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UNIVERSITY MATHEMATICS · ORIGINAL COURSE RESOURCE
Discrete Mathematics Complete Notes, Solved Examples, Proofs, Exercises & PDF Book
Discrete Math · Discrete Maths · Discrete Structures · Discrete Mathematical Structures
One complete 20-chapter university resource for AD/ADP Mathematics, BS Mathematics, BSCS, BSIT, Software Engineering, AI, Data Science, Cyber Security and Computer Engineering.
ABOUT THIS BOOK
Logic, proof, structures, counting and computation
This original The Math Hub course book develops precise logical and proof-writing foundations, then connects sets, relations, functions, induction, algorithms, number theory, combinatorics, probability, graphs, trees, Boolean algebra, automata and advanced discrete methods.
It contains fully solved worked examples and topic-wise exercises. It does not claim every exercise is solved.
What is Discrete Mathematics?
Discrete Mathematics studies distinct, countable objects rather than continuously varying quantities. Discrete Structures is a closely related course label, not automatically an identical syllabus.
DefinitionsTheorems and proofsWorked examplesPractice exercisesComputing connections
Why study this course?
For computing
Connect logic, algorithms, complexity, Boolean circuits, automata, cryptography, graphs and data structures.
For revision
Use chapter progression, worked examples, decision questions and honest exercise-scope notes.
Learning outcomes and skills developed
Students develop formal translation, proof selection, counterexamples, set and relation manipulation, induction, recursion, algorithm analysis, modular computation, combinatorial modelling, probability reasoning, graph algorithms, tree traversal, Boolean simplification and automata reasoning.
Recommended background: basic algebra, mathematical notation and willingness to write careful arguments. Institution-specific prerequisites may differ.
COURSE CONTENTS · EXACTLY 20 ENTRIES
Complete 20-chapter learning journey
Jump to any chapter for detailed coverage.
- Chapter 1: Mathematical Logic & PropositionsBook pages 1–39
- Chapter 2: Predicate Logic, Quantifiers & Rules of InferenceBook pages 40–78
- Chapter 3: Mathematical Reasoning & Methods of ProofBook pages 79–116
- Chapter 4: Set Theory, Cardinality & CountabilityBook pages 117–154
- Chapter 5: Relations, Equivalence Relations, Partial Orders & LatticesBook pages 155–187
- Chapter 6: Functions, Sequences, Summations & Discrete StructuresBook pages 188–219
- Chapter 7: Mathematical Induction, Strong Induction & RecursionBook pages 220–240
- Chapter 8: Algorithms, Correctness & Computational ComplexityBook pages 241–259
- Chapter 9: Number Theory & Modular ArithmeticBook pages 260–272
- Chapter 10: Counting Principles, Permutations & CombinationsBook pages 273–286
- Chapter 11: Advanced CombinatoricsBook pages 287–299
- Chapter 12: Recurrence Relations & Divide-and-ConquerBook pages 300–311
- Chapter 13: Generating FunctionsBook pages 312–322
- Chapter 14: Discrete ProbabilityBook pages 323–335
- Chapter 15: Graph TheoryBook pages 336–355
- Chapter 16: Graph Algorithms, Coloring, Matching & Network OptimizationBook pages 356–371
- Chapter 17: Trees, Traversals & Spanning TreesBook pages 372–387
- Chapter 18: Boolean Algebra, Logic Gates & Circuit MinimizationBook pages 388–407
- Chapter 19: Formal Languages, Automata & Models of ComputationBook pages 408–430
- Chapter 20: Advanced Topics in Discrete MathematicsBook pages 431–494
Detailed chapter-by-chapter coverage
01
BOOK PAGES 1–39
Chapter 1: Mathematical Logic & Propositions
Formal logic is the language of proof, algorithmic reasoning and digital logic.
What is taught
- Statements
- truth values
- connectives
- XOR
- implication
- biconditional
- truth tables
- equivalence
- tautology
- contradiction
- satisfiability
- CNF
- DNF and formal specifications
Learning outcomes
Translate statements, build truth tables and simplify propositions correctly.
