Functions

A function from a set $A$ to a set $B$ is a map from elements of $A$ to elements of $B$ which pairs each element of $A$ with exactly one element of $B$. If $f$ is such a function, we write \[f : A → B\] to indicate this. We call $A$ the domain of the function, and we call $B$ the codomain of the function. If $a ∈ A$, then we write \[f(a)\] to pick out the unique $b ∈ B$ that $f$ pairs $a$ up with. In case $f(a) = b$ (for some $a$ and $b$), we call $a$ the argument of the function $f$, and we call $b$ the value of the function $f$ on $a$.

Kind of like with relations, the definition of (unary) functions can be generalized to that for n-ary functions by considering the latter to be a map from the product of $n$ sets $A₁, ..., Aₙ$ to a set $B$. If $f$ is such an $n$-ary function, we write \[f : A_1 × ... × A_n → B\] to indicate this. Given $n$ arguments $a₁ ∈ A₁, ..., aₙ ∈ Aₙ$, we write \[f(a_1, ..., a_n)\] to pick out out the $b ∈ B$ that $f$ maps $a₁, ..., aₙ$ to.

Importantly, there is a corresponding notion of function composition. We need only consider functions from $A$ to $B$ a type of relation on $A$ and $B$—one that relations every element in $A$ to exactly one element in $B$. Given functions $f : A → B$ and $g : B → C$, their composition $g ∘ f : A → C$ is such that $(g ∘ f)(x) = g(f(x))$, for any $x ∈ A$.

--- title: "Functions" bibliography: ../../semantics.bib --- A function from a set $A$ to a set $B$ is a map from elements of $A$ to elements of $B$ which pairs each element of $A$ with exactly one element of $B$. If $f$ is such a function, we write $$f : A → B$$ to indicate this. We call $A$ the *domain* of the function, and we call $B$ the *codomain* of the function. If $a ∈ A$, then we write $$f(a)$$ to pick out the unique $b ∈ B$ that $f$ pairs $a$ up with. In case $f(a) = b$ (for some $a$ and $b$), we call $a$ the *argument* of the function $f$, and we call $b$ the *value* of the function $f$ on $a$. Kind of like with relations, the definition of (unary) functions can be generalized to that for *n*-ary functions by considering the latter to be a map from the product of $n$ sets $A₁, ..., Aₙ$ to a set $B$. If $f$ is such an $n$-ary function, we write $$f : A_1 × ... × A_n → B$$ to indicate this. Given $n$ arguments $a₁ ∈ A₁, ..., aₙ ∈ Aₙ$, we write $$f(a_1, ..., a_n)$$ to pick out out the $b ∈ B$ that $f$ maps $a₁, ..., aₙ$ to.             Importantly, there is a corresponding notion of *function composition*. We need only consider functions from $A$ to $B$ a type of relation on $A$ and $B$---one that relations *every* element in $A$ to exactly one element in $B$. Given functions $f : A → B$ and $g : B → C$, their composition $g ∘ f : A → C$ is such that $(g ∘ f)(x) = g(f(x))$, for any $x ∈ A$.