Function mapping

Mathematical statement

A function f:X→Y assigns exactly one output in Y to every input in X; the codomain Y need not equal the function's actual range.

Representative formula (exact TeX)

f:X\to Y

Assumptions

• X and Y are specified sets or compatible types.
• Every input in X has exactly one output under f.

Symbol scope

• f — function
• X — domain
• Y — codomain
• → — mapping arrow

Intuition, definition, and misconceptions

The core of a function is not a drawn curve but a rule that associates every permitted input with exactly one output.

The declaration f:X→Y specifies the domain and codomain. The set of outputs actually attained may be only a subset of Y.

The Lean theorem function_total_unique proves, for every f:X→Y and x:X, that there exists a unique y:Y satisfying f x=y.

A common mistake is to treat the codomain, range, and graph of a function as the same object, or to identify effectful or failure-prone program functions directly with pure total mathematical functions.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Set membership — Function inputs and outputs belong to the domain and codomain respectively.

Verification boundary

Lean formalizes the core total-function claim that every input has a unique output. Partial functions, side effects, randomness, and execution failure lie outside this statement.

formalized_total_function_core

• evidence-sha256-d999133c1ae711b73752af8ee4b1b3ce74ca135ff7c06aabbe9fc8a3186f434b

Next obligation

Add objects for range, injection, surjection, bijection, and partial functions, keeping the typed codomain distinct from the set of outputs actually attained.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.