Special Sections of Mathematics 1: Discrete Mathematics (Telenyk & Dorohyi, 2012)
A complete source map of a 72-page lecture course spanning sets, relations, functions, orders, elementary counting, relational data modelling, and trees. It is valuable as a curricular map but contains enough mathematical and database errors that every retained mechanism is independently re-derived and verified.
This 2012 lecture volume, compiled by Serhii F. Telenyk and Yaroslav Yu.
Dorohyi, is the admissible conceptual spine of the supplied SUM - 1 DM
course folder. It is not the whole course: automata and formal-language topics
appear in separate departmental homework material. It is also not a safe
authority by itself. The source is uncited, compact enough to skip hypotheses,
and contains several visibly typeset but mathematically false statements.
The source was read and visually inspected on every physical page. Its 72
native-text pages, 72 renders, and 72 coverage records reconcile exactly; the
original SHA-256 remained
a6d288b156e5a2ba790ab65808c702fc5507755b14f8046c6a531d7916dcc111.
What the volume covers #
| Physical pages | Durable subject | Evidentiary weight |
|---|---|---|
| 1–4 | Course identity; discrete models and representations | Curricular orientation |
| 5–13 | Route/graph modelling and a purported shortest-route procedure | Topic signal only; algorithm is not generally correct |
| 14–25 | Sets, membership, subsets, Boolean set operations, products | Useful explanatory structure after direct proof checking |
| 26–33 | Binary relations, relation matrices, functions, inverse mappings | Useful definitions after typing and convention checks |
| 34–49 | Relation properties, equivalence, partitions, orders, quotient structures | Strong topic map; several definitions/examples require correction |
| 50–55 | Permutations, arrangements, combinations | Topic map only where formulas are corrupted; re-derive independently |
| 56–60 | Relation powers and closure-like operations | Topic map; printed example is inconsistent |
| 61–71 | Relations as a database model, operations, keys, and normalization | Historical bridge only; replace definitions with primary database sources |
| 71–72 | Trees and quotient/order example | Tree topic signal; final example is not valid as printed |
The durable integration deliberately does not mirror this order. Logic and sets come first; relations then split into equivalence/order, graph, and relational-database branches; counting gets its own derivation-led chapter; traversal and shortest paths follow a separately verified graph model.
What remains useful #
The source repeatedly moves between four representations of one structure: set notation, relation/pair notation, matrices or tables, and directed diagrams. That representational fluency is worth preserving. A student should be able to tell whether two displays encode the same relation and which operation becomes easy in each form.
Its best curricular distinctions are also durable:
- a Cartesian product is a universe of typed pairs; a relation selects some of those pairs;
- a function is a relation with totality and uniqueness conditions;
- an equivalence relation groups elements into classes, whereas a partial order arranges comparable elements without collapsing them;
- logical relations and relational operations belong above the physical storage layout of a database;
- graph and tree representations are special structures, not merely pictures.
GTK keeps those distinctions but supplies formal hypotheses, counterexamples, proofs, and current implementation boundaries from independent sources.
Where the source fails #
| Location | Source problem | Corrective treatment |
|---|---|---|
| pp. 10–12 | The shortest-route procedure is not a correct general shortest-path algorithm | Replace with a relaxation-based taxonomy: BFS, DAG relaxation, Dijkstra, Bellman–Ford, Floyd–Warshall, and Johnson, each with explicit preconditions |
| p. 25 | Several set identities, including a De Morgan/difference expression, are malformed | Prove identities element by element and test boundary cases such as empty and universal sets |
| p. 34 | The transitivity condition contains an extra restriction not present in the standard definition | Use the quantified definition and distinguish transitivity from connectedness/totality |
| p. 39 | A rational/decimal order example reverses an inequality | Replace with a newly computed finite counterexample and verified order diagram |
| pp. 50–52 | Permutation/counting notation and formulas are corrupted or inconsistent | Derive formulas from the product rule, with explicit replacement/repetition assumptions |
| pp. 58–60 | A closure example is internally inconsistent | Compute reflexive, symmetric, and transitive closures independently and verify minimality |
| pp. 61–71 | Database material conflates logical relations with records/storage and gives stale or incorrect normalization language | Ground relational algebra in Codd and dependencies/normal forms in Armstrong, Bernstein, and Fagin; use current SQL documentation only for implementation semantics |
| pp. 71–72 | The final quotient/tree example is invalid as printed | Omit it; use independently designed quotient-poset and rooted-tree examples |
Overlapping or poorly spaced source text is visible on physical pp. 4 and 16. These defects occur in the PDF itself, not only in text extraction.
Visual audit #
All 72 pages were rendered and inspected at readable scale. Forty-two information-bearing visual groups were recorded, including region diagrams, relation matrices, mapping diagrams, order/Hasse-style drawings, trees, database tables, and long equation layouts.
No visual is copied into GTK. Several are attached to incorrect surrounding claims, and others have label overlap or source-specific notation. The retained visual requirements are instead implemented as independently checked, theme-aware diagrams:
- one set/product/relation/function transformation chain;
- one equivalence-class versus partial-order comparison;
- one representation comparison for a fixed graph;
- traversal and shortest-path proof diagrams with explicit direction/weights;
- relational-algebra and normalization dependency diagrams.
This preserves the explanatory role of the source's visual language without transferring its errors or layout limitations.
Source claims audit #
| Load-bearing source claim | Status | GTK treatment |
|---|---|---|
| Sets, products, relations, and functions form a dependency chain | current after corrected typing | Retained and re-derived in the sets/functions/relations chapter |
| Equivalence relations and partitions are two views of one quotienting mechanism | current after proof | Retained with both directions proved and quotient well-definedness conditions |
| Partial orders can be represented by reduced order diagrams | current with convention qualification | Retained with covers, least/minimal distinctions, and explicit orientation |
| Counting formulas can be selected by labels alone | error-prone | Replaced by a decision tree derived from ordered/unordered and replacement assumptions |
| The printed shortest-route method solves the general problem | error in source | Not cited or reproduced; replaced by independently verified algorithms |
| A relation is essentially a stored database table/record layout | outdated and conceptually wrong | Replaced by the logical relation/schema/constraint/algebra model and a separate SQL/physical boundary |
| Printed normalization descriptions are reliable design rules | needs correction | Replaced by formal FDs, closure, keys, lossless join, dependency preservation, 3NF/BCNF/4NF |
| Trees are a useful discrete representation | current but underspecified | Retained only through independently defined graph/tree structures |