Bijectivity and unique preimages

Mathematical statement

A total function f:A→B is bijective exactly when it is both injective and surjective, or equivalently when each fiber of b∈B has exactly one element. For finite sets, some bijection A→B exists exactly when |A|=|B|; equal cardinality does not make every specified f bijective.

Representative formula (exact TeX)

f:A\to B

Assumptions

• Testing a given function and asking whether some bijection exists are distinct questions.
• Cardinality-existence exercises use finite sets with nonnegative integer sizes; size zero on both sides permits the unique empty bijection.

Symbol scope

• f — total function to be checked
• A — domain
• B — codomain

Intuition, definition, and misconceptions

A finite function can be bijective, injective only, surjective only, or neither. Checking set sizes alone omits the actual mapping data.

A complete fiber profile lists all codomain targets in order, including zero-size fibers. All fibers must have size one for a bijection.

Equal finite cardinalities allow pairing to construct some bijection, but a different given function may fail to be bijective. A cardinality exercise asks only about existence and does not adjudicate a supplied function.

Common errors include inferring bijectivity of a given f directly from |A|=|B|, or checking only one of the two properties.

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

• Injectivity and distinct inputs — Bijectivity includes the injectivity condition.
• Surjectivity and codomain coverage — Bijectivity includes the surjectivity condition.

Verification boundary

Finite fibers and cardinalities can be recomputed; general bijection, equinumerosity, and inverse-existence results have no attached formal Evidence.

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.Bijective, existsUnique, and the finite bridge through Fintype.card and Equiv.

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.