Session 01 - Notation & Logic

Mathematics for Master Students

Dr. Tobias Vlcek

Introduction

About Me

Field: Optimising and simulating complex systems
Languages of choice: Julia, Python and Rust
Interest: Modelling, simulations, machine learning
Teaching: OR, algorithms, programming & mathematics
Contact: vlcek@beyondsimulations.com

I really appreciate active participation and questions: the more you ask, the more useful these sessions become for you!

Our Plan for Today

How this tutorial works and why it exists
How to make good use of AI while studying
Notation: the vocabulary of mathematics
Logic: the grammar of mathematical reasoning
Your homework until the next session

Tutorial Structure

Session	Topic	Date	Time	Room
1	Notation & Logic	Tue 8 Sep 2026	18:00–19:30	Auditorium
2	Sets	Tue 15 Sep 2026	18:00–19:30	Auditorium
3	Functions	Tue 22 Sep 2026	18:00–19:30	Zoom
4	Differentiation	Tue 29 Sep 2026	18:00–19:30	Zoom
5	Single Variable Optimization	Tue 6 Oct 2026	18:00–19:30	Auditorium
6	Systems of Linear Equations	Tue 13 Oct 2026	18:00–19:30	Auditorium
7	Mock Exam (voluntary!)	Tue 27 Oct 2026	14:30–16:00	Auditorium

Six sessions of 90 minutes plus a voluntary mock exam: a compact refresher of the mathematics used in your M.Sc. courses.

Why These Topics?

Your KLU courses assume you can read and use mathematics
Lecturers will not pause to explain notation or reasoning
The topics of this tutorial were chosen by checking where they appear in your first- and second-semester courses
The next two slides show the result of that mapping

Maths Topics in Your KLU Courses I

Semester 1

Course	Notation	Logic	Sets	Functions	Differentiation	Linear Systems	Optimization
Statistics & Econometrics	✓	✓	✓			✓
Business Logistics & SCM	✓			✓	✓		✓
Fundamentals of Business Analytics	✓			✓	✓	✓	✓
International Economics	✓			✓	✓	✓	✓
Strategic Issues in SCM	✓	✓				✓
Fundamentals of Data Science	✓	✓	✓	✓	✓

Maths Topics in Your KLU Courses II

Semester 2

Course	Notation	Logic	Sets	Functions	Differentiation	Linear Systems	Optimization
Transportation & Distribution	✓	✓	✓	✓	✓	✓	✓
Global Trends in Human Resources		✓				✓
Economics of Digital Transformation				✓	✓
Economics of Business Strategy	✓			✓	✓	✓	✓
Business Strategy & Sustainability				✓

Today’s topics, Notation and Logic, appear in almost every course.

How This Tutorial Works

Each session comes with four materials:

Lecture: what we discuss together in class
Tasks: homework problems with worked solutions
Self-Test: a short online quiz to check yourself
Cheatsheet: a compact reference to keep at hand

You work on the tasks between sessions. Every session starts with a recap and your questions.

No Grades, Just Preparation

This tutorial is not graded: it is purely preparatory
Nobody tracks your answers or your progress
The only goal: you follow your M.Sc. courses with ease
The mock exam in Session 7 is voluntary, and we work through it together

Ask any question you have, no matter how simple it may seem. If you are lost, even after something was explained multiple times, ask again: that is exactly what this tutorial is for!

Tutorial Materials

All materials are hosted online at math-master.tobiasvlcek.com
Lectures are available as web page, slides, and PDF
Tasks can be printed as a worksheet: solutions are on the website
If you find good additional resources, share them with the group!

Artificial Intelligence

How to Use AI in This Tutorial

Feel free to use AI tools to support your understanding
A tutorial chatbot is available directly on the tutorial website
It is designed to guide your thinking, not to hand you solutions
There is no graded exam here, so the rule is simple:

Use AI to learn, not to outsource your thinking. If the chatbot solves your homework, you have practised prompting, not mathematics.

The Tutorial Chatbot

Click the chat bubble on any page of the tutorial website
Or open the dedicated Math Helper page
Ask your question as specifically as possible: this gives the model enough context to help you
It is set up by me for this tutorial and will nudge you with hints instead of full solutions
Please don’t share personal information in the chat

LLMs and Mathematics

Large language models (LLMs) are pattern-based
They predict text, they do not compute
They are great at explaining concepts and rephrasing ideas
They can confidently produce wrong calculations

Always question the maths output of LLMs!

