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Mathematics · unit version 1

Orders, graph representations and certificates

Translate relations into orders, construct faithful representations, count directed walks and certify graph equivalence or difference.

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Readiness and preparation

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Does an implication in one direction prove two descriptions are equivalent?

Faithful representations must preserve and reflect the intended relation.

Compare reasoning and find a refresher

No. Prove both implications. For sets, show both inclusions; for a relation represented by another relation, verify each direction separately.

Review two-direction proofs

How does a matrix product combine intermediate states?

Counting walks needs a sum over each possible intermediate vertex.

Compare reasoning and find a refresher

Entry (i,j) of AB is the sum over t of A(i,t) times B(t,j). For adjacency counts, row i is departure and column j is arrival; the convention must stay fixed.

Review matrix multiplication

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Representations that preserve exactly the right information

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Lesson sequence

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  1. Distinguish strict and non-strict partial orders

    Lesson details and objectives

    Check order axioms, separate extreme-element notions and identify where finiteness matters.

    • Translate between strict and non-strict partial orders and prove the axioms in both directions.
    • Identify minimal, least, maximal and greatest elements from a specified relation.
    • Prove the finite unique-minimal theorem and diagnose its failure for an infinite domain.

    8 practice tasks.

  2. Represent orders with lower sets and cover diagrams

    Lesson details and objectives

    Prove faithful order representations, build product orders and locate the finite boundary of cover reconstruction.

    • Prove the principal lower-set map preserves and reflects order and is injective.
    • Verify componentwise product orders and distinguish them from strict-in-every-coordinate relations.
    • Reconstruct a finite order from its covers and explain why dense infinite orders break that mechanism.

    6 practice tasks.

  3. Count directed walks with matrix powers

    Lesson details and objectives

    Prove what each matrix entry counts, distinguish fixed-length walks from reachability, and justify a finite reachability test.

    • Construct directed adjacency matrices with an explicit source-row convention and expand a product through intermediate vertices.
    • Prove the walk-counting theorem for every nonnegative length, including the identity-matrix base case.
    • Justify reachability using powers through n−1 and distinguish walks, simple paths and probabilities.

    6 practice tasks.

  4. Verify graph isomorphisms and refute incomplete invariants

    Lesson details and objectives

    Supply a vertex bijection with an edge/nonedge certificate, or prove that an invariant separates two graphs.

    • Verify an explicit bijection preserves adjacency in both directions for finite simple undirected graphs.
    • Prove useful isomorphism invariants and distinguish necessary conditions from sufficient certificates.
    • Translate vertex relabeling into simultaneous adjacency-matrix row and column permutation.

    6 practice tasks.