Discrete Mathematics: The Computer Scientist's Toolkit
Signature Course No. 457 — A Signature AI Socratic Coaching Course.
No experience needed — your coach starts where you are.
A New Kind of Learning
Socratic coaching, delivered by an agentic AI coach.
Not a course you watch. An AI coach you work with.
The Lighthouse Coach begins by discovering how you think about the subject, teaches what is missing, and gives you realistic practice before asking you to prove that you can use it. You may test out of concepts you already understand, while capstones turn the learning into a practical plan, decision, conversation, or other real-world result.
The curriculum is deliberately authored; the AI delivers it through coaching. It is not an AI randomly generating lessons.
A coach that teaches by asking — not by lecturing
The Lighthouse Coach is an AI Socratic coach built on Cognosco Learning's Discrete Mathematics: The Computer Scientist's Toolkit course, in the Personal Effectiveness & Well-Being category. It does not play videos at you. It holds a conversation: it asks you a question, listens to your answer, names the misconception it hears, affirms what you got right, and asks the next question — the way a great coach works in the room.
Every claim in this course can be checked by the reader. A statement is true or false, a count is right or wrong, a proof holds or it does not — and none of those verdicts needs to be taken on trust. When a truth table is printed, you build it yourself. When a count is given, you recount it by a second route. When a proof takes a step, you ask what licenses that step before you read the next line.
It starts with saying exactly what you mean. Ordinary language is a poor instrument for exact claims, and computing cannot afford the guessing it invites. You will learn propositional and predicate logic — statements combined by not, and, or, implies, and if-and-only-if — and how a truth table settles what any combination means, in which rows it is true, and in which single row, if any, it fails.
It works in two movements. First, Module 1 — Logic, Structure, and Proof — work through Concepts 1–9, finishing with Specify and Prove a Small System; then, Module 2 — Counting, Graphs, and Number — work through Concepts 10–18, finishing with Count and Cost a Small Network.
Both capstones produce real artifacts — a complete logical and structural specification of a small lending locker, proved correct piece by piece as a specification and proof dossier; and a counted, graphed, and costed design for a six-office network, with every number derived and defended in a design dossier.
The honesty standard is part of the course. This course promises working capability and judgment in the discrete syllabus. It does not promise a grade in any university course, a result in any examination, or fluency without practice. Mathematics is learned by producing answers, not by recognizing them, and every concept here demands production: a table built, a proof written, a count derived and defended.
The course is self-paced and always on — roughly 30 minutes per concept, about 9 hours in full, taken in as many sittings as you like. You can type your answers, or talk out loud: the Voice Coach speaks its questions and listens to your spoken replies. Learners who already know the material can test out concept by concept.
Discrete Mathematics: The Computer Scientist's Toolkit is part of The Lighthouse Signature Series — the online counterpart to the one-day, in-person learning Cognosco Learning delivers.
Course Facts
One full concept, coached end to end — free: verify your email and we’ll send you a link to start.
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Diagnose → Learn & Practice → Prove → Capstone
Every learner moves through the same discipline a Cognosco instructor would insist on in the room.
Diagnose
The Coach starts where the course starts: with propositional logic. Write down one sentence from your own work that uses ‘or’ and say whether you meant at least one or exactly one; judge a program condition — ‘if the user is logged in, the dashboard loads’ — on a day when no user logged in, in table language; and state how many rows a truth table in four variables has, and how you know without listing them. Every concept opens with three questions that locate where you actually stand, and your answers shape how deeply the Coach works the concept with you. Everyone begins with Module 1 — no skipping the foundations you only think you have.
Learn & Practice
Work each concept in conversation with the Coach — spoken or written — then practice it against realistic situations until it holds up.
Prove
Every concept ends in proof: answer the check cold and demonstrate the skill. Know it already? Test out and take credit immediately.
Capstone
Finish with two capstones — Specify and Prove a Small System for Module 1 and Count and Cost a Small Network for Module 2 — that pull the whole course together, and earn the certificate.
What You'll Learn
Say it exactly, then check it exactly. Every concept pairs a way of stating a claim precisely — a formula, a definition, a quantified sentence — with a way of checking it: a truth table, a recount by a second route, a base case and a step. Precision without a check is decoration; a check without precision has nothing to hold onto.
Module 1 builds the language and the proof machine. Propositional Logic; Equivalence and Inference; Predicate Logic; Sets; Relations; Functions; Direct Proof and Contrapositive; Contradiction, Cases, and Counterexamples; and Induction.
