Session 01 - Notation & Logic
Mathematics for Master Students
Introduction
About Me
- Field: Optimizing 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 |
|---|---|---|
| 1 | Notation & Logic | TBD |
| 2 | Sets | TBD |
| 3 | Functions | TBD |
| 4 | Differentiation | TBD |
| 5 | Single Variable Optimization | TBD |
| 6 | Systems of Linear Equations | TBD |
| 7 | Mock Exam (voluntary!) | TBD |
. . .
Seven sessions of 90 minutes each: 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
Math Topics in Your KLU Courses I
Semester 1
| Course | Notation | Logic | Sets | Functions | Differentiation | Linear Systems | Optimization |
|---|---|---|---|---|---|---|---|
| Analytical Methods in Log. and SCM | ✓ | ✓ | ✓ | ✓ | |||
| Business Logistics in SCM | ✓ | ✓ | ✓ | ✓ | |||
| Business Analytics and Econometrics | ✓ | ✓ | ✓ | ✓ | ✓ | ||
| International Economics | ✓ | ✓ | ✓ | ✓ | ✓ | ||
| Strategic Issues in SCM | ✓ | ✓ | ✓ |
Math Topics in Your KLU Courses II
Semester 2
| Course | Notation | Logic | Sets | Functions | Differentiation | Linear Systems | Optimization |
|---|---|---|---|---|---|---|---|
| Data Science | ✓ | ✓ | ✓ | ✓ | ✓ | ||
| Transportation & Distribution | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |
| Global Trends in HR | ✓ | ✓ | |||||
| Economics of Digital Transformation | ✓ | ✓ | |||||
| Economics of Business Strategy | ✓ | ✓ | ✓ | ✓ | ✓ | ||
| Business Strategy and 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 self-diagnosis
. . .
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 beyondsimulations.github.io/Mathematics-Master
- 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 practiced 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
. . .
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?
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
. . .
\(\pi \approx \frac{22}{7}\), 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.
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: \(\sum_{t=1}^{12} (c_t + f) = \sum_{t=1}^{12} c_t + 12 \cdot f\)
. . .
Read \(\sum\) as a small loop: “for each \(t\) from 1 to 12, add up …”.
Your Turn: Evaluate a Sum
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.
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.
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: \(\neg(\neg p) = p\)
. . .
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 math 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 |
. . .
“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.
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 |
. . .
\(\Rightarrow\) 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
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
Negate: “All shipments arrive on time.”
. . .
“At least one shipment does not arrive on time.”
. . .
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 \(p\)
. . .
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
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 practicing
- 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
- Sydsaeter & Hammond: Essential Mathematics for Economic Analysis
- Jacques: Mathematics for Economics and Business
- Full recommendations on the tutorial’s literature page
. . .
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.