Applications and student guidance
Program conditions, digital logic and formal specifications.
How do I build a truth table without missing rows?Converse vs inverse vs contrapositive?CNF vs DNF — when is each useful?
02
BOOK PAGES 40–78
Chapter 2: Predicate Logic, Quantifiers & Rules of Inference
Predicate logic extends propositions to variables and domains used in definitions and databases.
What is taught
- Predicates
- domains
- truth sets
- universal/existential/unique quantifiers
- negation
- nested quantifiers
- inference rules
- instantiation
- generalization and resolution
Learning outcomes
Translate quantified language, negate it accurately and test argument validity.
Applications and student guidance
Theorem formulation, database queries, AI reasoning and formal methods.
How do I negate ∀ and ∃ statements?Why does quantifier order matter?Which rule of inference applies?
03
BOOK PAGES 79–116
Chapter 3: Mathematical Reasoning & Methods of Proof
Proof turns mathematical claims into reliable arguments.
What is taught
- Proof language
- direct proof
- contraposition
- contradiction
- cases
- exhaustion
- existence
- uniqueness
- constructive/nonconstructive proof
- counterexamples and proof critique
Learning outcomes
Choose, construct and critique rigorous proofs and disprove universal claims.
Applications and student guidance
Pure mathematics, algorithms, number theory and graph theory.
Which proof method should I use?Direct proof vs contrapositionWhen is contradiction simplest?
04
BOOK PAGES 117–154
Chapter 4: Set Theory, Cardinality & Countability
Sets are the basic language of relations, functions, probability and graphs.
What is taught
- Set notation
- subsets
- power sets
- operations
- indexed families
- partitions
- Cartesian products
- injections
- surjections
- bijections
- countability and Cantor diagonalization
Learning outcomes
Perform set operations and compare finite, countable and uncountable sizes.
Applications and student guidance
Databases, probability, formal languages and mathematical foundations.
How many elements does a power set have?Countable vs uncountable — what is the test?How does Cantor’s diagonal argument work?
05
BOOK PAGES 155–187
Chapter 5: Relations, Equivalence Relations, Partial Orders & Lattices
Relations formalize connection, equivalence and order.
What is taught
- Ordered pairs
- matrices
- digraphs
- relation properties
- closures
- Warshall’s algorithm
- equivalence classes
- quotient sets
- posets
- Hasse diagrams
- bounds
- meet
- join and lattices
Learning outcomes
Test properties, compute closures, draw Hasse diagrams and identify lattice structure.
Applications and student guidance
Databases, classification, scheduling, dependencies and Boolean structures.
How do I test a relation step by step?How do I draw a Hasse diagram?Minimal vs least — what is the difference?
06
BOOK PAGES 188–219
Chapter 6: Functions, Sequences, Summations & Discrete Structures
Functions and sequences model transformations and ordered discrete data.
What is taught
- Domains
- ranges
- injective/surjective/bijective functions
- composition
- inverses
- images
- preimages
- indicators
- floor/ceiling
- sequences
- strings
- bit strings and finite sums
Learning outcomes
Classify functions, work with sequences and manipulate finite sums and products.
Applications and student guidance
Algorithms, coding, combinatorics and data structures.
Injective vs surjective vs bijective?When does an inverse exist?How do I telescope a sum?
07
BOOK PAGES 220–240
Chapter 7: Mathematical Induction, Strong Induction & Recursion
Induction and recursion reason about integers, structures and recursive algorithms.
What is taught
- Base cases
- inductive hypotheses
- identities
- inequalities
- divisibility
- strong induction
- well-ordering
- recursive definitions
- structural induction
- termination and loop invariants
Learning outcomes
Select base cases, strengthen hypotheses and connect induction to verification.
Applications and student guidance
Recursive algorithms, data structures, number theory and program correctness.
Ordinary vs strong induction?How many base cases are needed?How are loop invariants related to induction?
08
BOOK PAGES 241–259
Chapter 8: Algorithms, Correctness & Computational Complexity
Discrete reasoning becomes computation through specifications, proof and resource analysis.
