\[ \newcommand{\tr}{\Rightarrow} \newcommand{\trs}{\tr^{\!\ast}} \newcommand{\rlnm}[1]{\mathsf{(#1)}} \newcommand{\rred}[1]{\xrightarrow{#1}} \newcommand{\rreds}[1]{\mathrel{\xrightarrow{#1}\!\!^*}} \newcommand{\cl}{\mathsf{Cl}} \newcommand{\pow}{\mathcal{P}} \newcommand{\matches}{\mathrel{\mathsf{matches}}} \newcommand{\kw}[1]{\mathsf{#1}} \]

Functions

A function $f : A \to B$ from $A$ to $B$ is a relation $f$ from $A$ to $B$ that has two special properties:

• it is total: for every $a \in A$ there exists at least one $b \in B$ such that $(a, b) \in f$
• it is single-valued: for every $a \in A$ there exists at most one $b \in B$ such that $(a, b) \in f$

Putting these together, $f$ is a function just if for every possible input $a \in A$ there exists a unique output $b \in B$ such that $(a, b) \in f$. We write $f(a)$ for this unique $b \in B$.

The set of all functions from $A$ to $B$ is defined to be the set

\[A \to B = \{ f \subseteq A \times B \mid f : A \to B \}\]

of relations from $A$ to $B$ which are total and single-valued.

Partial functions

Computer programs sometimes fail to halt. In order to model that possibility we sometimes relax the totality condition in the definition of a function.

A partial function $f : A ⇀ B$ from $A$ to $B$ is a relation from $A$ to $B$ that is single-valued, in that for every $a \in A$ there is at most one $b \in B$ such that $(a, b) \in f$.

Thus, given some $x \in A$ the partial function $f : A ⇀ B$ might be undefined at that particular input; we write $f(x) \uparrow$ to denote that.

Whenever $f$ is defined at $x \in A$ and is equal to $y \in B$, we write $f(x) \simeq y$. As $f$ is single-valued, there is at most one such $y$.

Kleene equality

In general, the notation $e \simeq e’$, which is called Kleene equality, means one of two things:

• both the expressions $e$ and $e’$ are undefined, or
• both $e$ and $e’$ have a well-defined value, and their value is equal

For example, we can say that \(\frac{1}{0} \simeq \frac{2}{0}\) because both expressions are undefined (division by zero is not well-defined—division is a partial function!).

However, \(\frac{1}{2} \not\simeq \frac{2}{2}\) as the value of both sides is well-defined, but unequal.

Identity function

For every set $A$ there exists an identity function which maps every element to itself:

\[\begin{aligned} & \textsf{id}_A : A \to A & \textsf{id}_A(x) = x \end{aligned}\]

Composition of functions

Given two functions $f : A \to B$ and $g : B \to C$, their composition $g \circ f : A \to B$ is defined by

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

Preimage

The preimage of a function $f : A \to B$ at $b \in B$ is the set

\[f^{-1}(\{ b \}) = \{ a \in A \mid f(a) = b \}\]

Warning. This is not to be confused with the inverse $f^{-1}$ of a function, which does not always exist (it only exists when $f$ is a bijection). We distinguish the preimage of $f$ by writing it using the singleton set ${ b }$, as above.

The preimage of a function $f$ at $b$ consists of the set of all solutions of the equation $f(x) = b$ for $x$.

Example

Let $f : \mathbb{N} \to \mathbb{B}$ be defined by

\[\begin{aligned} f(n) = \begin{cases} \top & \text{ if $n$ is even} \\ \bot & \text{ if $n$ is odd} \end{cases} \end{aligned}\]

Then we have that $f^{-1}({ \top })$ is the set of all even numbers, whereas $f^{-1}({ \bot })$ is the set of all odd numbers.