Function

A function is a machine that takes an input and produces exactly one output. The square button on a calculator is a function: put in 3, get out 9. Put in 5, get out 25. The rule is fixed, and every input gives a single, definite result. If a rule could give two different outputs for the same input, it would not be a function.

Formally, a function $f$ from a set $A$ to a set $B$ (written $f:A\to B$) assigns to each element of $A$ exactly one element of $B$. The set $A$ is called the domain (the inputs the function accepts) and $B$ is called the codomain (the set that contains all possible outputs). The requirement that each input maps to exactly one output is called well-definedness — it is what separates functions from relations in general, where an input might relate to many outputs or none at all.

Functions can be chained together. If $f:A\to B$ and $g:B\to C$, then the composition $g\circ f:A\to C$ applies $f$ first and $g$ second. Composition is one of the most pervasive operations in mathematics: it lets you build complicated transformations out of simple pieces.

Three properties classify how a function connects its domain and codomain:

• Injective (one-to-one): different inputs always give different outputs. No information is lost.
• Surjective (onto): every element of the codomain is hit by at least one input. Nothing in the target is missed.
• Bijective: both injective and surjective. The function pairs up elements of $A$ and $B$ perfectly, and it has an inverse that undoes it.

Functions appear everywhere in this vault’s formal architecture. Closure operators are functions that expand a structure to include everything implied by it. The semiotic universe is built from three such functions composed together, and its core object — the initial semiotic structure — is the least fixed point of that composition. Understanding functions and composition is the entry point to understanding that construction.

Related concepts

• Relation — a function is a special case of a relation
• Closure operator — a function with specific properties central to this vault
• Fixed point — an input that a function maps to itself
• Set theory — provides the sets that serve as domains and codomains
• Semiotic universe — built from composed functions on a Heyting algebra