What is taught
- Specifications
- preconditions
- postconditions
- pseudocode
- search
- sorting
- correctness
- termination
- growth
- O/Ω/Θ
- time/space complexity
- brute force
- greedy
- divide-and-conquer
- P and NP
Learning outcomes
Write pseudocode, prove correctness and compare algorithmic growth.
Applications and student guidance
Software development, data structures, optimization and complexity theory.
Linear vs binary search?Big-O vs Big-Theta vs Big-Omega?Brute force vs greedy vs divide and conquer?
09
BOOK PAGES 260–272
Chapter 9: Number Theory & Modular Arithmetic
Number theory supplies exact tools for modular computation and cryptography.
What is taught
- Divisibility
- Division Algorithm
- gcd
- Euclidean Algorithm
- Bézout
- primes
- factorization
- congruences
- inverses
- CRT
- Fermat
- Euler
- Wilson
- fast exponentiation and RSA
Learning outcomes
Solve modular equations, combine congruences and explain RSA’s mathematical skeleton.
Applications and student guidance
Cryptography, cyber security, hashing and computer arithmetic.
How do I find a modular inverse?When can a linear congruence be solved?How does RSA use modular arithmetic?
10
BOOK PAGES 273–286
Chapter 10: Counting Principles, Permutations & Combinations
Counting measures finite possibilities without listing them.
What is taught
- Sum
- product
- complement and division rules
- factorials
- permutations
- combinations
- coefficients
- Pascal
- binomial/multinomial theorems
- repetition
- multisets
- stars-and-bars and lattice paths
Learning outcomes
Choose rules, handle order/repetition and verify cases without omissions or double-counting.
Applications and student guidance
Probability, passwords, sampling and algorithm analysis.
Permutation or combination?Does order matter?When should I use stars-and-bars?
11
BOOK PAGES 287–299
Chapter 11: Advanced Combinatorics
Structural methods extend basic counting to restrictions and existence arguments.
What is taught
- Pigeonhole
- inclusion-exclusion
- surjections
- occupancy
- derangements
- restricted permutations
- double counting
- Stirling/Bell numbers
- monotone subsequences and rook numbers
Learning outcomes
Count surjections, construct double-counting proofs and model forbidden positions.
Applications and student guidance
Scheduling, allocation, coding, probability and enumerative combinatorics.
Pigeonhole or counting formula?When should I use inclusion-exclusion?When does double counting help?
12
BOOK PAGES 300–311
Chapter 12: Recurrence Relations & Divide-and-Conquer
Recurrences model earlier-value dependence, counting and recursive runtime.
What is taught
- Linear and higher-order recurrences
- characteristic roots
- repeated/complex roots
- nonhomogeneous methods
- iteration
- Fibonacci
- runtime recurrences
- recursion trees and Master Theorem
Learning outcomes
Classify and solve common recurrences and select a valid analysis method.
Applications and student guidance
Algorithms, population/financial models, strings and tilings.
Characteristic roots vs iteration vs Master Theorem?How are repeated roots handled?When does Master Theorem apply?
13
BOOK PAGES 312–322
Chapter 13: Generating Functions
Generating functions bridge sequences, algebra, recurrences and counting.
What is taught
- OGF basics
- coefficient correspondence
- shifts
- delays
- truncation
- differentiation
- integration
- convolution
- rational forms
- partial fractions
- recurrence solving
- partitions and EGF
Learning outcomes
Build OGFs, interpret convolution and solve selected recurrence/counting models.
Applications and student guidance
Enumerative combinatorics, coin change, partitions and labelled structures.
When should I use generating functions?OGF vs EGF?How does convolution arise?
14
BOOK PAGES 323–335
Chapter 14: Discrete Probability
Discrete probability attaches uncertainty to finite or countable models.
What is taught
- Experiments
- outcomes
- sample spaces
- events
- axioms
- conditional probability
- independence
- total probability
- Bayes
- PMFs
- CDF
- expectation
- variance and distributions
Learning outcomes
Build sample spaces and use conditional probability, expectation and standard distributions.
