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 f: \; A \to B is injective (one-to-one) if every element of the codomain is mapped to by at most one element of the domain. Notationally,
\forall x, y \in A, f(x)=f(y) \Rightarrow x=y\ or, equivalently,
\forall x,y \in A, x \neq y \Rightarrow f(x) \neq f(y).\

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,
\forall y \in B, \exists x \in A \text{ such that } y = f(x).\

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.

Illustration of bijection

Image via Wikipedia

Injective and surjective (bijective).

A non-surjective function. (This one happens t...

Image via Wikipedia

Injective and non-surjective.

A non-injective function (this one happens to ...

Image via Wikipedia

Non-injective and surjective.

An example of a partial function that is also ...

Image via Wikipedia

Non-injective and non-surjective.

About NICO MATEMATIKA

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.

  1. i don’t understand about this materi,, i so confused O.o ???

  2. You give a Wikipedia-like post on the three terminology.
    http://gowers.wordpress.com/2011/10/11/injections-surjections-and-all-that/
    Gowers has posted an excellent post about such matter.And Terence Tao made an good comment on Gowers post.

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