Key takeaways
- Discrete mathematics studies objects that are separate and countable, such as integers, true-or-false statements, finite sets and networks, rather than quantities that vary smoothly.
- Between them, MIT's 6.042J outline and Levin's textbook cover logic and proof, sets, functions and relations, number theory, sequences, counting, discrete probability and graph theory.
- MIT's course description says students should be able to use these methods later in algorithms, computability theory, software engineering and computer systems.
- Much of the subject asks for proofs rather than numbers, so writing definitions down and testing small cases matter more than memorising formulas.
- Start with logic, then proof techniques, because every later topic is stated and proved in that language.
Discrete mathematics is the mathematics of things that come in separate pieces: whole numbers, true-or-false statements, finite collections, arrangements, and networks of points joined by lines. The word "discrete" means individually separate and distinct, and Oscar Levin's open textbook starts from exactly that dictionary definition. The opposite is continuous mathematics, the world of calculus, where between any two values there are infinitely many more.
A quick way to feel the difference: the outputs of f(x) = x² form the interval [0, ∞), with no gaps anywhere. The number of siblings a person has forms something like {0, 1, 2, 3, 4}, and nobody has 1.7 siblings. Both are reasonable things to study. Discrete math studies the second kind.
The topics a first course covers
There is no single official syllabus, but published course outlines overlap heavily. MIT's Mathematics for Computer Science (6.042J), whose materials are free on OpenCourseWare, describes its coverage in thirds: fundamental concepts (definitions, proofs, sets, functions, relations), discrete structures (graphs, state machines, modular arithmetic, counting) and discrete probability theory. Levin's book names four main topics, combinatorics, sequences, symbolic logic and graph theory, and lists number theory, probability and others as also belonging under the discrete umbrella.
| Topic | The question it answers | A small example |
|---|---|---|
| Logic | When is a compound statement true? | "If it rains, the match is off" is false only when it rains and the match goes ahead |
| Proof | How do you show something holds in every case? | The sum of two odd numbers is always even |
| Sets, functions, relations | How are collections and connections described precisely? | A function that pairs every student with exactly one ID number |
| Number theory | How do integers behave under division? | 17 leaves remainder 2 when divided by 5 |
| Induction and recursion | How do you prove a pattern continues forever? | 1 + 3 + 5 + … + (2n − 1) = n² for every n |
| Counting | How many arrangements are there? | A 4-digit PIN has 10⁴ = 10,000 possibilities |
| Discrete probability | How likely is an outcome among countable ones? | The chance two dice total 7 is 6 out of 36 |
| Recurrences | How does a sequence defined by its own earlier terms grow? | Each term is the sum of the previous two |
| Graph theory | What can be said about networks of connections? | Can you draw every road of a map without lifting the pen? |
Why computer science leans on it
MIT's course description is explicit about the purpose: students should be able to use these methods later in the design and analysis of algorithms, computability theory, software engineering and computer systems. Some of the connections can be seen directly:
- Logic. A condition built from "and", "or" and "not" is a compound statement, so simplifying it, or showing two conditions always agree, is propositional logic.
- Induction. A statement about every input size n, not just the sizes you tested, is a statement about every natural number, which is what induction proves.
- Counting. Asking how many cases there are, such as the 10,000 four-digit PINs, is a counting problem.
- Modular arithmetic is the arithmetic of remainders; MathWorld's account of RSA public-key encryption is built on congruences.
- Graphs describe anything made of connections. The open textbook Applied Discrete Structures defines a simple undirected graph as a nonempty set of vertices together with a set of two-element subsets of it, the edges.
Four problems that show the flavour
None of these needs more than arithmetic, and each one introduces a whole area of the subject.
- Handshakes. Eight people each shake hands with every other person once. How many handshakes? Each handshake is a choice of 2 people from 8, so the count is 8 × 7 ÷ 2 = 28. Dividing by 2 corrects for counting each handshake from both ends, which is the central idea of permutations versus combinations.
- Odd sums. Add the first few odd numbers: 1, 1 + 3 = 4, 1 + 3 + 5 = 9, then 16 and 25. The totals are perfect squares. Checking five cases is evidence; proving it for every n takes induction.
- Two signs. "If this box is empty, the other sign is true" is a conditional statement, and deciding which box to open is a truth table problem.
- Three houses, three utilities. Can three houses each be joined to water, gas and electricity without any lines crossing on flat ground? No. That graph, called K₃,₃, cannot be drawn in the plane without a crossing, a fact from planar graph theory.
Is it hard?
In much of the subject an answer is an argument rather than a number: MIT's course lists definitions and proofs among its fundamental concepts. Three habits help with that shift: write the definition down before using it, try the claim on small cases, and learn which proof method a statement is asking for. The study habits in how to study for a discrete math exam are built around that.
The good news is that many claims can be checked by hand. A formula for counting arrangements can be tested by listing the arrangements for a small case; a claimed equivalence between two-variable statements can be tested with a four-row truth table. That makes self-checking practical when you study alone.
Where to start
Begin with logic, because every later topic is stated in its language, then move to proof techniques, then to sets and functions. After that the order matters less. Counting and probability go together, and graph theory can be studied almost independently once you can write a proof.
Our discrete math hub lists the guides in that order. If you want lessons and quizzes in the same sequence, Discrete Math: Study & Quiz is organised as ten units from logic to graph theory; the guide to the app explains how to use them. It is one of our education and brain-training apps.
Good to know
Frequently asked questions
What does discrete mean in discrete math?
Is discrete math harder than calculus?
Do you need calculus before taking discrete math?
What is discrete math used for in computer science?
Sources
- What is Discrete Mathematics? (Discrete Mathematics: An Open Introduction, 3rd edition, Oscar Levin) (opens in a new tab)Open Math Books — accessed
- 6.042J Mathematics for Computer Science, Spring 2015: Course description (opens in a new tab)MIT OpenCourseWare — accessed
- Applied Discrete Structures: Graphs, General Introduction (Al Doerr and Ken Levasseur) (opens in a new tab)discretemath.org — accessed
- RSA Encryption (opens in a new tab)Wolfram MathWorld — accessed
About the author
Reign Creative LLC — The independent app studio that builds the apps covered on this site and publishes every article here. On Google Play it publishes as Reign Collective Apps.
We use AI tools to help research, draft and edit our articles. Each article lists the sources its factual claims rely on, and details about our apps come from their live Google Play listings. No article on this site has been reviewed by an independent expert unless the reviewer is named on it.
No person at Reign Creative LLC has read this article in full. It was published after automated checks and an AI editorial review, which are described in our editorial policy and are not the same as a person reading every sentence.
Spotted an error? Tell us — how corrections are handled is set out in our editorial policy.