Applications and student guidance
Randomized algorithms, data science, reliability and networks.
Conditional probability vs independence?Pairwise vs mutual independence?PMF vs CDF?
15
BOOK PAGES 336–355
Chapter 15: Graph Theory
Graphs model networks, relationships and combinatorial structure.
What is taught
- Graph types
- vertices
- edges
- degree
- families
- subgraphs
- representations
- isomorphism
- walks
- trails
- paths
- cycles
- connectivity
- Euler/Hamilton structures and planarity
Learning outcomes
Represent graphs, analyze connectivity and apply Euler/Hamilton and planarity results.
Applications and student guidance
Networks, transportation, scheduling, social systems and circuits.
Walk vs trail vs path vs cycle?Euler vs Hamilton?How can I show a graph is nonplanar?
16
BOOK PAGES 356–371
Chapter 16: Graph Algorithms, Coloring, Matching & Network Optimization
Graph models become algorithms for routing and optimization.
What is taught
- BFS
- DFS
- DAGs
- topological sorting
- shortest paths
- Dijkstra
- Bellman-Ford
- Floyd-Warshall
- coloring
- matchings
- Hall/König
- flow networks and Ford-Fulkerson
Learning outcomes
Run traversals, select shortest paths and reason about matching and flow.
Applications and student guidance
Routing, scheduling, assignment, transportation and resource allocation.
BFS vs DFS?Dijkstra vs Bellman-Ford vs Floyd-Warshall?Matching vs max-flow?
17
BOOK PAGES 372–387
Chapter 17: Trees, Traversals & Spanning Trees
Trees model hierarchy and cycle-free structure.
What is taught
- Trees
- forests
- rooted/m-ary/binary trees
- traversals
- expression/decision trees
- prefix/Huffman codes
- spanning trees
- MSTs
- Prim
- Kruskal and Cayley
Learning outcomes
Analyze trees, perform traversals, build codes and find minimum spanning trees.
Applications and student guidance
File systems, search, compilers, compression and network design.
Preorder vs inorder vs postorder?Full vs complete vs balanced?Prim vs Kruskal?
18
BOOK PAGES 388–407
Chapter 18: Boolean Algebra, Logic Gates & Circuit Minimization
Boolean algebra connects logic with digital hardware.
What is taught
- Boolean laws
- duality
- functions
- truth tables
- minterms
- maxterms
- SOP/POS
- gates
- NAND/NOR
- adders
- K-maps and Quine-McCluskey
Learning outcomes
Move between tables, expressions and circuits and simplify them systematically.
Applications and student guidance
Digital circuits, architecture, control logic and hardware design.
SOP vs POS?Why are NAND and NOR universal?K-map vs Quine-McCluskey?
19
BOOK PAGES 408–430
Chapter 19: Formal Languages, Automata & Models of Computation
Automata model syntax, pattern recognition and computation.
What is taught
- Alphabets
- strings
- Kleene star
- languages
- regex
- grammars
- parse trees
- DFA
- NFA
- conversions
- PDA
- Turing machines
- decidability and halting
Learning outcomes
Manipulate languages, convert automata and distinguish decidable from recognizable.
Applications and student guidance
Compilers, pattern matching, protocols and formal verification.
DFA vs NFA?How do I convert NFA to DFA?Decidable vs recognizable?
20
BOOK PAGES 431–494
Chapter 20: Advanced Topics in Discrete Mathematics
The final chapter opens the foundations into modern discrete mathematics and theoretical computing.
What is taught
- Algebraic structures
- groups
- rings
- fields
- coding theory
- Hamming distance
- Burnside
- Pólya
- Ramsey
- probabilistic methods
- random walks
- Markov chains
- reductions
- SAT and computability
Learning outcomes
Recognize advanced structures, symmetry, coding, reductions and computational limits.
Applications and student guidance
Coding, cryptography, network science, randomized methods and complexity theory.
P vs NP vs NP-hard vs NP-complete?What does a polynomial reduction prove?How does diagonalization reveal limits?