For actual computation, tools like Wolfram Alpha are more reliable, as they use symbolic computation rather than pattern matching.

Notation

Why Mathematical Notation?

Mathematics is a language, with vocabulary and grammar:

It expresses complex ideas precisely and compactly
It is understood worldwide, independent of your mother tongue
A constraint like \(\sum_{t=1}^{12} c_t \leq B\) says in one line what takes a paragraph in English
Reading it fluently is a skill, and skills can be trained

The Number Sets

\[\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}\]

\(\mathbb{N} = \{1, 2, 3, \dots\}\): natural numbers (counting: units, trucks)
\(\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}\): integers (profits and losses)
\(\mathbb{Q} = \{\frac{p}{q} : p, q \in \mathbb{Z}, q \neq 0\}\): rational numbers (fractions)
\(\mathbb{R}\), real numbers: all points on the number line, including \(\sqrt{2}\) and \(\pi\)

Some books include \(0\) in \(\mathbb{N}\). In this tutorial, \(\mathbb{N} = \{1, 2, 3, \dots\}\), and we write \(\mathbb{N}_0\) if zero is included.

Your Turn: Rational or Not?

Question

Which of these numbers are rational: \(\quad 0.75, \quad \sqrt{2}, \quad \frac{7}{3}, \quad \pi\)?

\(0.75 = \frac{3}{4}\) and \(\frac{7}{3}\) are rational: both can be written as a fraction of integers
\(\sqrt{2}\) and \(\pi\) are irrational: they are real, but no fraction equals them exactly

Common Mistake

\(\pi \approx \frac{22}{7}\), but \(\pi \neq \frac{22}{7}\): an approximation is not an equality!

Interval Notation

Intervals describe connected ranges of real numbers:

Closed: \([a, b] = \{x \in \mathbb{R} : a \leq x \leq b\}\), endpoints included
Open: \((a, b) = \{x \in \mathbb{R} : a < x < b\}\), endpoints excluded
Half-open: \([a, b)\) includes \(a\) but excludes \(b\)
Unbounded: \([0, \infty)\), all non-negative reals

Square bracket \([\) means the endpoint is included, round parenthesis \((\) means it is excluded. Think of the bracket as “grabbing” the endpoint. \(\infty\) always gets a parenthesis: it is not a number you could include.

Your Turn: Write the Interval

Question

A warehouse accepts pallets weighing at least \(200\) kg and strictly less than \(1000\) kg. Write the admissible weight \(w\) as an interval.

\[w \in [200, 1000)\]

Common Mistake

“At least” includes the endpoint (square bracket), “strictly less than” excludes it (parenthesis). \((200, 1000]\) has the brackets on the wrong sides: it would reject a \(200\) kg pallet and accept a \(1000\) kg one.

Inequality Notation

Symbol	Meaning	Business example
\(<\)	strictly less than	\(x < 100\) (below the threshold)
\(\leq\)	less than or equal	\(x \leq 500\) (capacity limit)
\(\geq\)	greater than or equal	\(q \geq 20\) (minimum order quantity)
\(\neq\)	not equal to	\(\text{supply} \neq \text{demand}\)

Inequalities can be chained: \(\quad 0 \leq x \leq 500\)

means “\(x\) is at least \(0\) and at most \(500\)”, as an interval: \(x \in [0, 500]\).

Absolute Value

The absolute value \(|x|\) is the distance of \(x\) from zero:

\[|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}\]

\(|5| = 5\) and \(|-5| = 5\): distance is never negative
\(|a - b|\) is the distance between \(a\) and \(b\)
Business use: the forecast error \(|\text{demand} - \text{forecast}|\), we care about the size of the miss, not its direction

Sum Notation

The symbol \(\sum\) (capital sigma) compresses long sums:

\[\sum_{i=1}^{n} x_i = x_1 + x_2 + \dots + x_n\]

\(i\) is the index: it runs from the lower limit \(1\) to the upper limit \(n\)
\(x_i\) is the summand: the expression added up for each \(i\)
The index name is arbitrary: \(\sum_{i=1}^{n} x_i = \sum_{k=1}^{n} x_k\)

