An Introduction to Basic Set Theory Course Syllabus

Overview: This course provides a structured introduction to basic set theory, designed to build a strong foundation in discrete mathematics essential for computer science and related fields. Over eight modules, learners progress from fundamental concepts like sets and subsets to more complex ideas including relations, functions, and cardinality. Each module includes hands-on exercises to reinforce understanding and develop proof-writing skills. With approximately one week dedicated to each module, the course balances theoretical rigor with practical application, making abstract concepts tangible through Venn diagrams, combinatorial reasoning, and real-world problem solving. Estimated total time commitment: 40–50 hours.

Module 1: Fundamentals of Sets

Estimated time: 6 hours

• Definitions and basic notation of sets
• Roster form vs. set-builder notation
• Subsets and proper subsets
• Power sets and their construction

Module 2: Set Operations & Laws

Estimated time: 6 hours

• Union, intersection, and difference of sets
• Complement of a set
• Associative, commutative, and distributive laws
• Proof of De Morgan’s laws using Venn diagrams

Module 3: Cartesian Products & Tuples

Estimated time: 5 hours

• Ordered pairs and n-tuples
• Definition of Cartesian product
• Enumerating elements in product sets
• Cardinality of Cartesian products

Module 4: Relations on Sets

Estimated time: 6 hours

• Definition of relations and their domains and ranges
• Properties of relations: reflexive, symmetric, transitive
• Equivalence relations
• Partitions induced by equivalence relations

Module 5: Functions Between Sets

Estimated time: 6 hours

• Definition of functions and mappings
• Injections, surjections, and bijections
• Function composition
• Inverse functions and their existence conditions

Module 6: Introduction to Proofs

Estimated time: 7 hours

• Direct proof techniques in set theory
• Proof by contradiction
• Mathematical induction applied to set identities
• Proving basic cardinality formulas using induction

Module 7: Cardinality & Infinite Sets

Estimated time: 7 hours

• Finite vs. infinite sets
• Countable and uncountable sets
• Cantor’s theorem on power sets
• Comparing sizes of infinite sets

Module 8: Applications & Advanced Patterns

Estimated time: 7 hours

• Inclusion–exclusion principle
• Venn diagram problem solving for three sets
• Combinatorial reasoning with sets
• Applications in counting and logic

Prerequisites

• Familiarity with basic mathematical notation
• High school algebra
• Logical reasoning skills

What You'll Be Able to Do After

• Define and manipulate sets using formal notation
• Apply set operations and prove identities using laws
• Analyze relations and classify them as equivalence relations
• Determine properties of functions and construct inverses
• Use proof techniques to establish results in set theory