How to use this book
Read definitions → study a theorem or method → reproduce a worked example → attempt the nearby exercise without looking → revisit strategy notes → use the chapter summary for revision. Computing-oriented students should connect each structure to algorithms, data structures, logic or automata where relevant.
Pakistan University / Course Crosswalk
Curriculum and discovery context only—not endorsement or official university notes. Course titles, codes, semester placement and topic boundaries vary.
University of the Punjab
MATH-104 Discrete Mathematics; Graph Theory remains separately identified
Open supplied course context ↗University of Sargodha
MATH-5111 Discrete Mathematics, 3(3+0)
Open supplied course context ↗UCP
MT2343 Discrete Mathematics in Data Science, Computer Engineering and post-ADS context
Open supplied course context ↗UMT
MA-105 Discrete Mathematics in BS/Data Science and bridging context
Open supplied course context ↗Virtual University
MTH202 Discrete Mathematics; prerequisite N/A
Open supplied course context ↗LUMS
CS210 Discrete Mathematics
Open supplied course context ↗FAST-NUCES
Discrete Structures in BSCS, Software Engineering and AI
Open supplied course context ↗COMSATS
CSC102 Discrete Structures
Open supplied course context ↗HITEC / GIKI / UET Taxila
Discrete Structures/Mathematics course context
Open supplied course context ↗Broader institutional discovery context
Related mathematics or computing context may also be found at Quaid-i-Azam University, NUST, UET Lahore, GCU Lahore, BZU, IUB, University of Gujrat, University of Peshawar, BUITEMS, University of Karachi, NED, LUMS, FAST-NUCES, GIKI, HITEC, UOL, FCC, Superior, Riphah, Bahria and Air University. This is not a ranking or a claim that every institution teaches the same course.
Recommended Reference Books
Rosen — Discrete Mathematics and Its Applications ↗Epp — Discrete Mathematics with Applications ↗Grimaldi — Discrete and Combinatorial Mathematics ↗MIT Mathematics for Computer Science ↗Oscar Levin — Open Introduction ↗Hammack — Book of Proof ↗Applied Discrete Structures ↗Wilf — generatingfunctionology ↗
Free / legal further resources
MIT OCW, Oscar Levin, Applied Discrete Structures, Book of Proof, Applied Combinatorics and generatingfunctionology are included through the supplied lawful links above or in the locked dossier.
Frequently asked questions
What is Discrete Mathematics?
It studies distinct and countable structures such as logic, sets, graphs, algorithms, recurrences, probability and formal languages.
Is it the same as Discrete Structures?
The names overlap in many programmes, but course boundaries and codes vary; compare your current departmental scheme.
Why do computing students study it?
Logic, proofs, counting, graphs, algorithms, Boolean algebra and automata provide foundations for computing.
Do these notes include proofs and worked examples?
Yes. Fully solved worked examples are included, while some later exercises remain unsolved practice.
How many chapters are in this book?
There are exactly 20 numbered chapters, with content pages 1–494 and consolidated references at page 495.
Is this an official university textbook?
No. It is an original The Math Hub resource; university references are curriculum/discovery context only.
Can AD/ADP and BS students use it?
Yes, as broad undergraduate support, while matching the sequence with their own programme.
Does it require calculus or programming?
Basic algebra and careful reasoning are the main background. Programming helps with applications but is not required for every topic.
Where can I open the PDF?
Use View / Download PDF for the owner-hosted PDF or Backup Access for the supplied backup copy.
How should I study it?
Read definitions, reproduce a worked example, attempt the nearby exercise, review the strategy and revise from the chapter summary.
Does every Pakistani university use the same syllabus?
No. Titles, codes, placement and topic boundaries vary by university and programme.
Can MSc students use it?
It may support foundation or revision where relevant; it is not a universal MSc core resource.
Who prepared this resource?
Rana Ali Hasan and Mehreen Kanwal, both MPhil Mathematics, prepared it for The Math Hub.
Open the Complete Discrete Mathematics Book
Use the canonical HTML learning page first, then choose the supplied book access route.