Milestone map
Milestone map
3 milestones
Prove Results in Logic, Sets, and Four Proof Techniques
8–14 weeks
Study propositional logic (truth tables, logical equivalences), predicate logic (quantifiers), set theory (operations, power set), and four proof techniques: direct proof, contrapositive, contradiction, and proof by cases. Complete at least twenty proof exercises — not just read about them.
Proof required
Submit: a typed proof portfolio containing at least twenty completed proofs — at least five each of: direct proof, proof by contrapositive, proof by contradiction, and set identity proofs. Each proof must be fully written out with every step justified. A mathematician, CS lecturer, or logician must review the portfolio and confirm in writing that the proofs are formally valid.
What gets checked
- Portfolio contains at least twenty typed proofs — fully written out with every step justified
- All four technique categories have at least five proofs each
- A mathematician or CS lecturer has confirmed the proofs are formally valid
Common mistakes
- Portfolio of proof sketches rather than complete proofs — every step must be justified, not assumed
- Proofs that only use one technique — the portfolio requires variety across four technique types
Resources
Foundationstart here
Depthgo deeper
What a verifier looks for
- Are there at least twenty complete proofs — fully written out with every step justified?
- Are all four technique categories represented with at least five proofs each?
- Select three proofs at random and check every step is justified — no assumed steps
Prove Results in Induction, Combinatorics, and Graph Theory
8–12 weeks (after milestone 1)
Study mathematical induction (weak, strong, structural), combinatorics (permutations, combinations, binomial theorem, inclusion-exclusion), and graph theory (Eulerian circuits, Hamiltonian paths, graph colouring, trees). Complete at least fifteen proofs, including at least three genuine combinatorial proofs — proofs that argue from counting in two ways rather than algebraic manipulation.
Proof required
Submit: a typed proof portfolio of at least fifteen proofs — at least five induction proofs (including at least two strong induction), at least five combinatorics proofs (including at least three genuine combinatorial proofs — two-way counting, not algebraic), and at least five graph theory proofs. A mathematician or CS lecturer must confirm the proofs are formally valid.
What gets checked
- At least fifteen complete proofs across three topic areas
- At least three of the combinatorics proofs are genuine two-way counting arguments — not algebraic simplifications
- A mathematician or CS lecturer has confirmed the proofs are formally valid
Common mistakes
- Induction proofs that skip the inductive step — 'it clearly holds' is not a proof
- Combinatorics 'proofs' that are algebraic simplifications rather than two-way counting arguments
Resources
Foundationstart here
Depthgo deeper
What a verifier looks for
- Are the induction proofs complete — base case, inductive step, conclusion?
- Are the combinatorial proofs genuine two-way counting arguments — not algebraic simplifications?
- Are the graph theory proofs about specific graph properties — not generic existence arguments?
Apply Discrete Mathematics to a CS Problem and Defend
6–10 weeks (after milestone 2)
Choose a non-trivial CS problem where discrete mathematics is central: algorithm analysis using recurrence relations, a protocol correctness argument, or a security proof using modular arithmetic. Write a full mathematical analysis (at least 800 words). Present and defend the analysis to a mathematician or CS lecturer who challenges specific steps.
Proof required
Submit: a written analysis (at least 800 words) with all mathematical steps justified; and a recording or transcript of a live review session with a mathematician or CS lecturer where they challenged at least two specific steps in the analysis.
What gets checked
- Analysis addresses a specific CS problem — not a generic discrete mathematics exercise
- Every mathematical step is justified — no assumed steps
- A mathematician or CS lecturer challenged at least two specific steps in a documented live review
Common mistakes
- Choosing a problem that reduces to a textbook discrete mathematics exercise without genuine CS application
- Live review that is a walkthrough without challenge — the reviewer must push back on specific steps
Resources
Foundationstart here
Depthgo deeper
What a verifier looks for
- Is the problem a genuine CS application — not a pure mathematics exercise restated with computer terminology?
- Are all mathematical steps justified?
- Were at least two specific steps challenged in the documented live review?