Learn Discrete Mathematics unit by unit

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Units at a glance

Every unit in this course, what you will be able to do once you finish it, and roughly when it lands in a 14-week schedule.

Discrete Mathematics course map: 7 units
#	Unit	What you can do after it	Weeks
01	Logic	You can build truth tables, negate a statement correctly, and check whether an argument is valid.	Weeks 1-2
02	Proof methods	You can prove a claim directly, by contradiction, by contrapositive, or by induction.	Weeks 3-4
03	Sets and relations	You can work with unions, intersections and complements, and test a relation for being reflexive, symmetric or transitive.	Weeks 5-6
04	Counting	You can count arrangements and selections without listing them, using products, permutations and combinations.	Weeks 7-8
05	Recurrence relations	You can define a sequence in terms of its earlier terms and then find a closed formula.	Weeks 9-10
06	Graph theory	You can model connections as dots and lines, then find paths, trees and colourings.	Weeks 11-12
07	Number theory	You can work with divisibility, greatest common divisors, primes and clock arithmetic.	Weeks 13-14

Discrete Mathematics course units

Work in order for a complete course, or jump directly to the unit you need for an upcoming assessment.

Logic

Weeks 1-2

You can build truth tables, negate a statement correctly, and check whether an argument is valid. You can also translate an English sentence into symbols.

Build the full truth table for an if-then statement.
Negate 'all cats are grey' as 'some cat is not grey'.
Apply modus ponens and spot the converse mistake.

Worked example

When is 'if p then q' false?

Check all four rows of the truth table
Only the row with p true and q false fails

AnswerIt is false only when p is true and q is false

Most common mistakeReading 'if p then q' as 'if q then p': from 'if it rains, the ground is wet' concluding 'the ground is wet, so it rained', which any sprinkler disproves.

Explain logic Practice logic

Proof methods

Weeks 3-4

You can prove a claim directly, by contradiction, by contrapositive, or by induction. You can also knock a claim down with a single counterexample.

Prove odd plus odd is even by writing both numbers in 2k + 1 form.
Run an induction proof with a base case and an inductive step.
Disprove 'every prime is odd' by naming the number 2.

Worked example

Prove that the sum of two odd numbers is even

Let a = 2m + 1 and b = 2n + 1
a + b = 2m + 2n + 2 = 2(m + n + 1)

AnswerIt is 2 times a whole number, so it is even

Most common mistakeStopping after examples: 3 + 5 = 8 and 7 + 9 = 16 show the pattern but prove nothing about the infinitely many pairs you did not test.

Explain proof methods Practice proof methods

Sets and relations

Weeks 5-6

You can work with unions, intersections and complements, and test a relation for being reflexive, symmetric or transitive. You can also check whether a relation is a function.

Compute the size of a union with inclusion-exclusion.
Test whether 'is a sibling of' is transitive.
Build the equivalence classes of a relation.

Worked example

Set A has 20 members, set B has 15, and 6 are in both. How big is the union?

Size of union = 20 + 15 - 6
35 - 6

Answer29

Most common mistakeAdding the two sizes and stopping: reporting 35, which counts each of the 6 shared members twice.

Explain sets and relations Practice sets and relations

Counting

Weeks 7-8

You can count arrangements and selections without listing them, using products, permutations and combinations. You can also use the pigeonhole idea to prove something must repeat.

Decide whether order matters before choosing permutations or combinations.
Compute the number of 3-person teams from a group of 10.
Use the pigeonhole principle to show two of 13 people share a birth month.

Worked example

How many 3-person teams can be picked from 10 people?

Order does not matter, so use combinations
(10 x 9 x 8)/(3 x 2 x 1) = 720/6

Answer120

Most common mistakeUsing permutations by mistake: reporting 10 x 9 x 8 = 720, which counts the same three people once for each of their 6 orders.

Explain counting Practice counting

Recurrence relations

Weeks 9-10

You can define a sequence in terms of its earlier terms and then find a closed formula. You can also count the steps an algorithm takes this way.

Solve a(n) = 2a(n-1) with a(0) = 3.
Use the characteristic root method on a two-term recurrence.
Write the Towers of Hanoi move count as T(n) = 2T(n-1) + 1.