Sums in Business

Let \(c_t\) be your company’s cost in month \(t\). The total annual cost is

\[\sum_{t=1}^{12} c_t = c_1 + c_2 + \dots + c_{12}\]

Two rules you will use constantly:

Constant factors move out: \(\sum_{t=1}^{12} 0.9 \cdot c_t = 0.9 \sum_{t=1}^{12} c_t\)
Sums split over addition: \(\sum_{t=1}^{12} (c_t + f) = \sum_{t=1}^{12} c_t + \sum_{t=1}^{12} f = \sum_{t=1}^{12} c_t + 12 \cdot f\)

Read \(\sum\) as a small loop: “for each \(t\) from 1 to 12, add up …”.

Your Turn: Evaluate a Sum

Question

Compute \(\sum_{i=1}^{4} 2i\). What is the result?

Write out the loop, one term per index value:

\[\sum_{i=1}^{4} 2i = 2 \cdot 1 + 2 \cdot 2 + 2 \cdot 3 + 2 \cdot 4 = 2 + 4 + 6 + 8 = 20\]

Or move the constant out first: \(2 \sum_{i=1}^{4} i = 2 \cdot 10 = 20\).

Product Notation

The symbol \(\prod\) (capital pi) works like \(\sum\), but with multiplication:

\[\prod_{i=1}^{n} x_i = x_1 \cdot x_2 \cdot \dots \cdot x_n\]

Business example: capital grows for three years at rates \(r_1, r_2, r_3\):

\[K_3 = K_0 \cdot \prod_{t=1}^{3} (1 + r_t) = K_0 (1+r_1)(1+r_2)(1+r_3)\]

Growth compounds: that is why it is a product, not a sum.

Your Turn: Compounded Growth

Question

Revenue grows by \(10\,\%\), then \(20\,\%\), then \(5\,\%\) over three years. By what total percentage has it grown?

\[\prod_{t=1}^{3} (1 + r_t) = 1.10 \cdot 1.20 \cdot 1.05 = 1.386\]

so the total growth is \(38.6\,\%\).

Common Mistake

\(10 + 20 + 5 = 35\,\%\) is wrong: each year’s growth applies to the already grown revenue. Rates multiply as factors, they do not add.

Double Indices

Many logistics quantities depend on two things at once:

\(x_{ij}\) = quantity shipped from warehouse \(i\) to customer \(j\)
\(c_{ij}\) = transport cost per unit from \(i\) to \(j\)
So \(x_{23}\) is the flow from warehouse \(2\) to customer \(3\)
Convention: the first index is the origin (row), the second the destination (column)

Whenever you meet a variable, first ask: what do its indices mean? A well-defined model always states this explicitly.

Double Sums

With \(n\) warehouses and \(m\) customers:

Everything leaving warehouse \(i\): \(\quad \sum_{j=1}^{m} x_{ij}\)
Everything reaching customer \(j\): \(\quad \sum_{i=1}^{n} x_{ij}\)
Total transport cost over all routes:

\[\sum_{i=1}^{n} \sum_{j=1}^{m} c_{ij} \, x_{ij}\]

This is the objective function of the classic transportation problem. You will meet it again in Transportation & Distribution.

Your Turn: Total Transport Cost

Question

Two warehouses, two customers. Compute \(\sum_{i=1}^{2} \sum_{j=1}^{2} c_{ij} \, x_{ij}\).

	\(c_{i1}\)	\(c_{i2}\)		\(x_{i1}\)	\(x_{i2}\)
\(i = 1\)	2	5		10	0
\(i = 2\)	4	3		5	20

\[2 \cdot 10 + 5 \cdot 0 + 4 \cdot 5 + 3 \cdot 20 = 20 + 0 + 20 + 60 = 100\]

Bonus: everything leaving warehouse \(2\) is \(\sum_{j=1}^{2} x_{2j} = 5 + 20 = 25\). Same data, different index fixed.

Greek Letters

Greek letters are everywhere in your courses, a quick reference:

Letter	Name	Typical use
\(\alpha\)	alpha	significance level
\(\beta\)	beta	regression coefficient
\(\delta, \Delta\)	delta	(small) change, difference
\(\varepsilon\)	epsilon	error term, small number
\(\lambda\)	lambda	arrival rate

Letter	Name	Typical use
\(\mu\)	mu	mean
\(\pi\)	pi	\(3.14159\dots\), profit
\(\rho\)	rho	correlation
\(\sigma, \Sigma\)	sigma	std. deviation, sum
\(\theta\)	theta	generic parameter

Logic

Statements and Truth Values

A statement (proposition) is a declarative sentence that is either true or false, never both.

