Functions - mathSight

“Mathematics is the study of functions.”
— Vito Volterra

Theory

Functions are evidently important in mathematics. The definition of a function follows from set theory and it is worth exploring the derivation. Commonly a function is seen as something like ${f(x)=x^2}$ . However, their definition is grounded in set theory.

A function should be seen as an assignment of an element of a set ${A}$ to an element of a set ${B}$ with some restrictions. This idea can be stated as:

$\begin{equation*} f:A \rightarrow B \end{equation*}$

Where ${f}$ is the function. The image of ${f}$ is the element in the set ${B}$ given by ${f(x)}$ .

The set ${A}$ is the domain of the function ${f}$ . The range is the set of all elements of ${B}$ that are mapped to by elements of ${A}$ . The function ${f}$ forms a set of ordered pairs for which ${f}$ is true. Let ${R}$ be the range such that ${R \subseteq B}$ .

$\displaystyle \begin{array}{rcl} f = \{(x,y): x \in A \land y \in R\}\\ \implies (x,y)\in A \times R\\ \implies f \subseteq A \times B \end{array}$

A singled valued function can therefore be viewed as a set of ordered pairs for which ${f}$ is true. These ordered pairs are a subset of the cartesian product of ${A}$ and ${B}$ where ${f:A\rightarrow B}$ . Every element of ${A}$ must map to an element of ${B}$ . Further, to be a function, an element ${x}$ of ${A}$ must map to only one value ${y}$ in ${B}$ .

$\begin{equation*} \forall x\in A \; \exists! y\in B : (x,y)\in f \end{equation*}$

In the above definition, ${x}$ in ${A}$ need not map to an unique element ${y}$ of ${B}$ . If each ${x}$ does have a unique ${y}$ then the function is called injective.

Definition: Let ${A}$ and ${B}$ be sets and ${f:A\rightarrow B}$ . ${f}$ is injective if every ${x}$ maps to a unique ${y}$ .

$\displaystyle \begin{array}{rcl} (x_1,y) \in f \land (x_2,y) \in f \implies x_1 = x_2 \end{array}$

For the map ${f:A\rightarrow B}$ if the image of ${f}$ is all of ${B}$ then ${f}$ is called surjective.

Definition: Let ${A}$ and ${B}$ be sets and ${f:A\rightarrow B}$ . If every element ${y}$ in ${B}$ is the image of an element ${x}$ in ${A}$ then ${f}$ is surjective.

$\begin{equation*} \forall y \in B \; \exists \; x \in A : y = f(x) \end{equation*}$

Definition: If a function is both injective and surjective then the function is bijective

Composition of functions.

A composition of functions is to perform one function after another. Say $g$ and $h$ are functions. Then function composition gives a new function $f$ , for example, $f = g \circ h$ . This means $f$ is the function of $g$ applied after $h$ . This means $f(x) = g(h(x))$ . An alternative way of stating this is $[g\circ h](x)$ .

Consider the composition of three functions:

$[f\circ (g\circ h)](x) = f(g(h(x))) = [f\circ g](h(x)) = [(f\circ g)\circ h](x)$

Hence composition of functions is associative. Note that the composition of functions is not necessarily commutative.

Discussion

Functions needs note revolve around numbers. For example, in a monogynous society if a set of people A have partners in the set B then the partner function maps an element of A to B. If sets A and B are the same size then this would also be a bijective function. That is, there is a one to one map between each element of the two sets. If the society is not monogynous then the partner map is not a function as a member of set A could map to many members of set B which breaks the definition of a function.

If a function is bijective then there is an inverse function that maps an image back to the original element in the domain.

A common function in mathematics is $f(x) = x^2$ . This is a function that takes an element of the real numbers and maps it to only one element of the positive real numbers. However, this function is not injective. For example $f(x) = 4$ means $4$ maps to either $-2$ or $2$ . This reverse process is not a function. Therefore no inverse exists. There are many functions that are not injective, for example $f(x)= sin(x)$ .

Composition of functions is used in the chain rule of differentiation encountered in elementary calculus.

The bijective relationship is the core of understand the ‘size’ of a cardinal number. Two sets have the same size if a bijective map can be formed between the elements of each set. This means that we only need to define a standard set for each number and every other set can be compared to this to give a concept of ‘size’.