Unit 846: Discrete Mathematics Fundamentals

1. Introduction to Discrete Mathematics

• Overview of discrete mathematics
• Importance and applications in computer science and engineering
• Basic concepts: sets, functions, logic, proofs

2. Set Theory

• Definition and notation of sets
• Types of sets: finite, infinite, empty, universal
• Set operations: union, intersection, difference, complement
• Subsets and power sets
• Venn diagrams for visualization
• Cardinality and countability

3. Propositional Logic

• Propositions and truth values
• Logical connectives: AND, OR, NOT, IMPLIES, BICONDITIONAL
• Truth tables and their construction
• Logical equivalences and laws (De Morgan’s, distributive, associative, commutative)
• Conditional statements and implications
• Quantifiers introduction (brief overview)

4. Predicate Logic

• Predicates and quantifiers: universal (∀) and existential (∃)
• Translating statements into predicate logic
• Negations of quantified statements
• Logical equivalences involving quantifiers
• Applications in formal reasoning and proofs

5. Combinatorics

• Basic counting principles
• Permutations: definition and formulas
• Combinations: definition and formulas
• The principle of inclusion-exclusion
• The pigeonhole principle
• Applications in problem solving

6. Graph Theory

• Basic terminology: vertices, edges, degree
• Types of graphs: undirected, directed, weighted, bipartite
• Paths and cycles
• Connectivity and components
• Special graphs: complete, trees, planar
• Graph coloring and its importance

7. Number Theory

• Divisibility and division algorithm
• Prime numbers and fundamental properties
• Greatest common divisor (GCD) and Euclidean algorithm
• Modular arithmetic and congruences
• The Fundamental Theorem of Arithmetic

8. Recurrence Relations

• Introduction to recurrence relations
• Solving linear recurrence relations with constant coefficients
• Non-linear recurrence relations overview
• Generating functions as a solution tool
• Applications in counting and algorithm analysis

9. Trees and Binary Trees

• Definition and properties of trees
• Binary trees: structure and terminology
• Binary search trees (BSTs) and their applications
• Tree traversal algorithms: preorder, inorder, postorder
• Applications of trees in computer science

10. Cryptography

• Introduction to cryptography and importance
• Basic encryption techniques: symmetric and asymmetric
• Cryptographic algorithms overview
• RSA encryption: principles and mathematics
• Applications of discrete mathematics in data security