Discrete Mathematics Roadmap

The core theoretical foundation for Computer Science and its applications in information technology.

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Phase Main Topic Content & Learning Activities Objectives & Deliverables
1. Overview General Introduction
  • Role in Computer Science.
  • Practical applications.
  • Understand the importance and scope of Discrete Mathematics.
2. Reasoning Mathematical Logic and Propositions
  • Propositions and logical operators.
  • Laws of logic, truth tables.
  • Theorems, proofs (by contradiction, by induction).
  • Master the foundation of all reasoning and proofs.
3. Relations Sets and Relations
  • Set concepts, operations.
  • Relations, properties of relations.
  • Equivalence relations, order relations.
  • Learn to group objects and define relationships.
4. Mappings Functions and Mappings
  • Definition, domain, range.
  • Injective, surjective, bijective functions.
  • Composite functions, inverse functions.
  • Study the rules of correspondence between sets.
5. Computer Logic Boolean Algebra
  • Structure of Boolean algebra.
  • Representing and simplifying logical expressions.
  • Application in digital circuit design.
  • Explore the mathematical system of logic and its application in computers.
6. Arithmetic Discrete Arithmetic
  • Divisibility, prime numbers, GCD.
  • Euclidean algorithm.
  • Remainders, congruence, and applications (RSA cryptography).
  • Study the properties of integers and their application in cryptography.
7. Counting Combinatorics and Discrete Probability
  • Counting rules: sum, product.
  • Permutations, arrangements, combinations.
  • Discrete probability.
  • Learn counting techniques and analyze the likelihood of events.
8. Recursion Recurrence Relations and Generating Functions
  • Definition of recurrence relations.
  • Methods for solving linear recurrence relations.
  • Generating functions.
  • Model problems with self-repeating properties.
9. Networks Graph Theory
  • Graph concepts, paths, cycles.
  • Trees, spanning trees, binary trees.
  • Eulerian and Hamiltonian paths.
  • Foundation for modeling networks and relationships.
10. Theory Relational Algebra and Formal Languages
  • Formal languages, grammars.
  • Regular expressions.
  • Applications: finite automata, compilers.
  • Theoretical basis for databases and compilers.
11. Practice Applications in IT
  • Databases (sets, relations).
  • Cryptography (discrete arithmetic).
  • Search and AI (logic, graphs).
  • Summarize and connect learned knowledge with practical fields.

Core Mindsets for Discrete Mathematics

1. Logical Rigor & Precision

Every statement must be precise and every claim must be proven. Intuition is a guide, but a formal proof is the destination. There are no "almosts" in logic.

2. Think in Structures & Abstractions

Learn to see problems not as specific scenarios but as instances of abstract structures like sets, graphs, or relations. This allows you to apply powerful, general tools.

3. Deconstruct Problems

Complex challenges can often be broken down into smaller, discrete, and more manageable parts. Solve the parts, then combine the solutions methodically.

4. See the World Discretely

Train your mind to view the world in terms of distinct units, steps, and states. It's about counting, sequencing, and relating separate items, which is the foundation of computation.