Inverse functions of bijections

Mathematical statement

If f:A→B is bijective, its inverse g:B→A assigns each b its unique preimage and satisfies g(f(x))=x for x∈A and f(g(y))=y for y∈B. A finite bijection table can be inverted by exchanging ordered pairs. For nonbijective f, the exchanged table is not automatically an inverse on all of B.

Representative formula (exact TeX)

g(f(x))=x

Assumptions

• Inverse-table and evaluation exercises first establish that f is a bijection on the declared A→B.
• A candidate g is first validated as a total function B→A, then both identities are checked. The representative formula shows only one side.
• Affine examples use f:R→R with m≠0; inverse coefficients are given as reduced rational numbers with positive denominators.

Symbol scope

• g — inverse or candidate total function
• f — declared bijection
• x — permitted input in A

Intuition, definition, and misconceptions

Every target occurs exactly once in a finite bijection table, so exchanging pairs produces a total function B→A. The inverse of the empty bijection ∅→∅ is the empty function.

An inverse satisfies both identities. Candidate checks in this pulse verify each side and declare both types completely, without identifying a general one-sided inverse with a two-sided inverse. An inverse function is distinct from the reciprocal 1/f(x).

For f(x)=mx+b with m≠0, g(y)=(y−b)/m. In the form g(y)=uy+v, u=1/m and v=−b/m. Exercises compute exact rational coefficients from integer parameters; a machine proof of general invertibility is not claimed.

Common errors include calling an exchanged nonbijective table an inverse function, or interpreting f⁻¹ as 1/f.

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

• Bijectivity and unique preimages — An inverse requires a unique preimage for every target.
• Function composition — Composition expresses restoration in both directions.
• Affine functions and degree-one functions — The real affine example has nonzero slope.

Verification boundary

Explicit finite bijections allow verification of both restoration identities; general inverse existence, uniqueness, and type bridges await formalization.

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.LeftInverse, Function.RightInverse, Equiv.symm, and inverses of affine functions with nonzero slope.

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.