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Injective composition: the second function nee...

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A function is injective (one-to-one) if every possible element of the codomain is mapped to by at most one argument. Equivalently, a function is injective if it maps distinct arguments to distinct images. An injective function is an injection. The formal definition is the following.

The function f: A \to B is injective iff for all a,b \in A, we have f(a) = f(b) \Rarr a = b.
  • A function f : AB is injective if and only if A is empty or f is left-invertible; that is, there is a function g : f(A) → A such that g o f = identity function on A. Here f(A) is the image of f.
  • Since every function is surjective when its codomain is restricted to its image, every injection induces a bijection onto its image. More precisely, every injection f : AB can be factored as a bijection followed by an inclusion as follows. Let fR : Af(A) be f with codomain restricted to its image, and let i : f(A) → B be the inclusion map from f(A) into B. Then f = i o fR. A dual factorisation is given for surjections below.
  • The composition of two injections is again an injection, but if g o f is injective, then it can only be concluded that f is injective. See the figure at right.
  • Every embedding is injective.




injections, surjections and bijections

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. Read the rest of this entry



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In mathematics, the range of a function refers to either the codomain or the image of the function, depending upon usage. This ambiguity is illustrated by the function f that maps real numbers to real numbers with f(x) = x2. Some books say that range of this function is its codomain, the set of all real numbers, reflecting that the function is real-valued. These books call the actual output of the function the image. This is the current usage for range in computer science. Other books say that the range is the function’s image, the set of non-negative real numbers, reflecting that a number can be the output of this function if and only if it is a non-negative real number. In this case, the larger set containing the range is called the codomain.[1] This usage is more common in modern mathematics. Read the rest of this entry



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MATLAB adalah sebuah lingkungan komputasi numerikal dan bahasa pemrograman komputer generasi keempat. Dikembangkan oleh The MathWorks, MATLAB memungkinkan manipulasi matriks, pem-plot-an fungsi dan data, implementasi algoritma, pembuatan antarmuka pengguna, dan peng-antarmuka-an dengan program dalam bahasa lainnya. Meskipun hanya bernuansa numerik, sebuah kotak kakas (toolbox) yang menggunakan mesin simbolik MuPAD, memungkinkan akses terhadap kemampuan aljabar komputer. Sebuah paket tambahan, Simulink, menambahkan simulasi grafis multiranah dan Desain Berdasar-Model untuk sistem terlekat dan dinamik.

MATLAB ini kurang lebih sama dengan MAPLE yang digunakan untuk menjawab soal-soal matematika.

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Rational function

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In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input.

Informally, a function f assigns an output f(x) to every input x. The function has a limit L at an input p if f(x) is “close” to L whenever x is “close” to p. In other words, f(x) becomes closer and closer to L as x moves closer and closer to p. More specifically, when f is applied to each input sufficiently close to p, the result is an output value that is arbitrarily close to L. If the inputs “close” to p are taken to values that are very different, the limit is said to not exist.

Formal definitions, first devised in the early 19th century, are given below.

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