Discrete Mathematics (DM)
Zero-fluff mathematical foundations, rigorous algebraic proofs, set cardinality mechanics, binary relation closures, poset Hasse diagrams, lattice structures, formal logic inference trees, and graph algorithms for computer science undergraduates, systems engineers, and competitive examinations.
Set Theory & Cardinality
Well-defined set foundations, roster vs set-builder notation, the nested number system hierarchy (N ⊂ W ⊂ Z ⊂ Q ⊂ R ⊂ C), countable vs uncountable infinities, subset inclusion vs element membership, iterated power sets, algebraic set laws, and finite/infinite union & intersection cardinality theorems.
Binary Relations, Closures & Equivalence Classes
Cartesian products, binary relations over sets, matrix and directed graph representations, reflexive, symmetric, transitive, and antisymmetric properties, reflexive/symmetric/transitive closures (Warshall's algorithm), and equivalence relations with quotient partitions.
Functions, Bijections & the Pigeonhole Principle
Function mappings, domain, codomain, and range, injective (one-to-one), surjective (onto), and bijective classifications, inverse functions and composition algebra, generalized Pigeonhole Principle proofs, and Dirichlet's box distribution bounds.
Partially Ordered Sets (Posets) & Lattice Theory
Partial order relations, strict vs non-strict partial orders, Hasse diagram construction algorithms, maximal/minimal vs greatest/least elements, upper and lower bounds (LUB / GLB), lattice definitions, bounded, distributive, complemented, and Boolean lattices.
Group Theory & Algebraic Structures
Binary operations and closure, algebraic structures (Groupoids, Semigroups, Monoids, Groups, Abelian Groups), permutation groups and Cayley tables, cyclic groups and generators, subgroups and cosets, Lagrange's Theorem, rings, integral domains, and fields.
Propositional Logic & Inference Rules
Propositions and truth values, logical connectives (negation, conjunction, disjunction, implication, biconditional), truth tables, tautologies, contradictions, and contingencies, logical equivalences, Converse, Inverse, and Contrapositive, Conjunctive & Disjunctive Normal Forms (CNF/DNF), and formal rules of inference (Modus Ponens, Modus Tollens, Resolution).
First-Order Predicate Logic & Quantifiers
Predicates, universal (∀) and existential (∃) quantifiers, domain of discourse, quantifier negation laws, nested quantifier order-dependence, translating English statements into first-order logic, and formal validity analysis.
Combinatorics, Recurrence & Generating Functions
Counting principles (Rule of Sum and Rule of Product), permutations and combinations, combinations with repetition, linear homogeneous and non-homogeneous recurrence relations with constant coefficients, and ordinary generating functions.
Graph Theory Foundations, Paths & Connectivity
Graph definitions (undirected, directed, simple, multigraph), vertex degrees and Handshaking Lemma, graph representations (adjacency matrix, incidence matrix, adjacency list), subgraphs and isomorphisms, walks, paths, and cycles, connected components, cuts, and bridges.
Advanced Graph Theory: Trees, Planarity & Coloring
Eulerian paths and circuits (Euler's Theorem), Hamiltonian paths and cycles (Dirac's and Ore's conditions), tree characterizations and spanning trees, planar graphs, regions, and Euler's formula (V - E + R = 2), Kuratowski's Theorem, vertex coloring, and Welsh-Powell algorithm.