence of a bijection will demonstrate an equality-like relationship.

For instance, a homomorphism θ is a relationship between two algebraic objects (that is sets with some kind of multiplication defined for the purposes of this example) which preserves multiplication; that is θ(x) * θ(y) = θ(x * y) for all x and y. If a homomorphism is also a bijection then the two algebraic objects are described as isomorphic; they are so closely equivalent that consideration of the algebraic properties of one is equivalent to consideration of the algebraic properties of the other; in writing a proof about one you may as well be writing a proof about the other. Therefore a convenient proof technique to show the existence of some property in object A is to demonstrate a bijection between A and some other object B, and realize that someone else has already proven this property for object B.

Examples

Propositions


HomePage | Recent Changes | Preferences
This page is read-only | View other revisions
Last edited October 28, 2001 6:10 am by AxelBoldt (diff)
Search: