How to learn discrete mathematics

Discrete mathematics is a collection of topics rather than one linear subject, so it is easy to feel unsure where to begin. The order below groups the topics so that each stage uses ideas from the one before. Courses differ, so treat it as a sensible default and follow your syllabus if you have one.

Prefer worked walkthroughs? The guides cover truth tables, Boolean simplification, recurrence relations and induction step by step.

A method that works for most topics

1. Read the study guide on the topic page, focusing on the definitions.
2. Work each worked example by hand before reading the steps.
3. Make up a similar problem and solve it by hand, then check it with the calculator and compare the intermediate steps.
4. Try the “Test yourself” questions. If you miss one, find which definition you misapplied.
5. Return to the topic a few days later. Spaced repetition works far better than one long session.

Stage 1: Reasoning and proof

Everything else depends on reading statements precisely and justifying them. Start here even if it feels abstract.

• Propositional Logic: Truth tables, tautologies, equivalence and simplification.
• Predicate Logic: Quantifiers, variable binding, and translation.
• Proofs: Direct, contrapositive, contradiction, induction.
• Set Theory: Union, intersection, power sets, Venn diagrams.

Stage 2: Functions, relations and counting

These build the vocabulary of mappings and structure, and teach you how to count possibilities without listing them.

• Functions: Injections, surjections, bijections, composition.
• Relations: Properties, closures, equivalence classes.
• Counting: Permutations, combinations, pigeonhole principle.
• Number Theory: Primes, GCD/LCM, modular arithmetic, CRT.

Stage 3: Graphs and matrices

Graphs model networks and dependencies; matrices let you compute with them. Trees are the most heavily used special case.

• Graph Theory: Degree, handshaking, bipartite, Euler & Hamiltonian paths.
• Trees: Tree property checks and binary tree traversals.
• Graph Algorithms: BFS, DFS, Dijkstra, Prim's & Kruskal's MST.
• Matrices: Add, multiply, transpose, determinant, inverse.

Stage 4: Logic in hardware and computation

Here the earlier ideas become circuits, machines and measures of efficiency.

• Boolean Algebra: Truth tables, SOP/POS, Quine-McCluskey minimization.
• Logic Circuits: Half/full adders & subtractors, mux/demux.
• Automata: DFA / NFA simulation over an input string.
• Algorithms & Complexity: Big-O growth comparison, sorting & searching traces.

Stage 5: Sequences, chance and structure

These topics combine the earlier tools and are easiest once you are comfortable with proofs, counting and modular arithmetic.

• Recurrence Relations: Closed forms via characteristic equations.
• Probability: Conditional probability, Bayes' theorem, expectation.
• Algebraic Structures: Groups, monoids, semigroups from a Cayley table.
• Coding Theory: Hamming distance, parity, Hamming(7,4) codes.

Unsure about a term? The glossary defines the most common ones. Registered users can also try the quiz and practice problems.