[Home]Injective, surjective and bijective functions

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


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.



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Last edited October 28, 2001 6:10 am by AxelBoldt (diff)