# Bijection, injection and surjection

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.

• A function is injective (one-to-one) if every element of the codomain is mapped to by at most one element of the domain. Notationally,
or, equivalently,

An injective function is an injection.

• A function is surjective (onto) if every 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.) Notationally,

A surjective function is a surjection.

• A function is bijective (one-to-one and onto or one-to-one correspondence) if every 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 a bijection.

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 following diagrams.

 Image via Wikipedia Injective and surjective (bijective). Image via Wikipedia Injective and non-surjective. Image via Wikipedia Non-injective and surjective. Image via Wikipedia Non-injective and non-surjective.

Welcome to my blog. My name is Nico. Admin of this blog. I am a student majoring in mathematics who dreams of becoming a professor of mathematics. I live in Kwadungan, Ngawi, East Java. Hopefully in all the posts I can make a good learning material to the intellectual life of the nation. After the read, leave a comment. I always accept criticism suggestion to build a better me again .. Thanks for visiting .. : mrgreen:

Posted on October 4, 2011, in education and tagged , , , , , , , . Bookmark the permalink. 2 Comments.