Surjectivity and codomain coverage

Mathematical statement

A total function f:A→B is surjective exactly when, for every b∈B, there exists x∈A with f(x)=b; equivalently, the attained range f[A] equals the declared codomain B. Surjectivity does not require a unique preimage.

Representative formula (exact TeX)

f:A\to B

Assumptions

• The domain, codomain, and total function are explicit.
• A replacement codomain must contain every existing output for the same table to remain a valid total function.

Symbol scope

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

Intuition, definition, and misconceptions

For a finite table, the uncovered targets form B∖f[A]. Missing any target defeats surjectivity, while several inputs reaching one target do not.

For a specified t∈B, enumerate its preimages. The first preimage here means the numerically least input in the finite integer data; report none if the fiber is empty.

Changing the codomain from B to B2 may change surjectivity of the same output table. First check f[A]⊆B2, then ask whether f[A]=B2. Do not guess the codomain or discard attained outputs. The empty function ∅→∅ is surjective, while ∅→nonempty B is not.

Common errors include automatically identifying range with codomain, or using multiple preimages as an objection to surjectivity.

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 — Codomain and output eligibility are part of a function declaration.
• Quantifiers and explicit domains — Surjectivity requires a permitted preimage for every codomain element.

Verification boundary

Finite codomain coverage can be recomputed completely; general surjectivity and preimage-selection formal obligations remain open.

not_formalized

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

Next obligation

Bind Function.Surjective, range, and legality of codomain changes.

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.