An Introduction to Basic Set Theory

Module 1: Fundamentals of Sets

⏳ 1 week

• Topics: Definitions, notation, roster vs. set‐builder forms

• Hands-on: Create and classify sets; construct power sets for small finite examples

Module 2: Set Operations & Laws

⏳ 1 week

• Topics: Union, intersection, difference, complement; associative, commutative, and distributive laws

• Hands-on: Prove De Morgan’s laws and simplify set expressions with Venn diagrams

Module 3: Cartesian Products & Tuples

⏳ 1 week

• Topics: Ordered pairs, n-tuples, product sets, and their sizes

• Hands-on: Enumerate Cartesian products for given finite sets and count elements

Module 4: Relations on Sets

⏳ 1 week

• Topics: Definitions of relations, domains, ranges, properties (reflexive, symmetric, transitive)

• Hands-on: Determine whether sample relations are equivalence relations and partition sets accordingly

Module 5: Functions Between Sets

⏳ 1 week

• Topics: Functions, injections, surjections, bijections, inverse functions, composition

• Hands-on: Classify given mappings as injective/surjective/bijective and construct inverses

Module 6: Introduction to Proofs

⏳ 1 week

• Topics: Direct proof, proof by contradiction, proof by induction in the context of sets

• Hands-on: Prove basic set identities and use induction to establish formulas for cardinalities

Module 7: Cardinality & Infinite Sets

⏳ 1 week

• Topics: Finite vs. infinite sets, countability, Cantor’s theorem on power sets

• Hands-on: Show that the power set of ℕ has strictly greater cardinality than ℕ itself

Module 8: Applications & Advanced Patterns

⏳ 1 week

• Topics: Inclusion–exclusion principle, Venn‐diagram problem solving, beginnings of combinatorial set theory

• Hands-on: Solve counting problems using inclusion–exclusion and construct Venn diagrams for three sets

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