Worked example

Solve a(n) = 2a(n-1) with a(0) = 3

a(1) = 6, a(2) = 12, a(3) = 24
Each step doubles the one before

Answera(n) = 3 times 2 to the power n

Most common mistakeWriting a(n) = 2 to the power n and losing the starting value: that formula gives a(0) = 1 when the sequence actually starts at 3.

Explain recurrence relations Practice recurrence relations

Graph theory

Weeks 11-12

You can model connections as dots and lines, then find paths, trees and colourings. You can also decide whether a walk using every edge exists.

Use vertex degrees to test for an Euler circuit.
Run Dijkstra's algorithm on a small weighted graph.
Find a minimum spanning tree with Kruskal's algorithm.

Worked example

A graph has vertex degrees 2, 2, 3, 3. Is there an Euler circuit?

An Euler circuit needs every degree to be even
Two vertices have odd degree

AnswerNo circuit, but an Euler path exists between those two odd vertices

Most common mistakeCounting edges instead of degrees: the total number of edges tells you nothing here, since the test is on each vertex's own degree.

Explain graph theory Practice graph theory

Number theory

Weeks 13-14

You can work with divisibility, greatest common divisors, primes and clock arithmetic. You can also solve the simple congruences that coding schemes rely on.

Run the Euclidean algorithm to find a greatest common divisor.
Reduce a large power in modular arithmetic by finding its cycle.
Find the inverse of 3 modulo 7.

Worked example

Find the greatest common divisor of 252 and 105

252 = 2(105) + 42
105 = 2(42) + 21, then 42 = 2(21) + 0

Answer21

Most common mistakeStopping at the first remainder: reporting 42 as the answer, even though 42 does not divide 105 evenly.

Explain number theory Practice number theory

Prepare for Discrete Mathematics practice

Start with the earliest uncertain prerequisite

Before timing yourself, check whether you can explain Logic from a blank page. Then connect it to Proof methods. If either explanation depends on copying a formula, review the unit first and complete two untimed examples.

Use tools to verify, not to choose the method for you

The Discrete math calculator can test calculations and representations used in Discrete Mathematics. Make the setup yourself, predict the sign or scale, and compare the tool result with that prediction. Use the formula library to check conditions as well as notation.

Know when to move to the full test

Move from Discrete Mathematics practice to the complete course test after you can correct a missed problem without reopening the worked answer. Record the earliest wrong decision—not only the final score—so the next study session has a precise target.

Before and after the syllabus

Learn the ideas, then practise them

The unit list tells you what is covered. These two pages are where the method is explained and where you find out whether it stuck.

GuidesLinear Algebra explainedConcepts, worked methods and checks behind this course.

PracticeMixed Discrete Math questionsAnswer review after every attempt.

ReferenceFormulas for this courseConditions and a worked example for each one.

Assess

Take the complete Discrete Mathematics test

Begin test →

Reference

Review essential formulas

Open library →

Calculate

Use the Discrete math calculator

Open tool →

Plan

Prepare around your exam date

Build plan →

Questions about the Discrete Mathematics course

Where should I start?

Start with Logic if you are following the full sequence. If that unit feels automatic, open the Discrete Mathematics practice page, choose mixed review, and let the first errors identify the earliest prerequisite to revisit.

How do I know I am ready for the course test?

You are ready when you can choose a method without a hint, show the governing steps, and explain why the result is reasonable. Use the complete Discrete Mathematics test only after you can correct practice errors from a blank page.

Which calculator supports this course?

The Discrete math calculator supports the calculations and representations used in this course. Use it to test or visualize a result after making your own setup.

Learn Discrete Mathematics unit by unit

Units at a glance

Discrete Mathematics course units

Logic

Worked example

Proof methods

Worked example

Sets and relations

Worked example

Counting

Worked example

Recurrence relations

Worked example

Graph theory

Worked example

Number theory

Worked example

Prepare for Discrete Mathematics practice

Start with the earliest uncertain prerequisite

Use tools to verify, not to choose the method for you

Know when to move to the full test

Learn the ideas, then practise them

Other university courses

Probability

Mathematical Statistics

Numerical Methods

Take the complete Discrete Mathematics test

Review essential formulas

Use the Discrete math calculator

Prepare around your exam date

Questions about the Discrete Mathematics course