Quotient set

Mathematical statement

X/R is the set of all distinct equivalence classes, not a set of selected representatives. The classes cover X and are pairwise disjoint. The canonical projection sends each element to its class. Any quotient operation still needs a separate proof of independence from representative choice.

Representative formula (exact TeX)

Q(X)

Assumptions

• The finite carrier and certified equivalence relation are complete.
• Classes are normalized as sorted element arrays and deduplicated.

Symbol scope

• Q — quotient set
• R — equivalence relation

Intuition, definition, and misconceptions

List all distinct full classes.

The canonical projection value is a class, not a representative.

Coverage and pairwise disjointness give a partition; quotient operations separately require well-definedness.

A list of representatives is not the quotient set, and operations cannot be inherited without checking representative independence.

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

• Equivalence class — Quotient elements are distinct complete classes.
• Set — The quotient itself is a set.
• Function mapping — The canonical projection is a total map.
• Equivalence relation — All classes come from the same relation.

Verification boundary

Only finite quotient sets and their projections are constructed; no group or ring operations are automatically assigned.

not_formalized

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

Next obligation

Add quotient rings of finite rings with representative-independent operations.

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.