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Discrete Mathematics

5 mini courses, 20 lessons: logic, proofs, sets, counting, graphs and number theory, with Python

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20 lessons · ~3 hours · 30-question final exam · pass mark 70%

About the course

What you’ll learn

The maths behind algorithms, databases, cryptography and coding interviews, taught with examples you can run. Five one-hour mini courses cover logic and proofs, sets, relations and functions, counting and probability, graphs and trees, and number theory up to how RSA encryption works. Every idea is tried in Python, so you see it work instead of just memorising it. Pass every lesson quiz and the final exam to earn a verifiable certificate.

LogicProofsSets and functionsCountingGraph theoryNumber theoryProblem solving

Part of the career track: Computer Science Foundations

How the certificate works

  1. Read each lesson’s study guide, then the full lesson and any notebooks or exercises.
  2. Pass every lesson quiz with 70% or more.
  3. Pass the final exam: 30 questions in 45 minutes.
  4. Get a certificate with a public verification link, and add it to LinkedIn.

Syllabus

20 lessons in 5 mini courses

Each mini course takes about an hour. Already know one? Test out in the app and skip it.

1. Logic and proofs

Write logical statements precisely and prove simple claims.

1
Propositions and Connectives
10 min · A proposition is a statement that is true or false.
2
Truth Tables and Equivalences
10 min · A truth table lists all 2ⁿ input combinations for n variables.
3
Predicates and Quantifiers
10 min · A predicate P(x) becomes true or false once x is given.
4
Proofs
12 min · Examples are not proofs: a proof must cover every case.

2. Sets, relations and functions

Work with sets, relations and functions, and prove things by induction.

5
Sets
10 min · A set has distinct items and no order; |A| is its size.
6
Relations
10 min · A relation from A to B is a subset of A × B; a database table is a relation.
7
Functions
10 min · A function gives exactly one output for every input.
8
Mathematical Induction
12 min · Induction: base case plus inductive step proves a claim for all n.

3. Counting

Count arrangements and choices without listing them all.

9
The Sum and Product Rules
10 min · Product rule: m choices then n choices gives m × n.
10
Permutations and Combinations
12 min · Permutations count arrangements (order matters): n! for n items.
11
The Pigeonhole Principle
10 min · More than n items in n boxes means some box has two.
12
Counting and Probability
10 min · With equally likely outcomes, probability = favourable ÷ total.

4. Graphs and trees

Model networks as graphs and reason about paths, trees and colourings.

13
Graphs: Vertices and Edges
10 min · A graph has vertices joined by edges; edges can be directed and weighted.
14
Trees and Spanning Trees
10 min · A tree is connected with no cycles; n vertices means n − 1 edges.
15
Euler and Hamilton Paths
10 min · Euler path: every edge once; Euler circuit: also returns to the start.
16
Graph Colouring
10 min · Colour vertices so no edge joins two of the same colour; the minimum is the chromatic number.

5. Number theory and cryptography

Use modular arithmetic, primes and gcd, and explain how RSA works.

17
Modular Arithmetic
10 min · a mod n is the remainder; a ≡ b (mod n) means equal remainders.
18
Primes and the GCD
10 min · A prime has exactly two divisors; every number factors into primes in one way.
19
How RSA Encryption Works
12 min · Public key locks, private key unlocks.
20
Algebra and Error-Correcting Codes
10 min · A group: associative operation, identity, inverses (clock arithmetic mod 12).

Written by Vikosha. Lessons, questions and exercises are original. Licence: Original Vikosha course..