Surjection

Surjective composition: the first function nee...

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A function is surjective (onto) if every possible image is mapped to by at least one argument. In other words, every element in the codomain has non-empty preimage. Equivalently, a function is surjective if its image is equal to its codomain. A surjective function is a surjection. The formal definition is the following.

The function f: A \to B is surjective iff for all b \in B, there is a \in A such that f(a) = b.
  • A function f : AB is surjective if and only if it is right-invertible, that is, if and only if there is a function g: BA such that f o g = identity function on B. (This statement is equivalent to the axiom of choice.)
  • By collapsing all arguments mapping to a given fixed image, every surjection induces a bijection defined on a quotient of its domain. More precisely, every surjection f : AB can be factored as a projection followed by a bijection as follows. Let A/~ be the equivalence classes of A under the following equivalence relation: x ~ y if and only if f(x) = f(y). Equivalently, A/~ is the set of all preimages under f. Let P(~) : AA/~ be the projection map which sends each x in A to its equivalence class [x]~, and let fP : A/~ → B be the well-defined function given by fP([x]~) = f(x). Then f = fP o P(~). A dual factorisation is given for injections above.
  • The composition of two surjections is again a surjection, but if g o f is surjective, then it can only be concluded that g is surjective. See the figure at right*.

 

SEE ALSO ::

injections, surjections and bijections

Posted on October 6, 2011, in education and tagged , , , , , , , . Bookmark the permalink. 2 Comments.

  1. Kasih arti bahasa indonesianya napa nic, biar kita bisa lebih ngerti dan paham ,,

  1. Pingback: Bijection « Nico For Math

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