Core proof track
Sets & Functions (Proof-Oriented)
Work through these bite-sized lessons. Each subsection has reading, notebook prompts, and an optional AI critique to refine your understanding.
Recommended: move in order, but you can jump around if you’re reviewing specific ideas.
Subsections
Click into a subsection to read, practice, and get feedback on your explanation.
Basic Set Operations
Set operations like union, intersection, and complement are defined element-wise: a point belongs to the result exactly when it satisfies the defining condition.
Proving Set Equalities
To prove A = B, show A ⊆ B and B ⊆ A by starting with an arbitrary element and chasing membership through definitions.
What Makes a Function a Function?
A function is a rule assigning each element of the domain exactly one element of the codomain; domain and codomain matter in proofs.
Injective, Surjective, Bijective
Injective means no two inputs share an output; surjective means every codomain element is hit; bijective means both.
Inverse Functions & Proofs
Inverse functions exist exactly for bijections; to prove a function has an inverse, you show it is bijective and verify the two-sided inverse identities.
Constructing Functions in Proofs
Constructing functions (especially injections, surjections, and bijections) is a central proof technique for comparing sizes of sets and showing structural relationships.