Equivalence relation

Mathematical statement

A relation R on X is an equivalence relation when every x satisfies xRx, xRy implies yRx, and xRy together with yRz implies xRz. This slice checks each obligation on a complete finite carrier and complete set of relation pairs.

Representative formula (exact TeX)

E(R)

Assumptions

• The carrier is labeled 0 through n−1 and the relation is fully listed as distinct ordered pairs.
• Equivalence classes are emitted only after all three checks pass; one counterexample suffices to refute its corresponding axiom.

Symbol scope

• E — equivalence-relation record
• R — binary relation

Intuition, definition, and misconceptions

Reflexivity checks every diagonal pair.

Symmetry checks the reverse of every relation pair.

Transitivity checks every composable pair of relation pairs.

A grouped-looking relation diagram, or reflexivity and symmetry alone, does not establish equivalence.

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

• Binary relations and set projections — An equivalence relation is a relation on a fixed carrier.
• Logical connectives — The axioms use implication and conjunction.
• Quantifiers and explicit domains — The three axioms quantify over all carrier elements.

Verification boundary

Equivalence classes are derived only after full qualification.

not_formalized

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

Next obligation

Add equivalence classes, quotient sets and cardinality comparisons.

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.