Discrete mathematics guides
Short, step-by-step guides for the questions students ask most often. Every worked example is checked by an automated test, and each guide links to the free calculator that lets you confirm your own answers. Looking for a study order instead? See the learning path, or look a term up in the glossary.
Tools
Transitive Closure Calculator: Step-by-Step with Warshall's Algorithm
Free transitive closure calculator: enter a set and a relation to get the transitive, reflexive or symmetric closure with every step explained. Try it now.
Predicate Logic Translator: English to Logic Converter
Free predicate logic translator: type an English sentence and get its ∀ and ∃ formula, the predicates, a plain-English reading and the steps. Try it now.
Step-by-step guides
How to Make a Truth Table (Step-by-Step with Examples)
Learn how to make a truth table in five steps, with worked examples for AND, OR, implication and three-variable formulas, then check it with our calculator.
How to Simplify Boolean Expressions (Laws, K-Maps and Examples)
Learn how to simplify Boolean expressions with the algebra laws, Karnaugh maps and Quine–McCluskey, with worked examples you can check in our free calculator.
How to Find a Recurrence Relation: 5 Methods with Examples
Learn how to find a recurrence relation from a counting problem, algorithm or rule. Five methods with worked examples, plus how to check and solve the result.
Mathematical Induction: How to Prove by Induction (with Examples)
Learn how mathematical induction works in three steps, with worked examples for sums, inequalities and divisibility, a write-up template and common mistakes.
The Euclidean Algorithm Explained (Find the GCD Step by Step)
Learn the Euclidean algorithm with worked examples: find a GCD in a few divisions, get the LCM, and use the extended version to write gcd(a, b) = ax + by.
Permutations vs Combinations: Formulas and Worked Examples
Know when to use permutations and when to use combinations. Clear formulas, a one-question test, repetition cases and worked examples you can verify.
Dijkstra's Algorithm Explained with a Step-by-Step Example
See how Dijkstra's algorithm finds shortest paths: the idea, a full worked table, why negative weights break it, and its running time. Try your own graph.
Big O Notation Explained: Simple Rules and Worked Examples
Understand Big O notation with the formal definition, a short proof, five simple rules, a growth table and loop examples for finding an algorithm's complexity.
Propositional Logic Explained: Connectives, Equivalences and Inference
A clear introduction to propositional logic: the five connectives, converse and contrapositive, De Morgan's laws, and valid versus invalid arguments.
Discrete Mathematics Formulas: A Cheat Sheet with Examples
A practical cheat sheet of discrete mathematics formulas for logic, sets, counting, sums, number theory, graphs and functions, each with a numeric example.