Skills

Apply Discrete Mathematics Through Formal Proofs

8 weeks · 0 milestones

Produce written solutions to 30+ proof exercises spanning propositional logic, mathematical induction, graph theory, and combinatorics — not just correct answers, but proofs with valid structure (base case, inductive step, conclusion) that a mathematician would accept. Each proof must show your reasoning, not just the result. Proof: the submitted exercise set reviewed by a CS lecturer or maths researcher who also presents 2–3 unseen claims during the review and asks you to prove or disprove them live — the live reasoning is what demonstrates understanding rather than memorisation.

Part of: Computer Science & Data · Proves for: Engineers, UG

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

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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?

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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?

You'll sign in first, then come straight back here.

Part of

Computer Science Fundamentals →

Related outcomes

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Skills

Apply Discrete Mathematics Through Formal Proofs

8 weeks · 0 milestones

Part of: Computer Science & Data · Proves for: Engineers, UG

Milestone map

3 milestones

Prove Results in Logic, Sets, and Four Proof Techniques

8–14 weeks

Proof required

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

You'll sign in first, then come straight back here.

Prove Results in Induction, Combinatorics, and Graph Theory

8–12 weeks (after milestone 1)

Proof required

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?

You'll sign in first, then come straight back here.

Apply Discrete Mathematics to a CS Problem and Defend

6–10 weeks (after milestone 2)

Proof required

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?

You'll sign in first, then come straight back here.

Part of

Computer Science Fundamentals →