Functions - Essential Dictionary I

Mathematical Writing - Vivaldi Franco 2014

Functions
Essential Dictionary I

Functions are everywhere. Whenever a process transforms a mathematical object into another object, there is a function in the background. ’Function’ is arguably the most used word in mathematics.

A function consists of two sets together with a rule5 that assigns to each element of the first set a unique element of the second set. The first set is called the domain of the function and the second set is called the co-domain. A function whose domain is a set $$A$$ may be called a function over $$A$$ or a function defined on $$A$$. The terms map or mapping are synonymous with function. The term operator is used to describe certain types of functions (see below).

A function is usually denoted by a single letter or symbol, such as $$f$$. If $$x$$ is an element of the domain of a function $$f$$, then the value of $$f$$ at $$x$$, denoted by $$f(x)$$ is the unique element of the co-domain that the rule defining $$f$$ assigns to $$x$$. The notation

$$\begin{aligned} f:A\rightarrow B\quad \, x\mapsto f(x) \end{aligned}$$

(2.16)

indicates that $$f$$ is a function with domain $$A$$ and co-domain $$B$$ that maps $$x\in A$$ to $$f(x)\in B$$. The symbol $$x$$ is the variable or (the argument) of the function. The symbols $$\rightarrow $$ and $$\mapsto $$ have different meanings, and should not be confused. The function

$$ I_A:A\rightarrow A\qquad x\mapsto x $$

is called the identity (function) on $$A$$. When explicit reference to the set $$A$$ is unnecessary, the identity is also denoted by Id or $$1\!\!1$$.

In Definition (2.16) the symbols used for the function’s name and variable are inessential; the two expressions

$$ f:\mathbb {R}{\backslash }\{0\}\rightarrow \mathbb {R}\quad x\mapsto \frac{1}{x} \qquad x:\mathbb {R}{\backslash }\{0\}\rightarrow \mathbb {R}\quad f\mapsto \frac{1}{f} $$

define exactly the same function (even though the rightmost expression breaks just about every rule concerning mathematical notation—see Sect. 6.2).

Let us use the word ’function’ in short expressions. These are function definitions:

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We surmise that the function in item 2 is defined over the real numbers. Item 3 is a much-studied function in number theory. The set of values assumed by the function in item 4 is the closed interval $$[0,\uppi ]$$.

Functions of several variables are defined over cartesian products of sets. For example, the function

$$ f:\mathbb {Z}\times \mathbb {Z}\rightarrow \mathbb {N}\qquad (x,y)\mapsto \text{ gcd }(x,y) $$

depends on two integer arguments, and hence is defined over the cartesian product of two copies of the integers. This definition requires a value for $$\text{ gcd }(0,0)$$, which normally is taken to be zero.

Let $$f:A\rightarrow B$$ be a function. The set

$$\begin{aligned} \{(x,f(x))\in A\times B\,:\, x\in A\} \end{aligned}$$

(2.17)

is called the graph of $$f$$. A function is completely specified by three sets: domain, co-domain and graph. We can now reformulate the definition of a function, replacing the imprecise term ’rule’ with the precise term ’graph’. We write a formal definition.

DEFINITION. A function $$f$$ is a triple $$(X,Y,G)$$ of non-empty sets. The sets $$X$$ and $$Y$$ are arbitrary, while $$G$$ is a subset of $$X\times Y$$ with the property that for every $$x\in X$$ there is a unique pair $$(x,y)\in G$$. The quantity $$y$$ is called the value of the function at $$x$$, denoted by $$f(x)$$.

We see that, besides sets, the definition of a function requires the constructs of ordered pair and triple. It turns out that these quantities can be defined solely in terms of sets (see Exercise 2.14). So, to define functions, all we need are sets after all.

Given a function $$f:A\rightarrow B$$, and a subset $$X\subset A$$, the set

$$\begin{aligned} f(X)\mathop {=}\limits ^\mathrm{def}\{f(x)\,:\, x\in X\} \end{aligned}$$

(2.18)

is called the image of $$X$$ under $$f$$. The assignment operator gives meaning to the symbolic expression $$f(X)$$, which otherwise would be meaningless, since we stipulated that the argument of a function is an element of the domain, not a subset of it. Thus $$\sin (\mathbb {R})$$ is the closed interval $$[-1,1]$$.

Clearly, $$f(X)\subset B$$, and $$f(A)$$ is the smallest set that can serve as co-domain for $$f$$. The set $$f(A)$$ is often called the image or the range of the function $$f$$. This term is sometimes used to mean co-domain, which should be avoided. A constant is a function whose image consists of a single point.

