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Mathematical Foundations·Intermediate·5 lessons·~280 min
The mathematical scaffolding that makes formal reasoning possible
What you'll learn
Lessons
Introduces sets as collections defined by their members, explains the axiom of extensionality, distinguishes membership from subset, discusses the empty set, and shows why naive unrestricted comprehension collapses into Russell's paradox.
Introduces the standard set operations — union, intersection, complement, difference, symmetric difference, power set, and Cartesian product — and teaches students to verify results by element-chasing and visualize simple cases using Venn diagrams.
Defines a relation as a subset of a Cartesian product, introduces reflexivity, symmetry, transitivity, and antisymmetry, explains equivalence relations and the partitions they induce, and discusses order relations.
Defines a function as a well-defined relation, introduces injective, surjective, and bijective functions, develops cardinality for finite and infinite sets, states the pigeonhole principle, and presents Cantor's diagonal argument in intuitive form.
Integrates the unit by showing how set theory provides the semantic foundation for categorical and predicate logic. Students translate categorical and quantified arguments into set-theoretic form and verify them structurally using union, intersection, subset, and function reasoning.
How to study
Each lesson opens with a guided walkthrough — read it before the activity.
Look at why each step follows, not just what the answer is.
Know which rule applies and what would make the response weak before you start.
Optional context for the unit. Each lesson surfaces the concepts and rules it uses — these are here when you want the bigger picture.
A well-defined collection of distinct objects, considered as a single mathematical object; the objects are called elements or members of the set.
An object belonging to a set; if x is an element of the set A, we write x ∈ A, and if it is not, we write x ∉ A.
A set A is a subset of a set B, written A ⊆ B, if every element of A is also an element of B; A is a proper subset if in addition A ≠ B.
The power set of a set A, written P(A) or 2^A, is the set whose elements are all the subsets of A, including the empty set and A itself.
The Cartesian product A × B of two sets is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B.
A binary relation from A to B is a subset R ⊆ A × B; elements (a, b) ∈ R are usually written a R b.
A function f: A → B is a relation from A to B such that for every a ∈ A there is exactly one b ∈ B with (a, b) ∈ f; we write f(a) = b for this unique value.
A relation R on A that is reflexive, symmetric, and transitive; each equivalence relation partitions A into disjoint equivalence classes.
A measure of the size of a set; two sets have the same cardinality when there exists a bijection between them, and a set is countable when it has the same cardinality as the natural numbers.