Discrete Mathematics
“Master mathematical logic, first-order quantifiers, relation closures, POSET lattices, cyclic groups, combinatorics, recurrence relations, planar graphs, and chromatic coloring.”
A rigorous, foundational study notebook covering the mathematical structures that underpin all of computer science and theoretical computation: propositional logic equivalences, first-order predicate quantifiers, relation counting, Hasse diagrams, complemented distributive lattices, group theory and Lagrange's theorem, generating functions, Euler's planar formula, and Hall's matching theorem for B.Tech semester exams and top-rank GATE CS/IT & DA preparation.
Structured Notes & Exam Lens
A · B.Tech Structured Study Sequence
5 Sections · Full CurriculumMathematical Logic & Predicate Calculus
Truth tables, conditional implications, logical equivalences, DNF/CNF canonical normal forms, inference rules, resolution, and first-order predicate logic with nested quantifiers.
Tautologies, DNF/CNF canonical expansions, inference rules, and resolution refutation.
Universal and existential scope, negation duality, nested quantifier order, and English specification translations.
Sets, Relations, POSETs & Lattices
Equivalence relations, relation closures, exact counting of relation types on finite sets, partial orders, Hasse diagrams, and distributive/complemented lattices.
Reflexive, symmetric, transitive closures, counting relation types on finite sets, and partition Bell numbers.
Partial orders, topological sorting, LUB/GLB existence, distributive and complemented lattices, and Boolean algebras.
Algebraic Structures & Group Theory
Semigroups, monoids, groups, abelian groups, subgroups, Lagrange's subgroup coset theorem, and cyclic group generators.
Group axioms, abelian groups, permutation groups, subgroup criteria, and Lagrange's coset theorem.
Cyclic group structure, generator count with Euler's totient phi(n), element order divisibility, and quotient structures.
Combinatorics & Recurrence Relations
Pigeonhole Principle, stars and bars counting, Inclusion-Exclusion, derangements, and solving linear recurrence relations via characteristic roots and generating functions.
Generalized pigeonhole bounds, combinations with repetition, inclusion-exclusion, and derangements formula.
Linear homogeneous recurrence with distinct/repeated roots, non-homogeneous particular solutions, and OGF power series.
Graph Theory, Connectivity, Planarity & Coloring
Handshaking lemma, bipartite graphs, Eulerian circuits, Hamiltonian cycles, Euler's formula V - E + F = 2, Kuratowski's theorem, vertex coloring, and Hall's Marriage Theorem.
Degree sum theorem, bipartite characterizations, vertex/edge connectivity, and Whitney's inequality.
Eulerian circuit criteria, Dirac/Ore Hamiltonian theorems, Euler's formula V - E + F = 2, and Kuratowski minors.
Vertex coloring, Four Color Theorem, chromatic polynomial counting, and Hall's Marriage Theorem.
B · GATE 2027 Syllabus Scope
Official syllabus scope and related lesson coverage; topics outside the syllabus are marked as enrichment.
- •Mathematical logic: Propositional logic, connectives, equivalences, normal forms (DNF, CNF), inference rules, first-order predicate logic, quantifiers
- •Sets, relations, and functions: Operations, equivalence relations, partitions, closures, POSETs, Hasse diagrams, lattices, Boolean algebras
- •Algebraic structures: Semigroups, monoids, groups, abelian groups, subgroups, Lagrange's theorem, cyclic groups, generators
- •Combinatorics: Counting principles, permutations and combinations, stars and bars, Pigeonhole Principle, Inclusion-Exclusion, derangements
- •Recurrence relations: Linear recurrence relations with constant coefficients, characteristic equation roots, generating functions
- •Graph theory: Paths, cycles, Handshaking lemma, bipartite graphs, Eulerian graphs, Hamiltonian graphs, planar graphs (Euler formula, Kuratowski theorem), graph coloring, chromatic numbers, matching
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Authoritative Textbooks & Video Lectures
All notes in this notebook cross-reference standard academic literature. We do not pretend student notes substitute for classical texts — they guide you through them.
Curated lectures from verified educators (Gate Smashers, Neso Academy, Knowledge Gate, and MIT OCW) selected specifically for difficult concept visualizations.
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