The notation (2.18) is suggestive and widely used. However, in computer algebra, the quantities $$f(x)$$ and $$f(X)$$ (with $$x$$ an element and $$X$$ a subset of the domain, respectively) are written with a different syntax, e.g., f(x) and map(f,X) with Maple.

A function is said to be injective (or one-to-one) if distinct points of the domain map to distinct points of the co-domain. A function is surjective (or onto) if $$f(A)=B$$, that is, if the image coincides with the co-domain. A function that is both injective and surjective is said to be bijective.

For any non-empty subset $$X$$ of the domain $$A$$, we define the restriction of $$f$$ to $$X$$ as

$$ \left. f\right| _X:X\rightarrow B\qquad x\mapsto f(x). $$

Given two functions $$f:A\rightarrow B$$ and $$g:B\rightarrow C$$, their composition is the function

$$\begin{aligned} g\circ f:A\rightarrow C \qquad x\mapsto g(f(x)). \end{aligned}$$

(2.19)

The notation $$g\circ f$$ reminds us that $$f$$ acts before $$g$$. The image $$g(f(x))$$ of $$x$$ under $$g\circ f$$ is denoted by $$(g\circ f)(x)$$, where the parentheses isolate $$g\circ f$$ as the function’s symbolic name. The hybrid notation $$g\circ f(x)$$ should be avoided.

If $$f:A\rightarrow B$$ is a bijective function, then the inverse of $$f$$ is the function $$f^{-1}:B\rightarrow A$$ such that

$$ f^{-1}\circ f=1\!\!1_A \qquad f\circ f^{-1}=1\!\!1_B $$

where $$1\!\!1_{A,B}$$ are the identities in the respective sets. A function is said to be invertible if its inverse exists. If $$f:A\rightarrow B$$ is injective, then we can always define the inverse of $$f$$ by restricting its domain to $$f(A)$$ if necessary. In absence of injectivity, it may still be possible to construct an inverse by a suitable restriction of the function. Thus the arcsine may be defined by restricting the sine to the interval $$[-\uppi /2,\uppi /2]$$.

Let $$f:A\rightarrow B$$ be a function, and let $$C$$ be a subset of $$B$$. The set of points

$$\begin{aligned} f^{-1}(C)\mathop {=}\limits ^\mathrm{def}\{x\in A \,:\, f(x)\in \ C\} \end{aligned}$$

(2.20)

is called the inverse image of the set $$C$$.

Since the definition of inverse image does not involve the inverse function, the inverse image exists even if the inverse function does not. These two concepts must be distinguished carefully. When the reciprocal of a function comes into play, things get very confusing, since we now have three unrelated objects represented by closely related notation:

$$ f^{-1}(x)\qquad f^{-1}(\{x\})\qquad f(x)^{-1}. $$

The first expression is well-defined if $$x$$ belongs to the image of $$f$$ and $$f$$ is invertible there. In the second expression there is no condition on $$f$$, and $$x$$ need only be an element of the co-domain. In the third expression the point $$x$$ must belong to the domain of $$f$$, and $$f(x)$$ must be non-zero. Thus

$$ \sin ^{-1}(1)=\frac{\uppi }{2} \quad \,\sin ^{-1}(\{1\})=\frac{\uppi }{2}+2{\uppi }\,\mathbb {Z}\quad \, \sin (1)^{-1}=\csc (1). $$

In the first expression we tacitly assume that $$\sin ^{-1}=\arcsin : [-1,1]\rightarrow [-\uppi /2,\uppi /2]$$. In the third expression the symbol $$\csc $$ denotes the co-secant ($$\csc (x)=1/\sin (x)$$), defined in the domain $$\mathbb {R}{\backslash }\uppi \mathbb {Z}$$.

With a judicious use of definite and indefinite articles, we can specify a function’s type without committing ourselves to a specific object.

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In item 2, we infer that the function maps its domain into itself. Functions of type 3 will be considered in the next section to define cardinality of sets. Functions of type 4 are called involutions (e.g., $$x\mapsto -x$$, over a suitable domain).

Writing about real functions is considered in Chap. 5.

Exercise 2.5

Turn symbols into words. [ $$\not \varepsilon $$ ]

Exercise 2.6

Explain clearly and plainly. [ $$\not \varepsilon ,\,50$$ ] 6

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Exercise 2.7

I have two finite sets and a function between them. I am able to compute the value of the function at each point of the domain, and to count and compare the elements of these sets. I need explicit instructions for answering the following questions. [ $$\not \varepsilon ,\,50$$ ]

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Exercise 2.8

Answer each question as clearly as you can.

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