Module 2 counts, connects, and computes. Counting Basics; Inclusion–Exclusion and Pigeonhole; Binomial Counting and Identities; Recurrence Relations; Graphs and Degrees; Trees and Traversals; Circuits, Connectivity, and Coloring; Divisibility and Modular Arithmetic; and Finite Probability and Expectation.
And it replaces listing with structure throughout. An equivalence that simplifies a formula, a bijection that counts a set by counting a simpler one, a recurrence that counts stage by stage, a graph invariant that settles a question without tracing every path — listing cases is the method of last resort, and a proof is a machine for settling all cases at once.
What you will learn
- Translate statements into propositional and predicate logic, build truth tables, and test equivalence and validity
- Work with sets, relations, and functions, and classify each by its defining properties
- Prove claims by direct proof, contrapositive, contradiction, cases, and induction — and refute false claims by counterexample
- Count arrangements using the sum and product rules, permutations, combinations, inclusion–exclusion, and the pigeonhole principle
- Use binomial identities and recurrence relations to count and to describe growth
- Model networks as graphs and reason about degrees, paths, trees, traversals, circuits, and colorings
- Apply divisibility, modular arithmetic, and finite probability with exact, checkable computation
Everything in the course
- All 18 concepts of Discrete Mathematics: The Computer Scientist's Toolkit, in 2 modules
- A starting-point check first — the Coach finds where you actually stand before teaching
- Learn, Practice, Prove for every concept — no passive video watching
- A test-out on every concept: prove you know it and move on
- The Voice Coach: talk with the Coach out loud — it speaks, listens, and replies
- A personal Learner Record of your work across the course
- Two capstone exercises: Specify and Prove a Small System and Count and Cost a Small Network
- A printable session summary whenever you pause
- The Lighthouse Coach completion certificate (Signature Series)
- Your progress is saved as you go — stop anytime, resume where you left off
Built for the people computing asks to be exact
Computer science students — meeting the discrete syllabus as a required core subject and needing it as a working tool, not a hurdle to forget; and programmers and engineers returning to foundations — who write conditions, design data models, and estimate search spaces daily, and want the reasoning underneath those moves made explicit and reliable.
Data and systems practitioners — who meet sets, relations, and graphs in databases, networks, and permissions, and need to reason about them exactly.
Self-directed learners with school algebra — who can manipulate symbols, read a table, and follow an argument, and who want the university discrete course's core without enrolling. This course assumes comfort with ordinary arithmetic and basic algebra. It assumes no calculus, no statistics, and no prior proof experience — proof craft is taught here, from the first direct argument onward.
Prefer to learn live and in person? The same material anchors Cognosco's one-day programs — browse the in-person catalog.
Start with Propositional Logic
$195 — introductory price. One payment. No subscription, no password to manage.
Here is exactly what happens when you enroll:
- Pay securely with Stripe — you are taken straight into the course
- Create your learner profile and begin the Module 1 starting-point check
- Save the access key you are given — it restores your enrollment on any device
- Learn at your pace; your progress is stored in your browser on your device
For the full voice experience, use Chrome and allow the microphone when asked. Typing your answers works everywhere, all the time. Questions before you enroll? Email hello@cognoscolearning.com.
Before you enroll
What is a Socratic coaching session?
A conversation, not a video. Your coach asks a question; you answer — out loud by voice, or by typing — and the coach responds to what you said: naming the misconception it hears, affirming what you got right, and asking the next question. You advance only when you have mastered each concept, so a finished course means the learning held.
How long does it take — and how long do I have?
About 30 minutes per concept. Learn in sittings of any length; your progress is saved as you go. You have a full year from purchase to finish, and you can revisit and retake within that year.
What does the certificate verify?
That you completed every concept and both capstones of this course. Anyone can confirm a Cognosco certificate online — enter its certificate ID on our verification page.
What if it isn't for me — what are the refunds?
If you have not started, cancel for a full refund within seven days of purchase. Once you have started, there is no cash refund — stop before finishing and we credit the full amount you paid toward any other course in the library (credit valid one year from your original purchase). Once you complete the course, no refund or credit applies.
More Signature Courses are coming
Discrete Mathematics: The Computer Scientist's Toolkit is Signature Course No. 457. Leave your email and we'll send you news of future Signature Courses and Lighthouse updates as they are announced — nothing else.