“Hamburg is a port city.”, a true statement
“\(2 + 2 = 5\)”, a false statement
“Last week, every shipment arrived on time.”, a statement; whether it is true depends on the facts

Logic does not care what a statement is about, only that it has exactly one truth value.

What Is Not a Statement?

“When can we take a break?”, a question has no truth value
“Load the truck!”, a command has no truth value
“\(x + 2 = 2x\)”, an open sentence: true for \(x = 2\), false otherwise

An open sentence becomes a statement once every variable is fixed or quantified: that is why defining your variables matters so much.

Negation

The negation \(\neg p\) (“not \(p\)”) flips the truth value of \(p\):

\(p\): “The warehouse is full.” \(\quad \neg p\): “The warehouse is not full.”
If \(p\) is true, \(\neg p\) is false, and vice versa
Double negation cancels: \(\neg(\neg p) = p\)

Common Mistake

The negation of “the warehouse is full” is not “the warehouse is empty”! Between full and empty there is a lot of room: the negation only claims “not full”.

Conjunction and Disjunction

\(p \wedge q\) (“\(p\) and \(q\)”) \(\qquad\qquad\) \(p \vee q\) (“\(p\) or \(q\)”)

\(p\)	\(q\)	\(p \wedge q\)	\(p \vee q\)
T	T	T	T
T	F	F	T
F	T	F	T
F	F	F	F

The mathematical “or” is inclusive: \(p \vee q\) is also true when both hold. “Dessert or coffee?” in maths means: taking both is fine, too.

De Morgan’s Laws

Negation flips “and” into “or”, and vice versa:

\[\neg(p \wedge q) \Leftrightarrow \neg p \vee \neg q \qquad\qquad \neg(p \vee q) \Leftrightarrow \neg p \wedge \neg q\]

Business example: “The shipment is on time and complete.”

Its negation: “The shipment is late or incomplete”: one failure is enough.

You will meet the same pattern again in a moment: negating a statement swaps \(\forall\) and \(\exists\), just like it swaps \(\wedge\) and \(\vee\).

Implication

\(p \Rightarrow q\): “if \(p\), then \(q\)”, \(p\) is the premise, \(q\) the conclusion.

\(p\)	\(q\)	\(p \Rightarrow q\)
T	T	T
T	F	F
F	T	T
F	F	T

Think of a promise

“If your order exceeds €500 (\(p\)), shipping is free (\(q\)).” The promise is only broken when the order exceeds €500 but shipping is charged: the row \(p\) = T, \(q\) = F. If \(p\) is false, nothing was promised.

Your Turn: A Broken Promise?

Question

“If the order exceeds €500, shipping is free.” Which of these cases breaks the promise?

(a) Order €600, shipping charged	(b) Order €300, shipping charged
(c) Order €300, shipping free	(d) Order €600, shipping free

Only (a): premise true, conclusion false, the one row where \(p \Rightarrow q\) is F.

Common Mistake

Cases (b) and (c) do not break the promise: for an order of €300, nothing was promised at all. An implication with a false premise is true, whatever the conclusion.

Equivalence

\(p \Leftrightarrow q\): “\(p\) if and only if \(q\)”: both have the same truth value, i.e. \(p \Rightarrow q\) and \(q \Rightarrow p\).

\(p\)	\(q\)	\(p \Leftrightarrow q\)
T	T	T
T	F	F
F	T	F
F	F	T

Common Mistake

\(\Rightarrow\) and \(\Leftrightarrow\) are not interchangeable: \(x = 2 \Rightarrow x^2 = 4\) holds, but \(x^2 = 4 \Rightarrow x = 2\) does not (it could be \(-2\)).

Necessary and Sufficient Conditions

Given \(p \Rightarrow q\):

\(p\) is a sufficient condition for \(q\): \(p\) alone guarantees \(q\)
\(q\) is a necessary condition for \(p\): without \(q\), \(p\) is impossible

“If someone is a millionaire (\(p\)), then they are rich (\(q\)).”

