Bijection, injection and surjection
Properties of mathematical functions / From Wikipedia, the free encyclopedia
Dear Wikiwand AI, let's keep it short by simply answering these key questions:
Can you list the top facts and stats about Bijection, injection and surjection?
Summarize this article for a 10 year old
In mathematics, injections, surjections, and bijections are classes of functions distinguished by the manner in which arguments (input expressions from the domain) and images (output expressions from the codomain) are related or mapped to each other.
A function maps elements from its domain to elements in its codomain. Given a function :
- The function is injective, or one-to-one, if each element of the codomain is mapped to by at most one element of the domain, or equivalently, if distinct elements of the domain map to distinct elements in the codomain. An injective function is also called an injection.[1] Notationally:
- or, equivalently (using logical transposition),
- The function is surjective, or onto, if each element of the codomain is mapped to by at least one element of the domain. That is, the image and the codomain of the function are equal. A surjective function is a surjection.[1] Notationally:
- The function is bijective (one-to-one and onto, one-to-one correspondence, or invertible) if each element of the codomain is mapped to by exactly one element of the domain. That is, the function is both injective and surjective. A bijective function is also called a bijection.[1][2][3][4] That is, combining the definitions of injective and surjective,
- where means "there exists exactly one x".
- In any case (for any function), the following holds:
An injective function need not be surjective (not all elements of the codomain may be associated with arguments), and a surjective function need not be injective (some images may be associated with more than one argument). The four possible combinations of injective and surjective features are illustrated in the adjacent diagrams.