Functions

Se basa en

Requisitos previos obligatorios, hasta el fondo: 6

Aprenderás a

• Decide whether a rule, table or arrow diagram defines a function from a given domain to a given codomain.
• Identify the domain, codomain and image of a function, and explain why the image can be smaller than the codomain.
• Describe a function as a set of ordered pairs and classify functions as injective, surjective or bijective.
• Count the functions from a finite set A to a finite set B, and the injections when |A| ≤ |B|.

Definición

• A function f : A → B assigns to each element a of the domain A exactly one element f(a) of the codomain B.
• Formally, a function from A to B is a subset f ⊆ A × B such that for each a ∈ A there is exactly one b ∈ B with (a, b) ∈ f; the domain and codomain are part of the data.
• The image of f is the set {f(a) : a ∈ A}, a subset of the codomain.
• f is injective if f(a) = f(a′) implies a = a′, surjective if every b ∈ B equals f(a) for some a ∈ A, and bijective if it is both.

Teorema

• There are |B|^|A| functions from a finite set A to a finite set B; in particular there is exactly one function from ∅ to any set.
• If |A| = m ≤ n = |B|, there are n(n − 1)⋯(n − m + 1) injections from A to B.
• For finite sets with |A| = |B|, a function A → B is injective if and only if it is surjective.
• A function has an inverse function exactly when it is bijective.

Errores frecuentes

• Thinking a function must be given by a formula; any rule assigning exactly one output to each input is a function.
• Believing every element of the codomain must be an output; that is surjectivity, not part of the definition.
• Thinking a function may not send two different inputs to the same output.
• Ignoring the codomain: n ↦ 2n is surjective as a function from ℤ to the even integers, but not as a function from ℤ to ℤ.

Conduce a

Descubrimientos opcionales

• Products by maps: a first universal property A function into a Cartesian product A × B is exactly a pair of functions, one into A and one into B: the first example of a universal property.Sendero del Asombro →
• Programming for mathematics Programming languages also have 'functions', but a program function may read input, change state or fail to return, so it need not be a mathematical function.

Itinerarios de aprendizaje

• Conventional K–12 progression
• Accelerated secondary progression
• Adult beginner pathway

Lecturas complementarias

• I. M. Gelfand, E. G. Glagoleva and E. E. Shnol, Functions and Graphs