Being a millionaire is sufficient for being rich: no further criteria needed
Being rich is necessary for being a millionaire, but you can be rich without being a millionaire

Your Turn: Necessary or Sufficient?

Question

“A container clears the port only if its customs declaration is valid.” Is a valid declaration a necessary or a sufficient condition for clearance?

Necessary. The sentence says: clearance \(\Rightarrow\) valid declaration. Without a valid declaration, clearance is impossible.

Common Mistake

It is not sufficient: a valid declaration alone does not guarantee clearance, an inspection or unpaid duties can still stop the container. “Only if” points to a necessary condition, “if” to a sufficient one.

Quantifiers

Two symbols turn open sentences into statements:

\(\forall\), for all: \(\quad \forall x \in \mathbb{R}: x^2 \geq 0\)
\(\exists\), there exists: \(\quad \exists n \in \mathbb{N}: n > 10^6\)

In business language:

“Every route is profitable” \(\;\to\; \forall\) statement
“Some supplier delivers late” \(\;\to\; \exists\) statement

Negating Quantified Statements

Negation swaps the quantifier and negates the inside:

\[\neg\,(\forall x: P(x)) \;\Leftrightarrow\; \exists x: \neg P(x)\]

\[\neg\,(\exists x: P(x)) \;\Leftrightarrow\; \forall x: \neg P(x)\]

To disprove “all routes are profitable”, one loss-making route suffices
To disprove “some supplier delivers late”, you must check every supplier

Your Turn: Negate a Statement

Question

Negate: “All shipments arrive on time.”

“At least one shipment does not arrive on time.”

Common Mistake

The negation is not “no shipment arrives on time”: that is far too strong. One single late shipment already makes the original statement false.

Common Pitfall: The Converse Error

We know: “If a shipment is delayed (\(p\)), the customer complains (\(q\)).”

Today, a customer complained. Was the shipment delayed?

Tempting, but we cannot conclude that
Complaints can have other causes: damaged goods, wrong invoice, …
From \(p \Rightarrow q\) and \(q\), nothing follows about \(p\)

Common Mistake

Concluding \(p\) from \(p \Rightarrow q\) and \(q\) is the converse error (“affirming the consequent”), one of the most frequent reasoning errors in business and in exams.

What You May Conclude

From \(p \Rightarrow q\) (“delayed \(\Rightarrow\) complaint”):

Derived statement	Form	Valid?
No complaint \(\Rightarrow\) not delayed	contrapositive \(\neg q \Rightarrow \neg p\)	✓
Complaint \(\Rightarrow\) delayed	converse \(q \Rightarrow p\)	✗
Not delayed \(\Rightarrow\) no complaint	inverse \(\neg p \Rightarrow \neg q\)	✗

The contrapositive is always equivalent to the original implication: flip and negate. Flipping alone (converse) or negating alone (inverse) breaks the logic.

Closing

Key Takeaways

Mathematical notation is a precise, universal language
Number sets form a hierarchy: \(\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}\)
\(\sum\) and \(\prod\) are compact loops for adding and multiplying
Indices like \(x_{ij}\) carry meaning: always check their definition
Implication is a one-way street: beware the converse error
Negating \(\forall\) gives \(\exists\), and vice versa

That’s it for today.

Any Questions?

Until the Next Session

Work through the Tasks: problems with worked solutions
Check yourself with the Self-Test quiz
Keep the Cheatsheet next to you while practising
Note down anything unclear: we start next session with your questions

The tasks are where the actual learning happens: the lecture only opens the door.

Preview: Session 02: Sets

What sets are and how to describe them
Operations: union, intersection, difference, complement
Venn diagrams as a thinking tool
Why sets are the foundation for functions in Session 03

See you there, and bring your questions!

Literature & Further Reading

These sessions cover the essentials: textbooks offer more depth and practice
Start with Sydsæter et al. (2012) or Jacques (2015); full recommendations on the tutorial’s literature page

Jacques, Ian. 2015. Mathematics for Economics and Business. 8th ed. Always Learning. Pearson.

Sydsæter, Knut, Peter J. Hammond, and Arne Strøm. 2012. Essential Mathematics for Economic Analysis. 4th ed. Pearson.

Mathematics also has formal proof techniques: direct proof, proof by contradiction, and induction. They are beyond the scope of this tutorial, but the textbooks above cover them if you are curious.