Domain, codomain, and range

Mathematical statement

For a total function f:A→B, A is the set of all permitted inputs and B is the declared output set. The range f[A]={f(x):x∈A} is a subset of B and need not equal B.

Representative formula (exact TeX)

f:A\to B

Assumptions

• A and B are explicitly stated, and f is a total function assigning each input in A exactly one output in B.

Symbol scope

• f — total function
• A — domain
• B — codomain

Intuition, definition, and misconceptions

The domain specifies which inputs are permitted, while the codomain specifies where outputs must lie. The range collects only outputs that actually occur, with repeated outputs listed once.

The same input-output table can be compatible with different codomains, but the full declared functions f:A→B are then different data. The codomain must not be guessed from the table alone.

A finite table can be checked input by input for totality before A, B, and the range are read separately. A table with a missing input, multiple outputs for one input, or an output outside B is not the total function declared by this entry.

A common error is to identify the codomain with the attained range or compute the range of a claimed total function from a table that leaves an input undefined.

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

• Function mapping — The three set roles are defined from the input and output conditions of a total function.
• Set — The domain, codomain, and range are all explicit sets.

Verification boundary

A finite explicit table can be checked completely, but this does not replace formal proofs about general functions or infinite sets.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Bind Set.range, image sets, and the domain/codomain roles of function types.

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.