Session 01-02 - Language, Sets, and Number Systems

Section 01: Mathematical Foundations & Algebra

Dr. Nikolai Heinrichs & Dr. Tobias Vlćek

Your Confidence

Your Confidence in Topics

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Please make sure the 'data-folder' directory and 'data.csv' file exist.

That’s a good foundation for this course! :)

Entry Quiz

Quick Morning Check

Complete on paper - we’ll review together

Calculate: \(\frac{2}{3} + \frac{3}{4}\)
Simplify: \(x^2 \cdot x^3\)
What is 15% of 240?
Solve: \(|x| = 5\)

Ready? Let’s see how you did!

Number Systems

The Number Hierarchy

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

\(\mathbb{N}\) = {1, 2, 3, …} Natural numbers
\(\mathbb{Z}\) = {…, -2, -1, 0, 1, 2, …} Integers
\(\mathbb{Q}\) = \(\{\frac{p}{q} : p, q \in \mathbb{Z}, q \neq 0\}\) Rationals
\(\mathbb{I}\) = Numbers that cannot be expressed as fractions Irrationals
\(\mathbb{R}\) = All points on the number line Reals

Some books include 0 in \(\mathbb{N}\), denoted \(\mathbb{N}_0\). For this course, we define \(\mathbb{N} = \{1, 2, 3, ...\}\). The set including zero is denoted \(\mathbb{N}_0\).

Set Theory Basics

What is a Set?

A set is a well-defined collection of distinct objects.

Notation:

Roster: \(A = \{1, 2, 3, 4, 5\}\)
Set-builder: \(B = \{x \in \mathbb{N} : x < 6\}\)
Interval: \(C = [0, 1] = \{x \in \mathbb{R} : 0 \leq x \leq 1\}\)

Common Mistake

\(\{1, 2, 2, 3\} = \{1, 2, 3\}\) — Sets contain only distinct elements!

Interval Notation

Interval notation uses brackets and parentheses to show whether endpoints are included or excluded:

Closed Intervals: Both endpoints included

\([a, b] = \{x \in \mathbb{R} : a \leq x \leq b\}\)
Example: \([1, 5]\) includes both 1 and 5

Open Intervals: Both endpoints excluded

\((a, b) = \{x \in \mathbb{R} : a < x < b\}\)
Example: \((1, 5)\) excludes both 1 and 5

Mixed Intervals: One endpoint included, one excluded

\([a, b) = \{x \in \mathbb{R} : a \leq x < b\}\) (includes \(a\), excludes \(b\))
\((a, b] = \{x \in \mathbb{R} : a < x \leq b\}\) (excludes \(a\), includes \(b\))

Mathematical Language

Why Mathematical Notation?

Mathematics is a universal language that allows us to:

Express complex ideas precisely
Communicate without ambiguity
Solve problems systematically
Build logical arguments

Clear notation prevents costly misunderstandings in contracts, financial models, and data analysis.

Essential Symbols - Sets

Symbol	Meaning	Example
\(\in\)	Element of	\(3 \in \mathbb{N}\)
\(\notin\)	Not element of	\(\pi \notin \mathbb{Q}\)
\(\subset\)	Subset	\(\mathbb{N} \subset \mathbb{Z}\)
\(\subseteq\)	Subset or equal	\(A \subseteq A\)
\(\cup\)	Union	\(A \cup B\)
\(\cap\)	Intersection	\(A \cap B\)
\(\emptyset\)	Empty set	\(A \cap B = \emptyset\)

Essential Symbols - Logic

Symbol	Meaning	Example
\(\forall\)	For all	\(\forall x \in \mathbb{R}: x^2 \geq 0\)
\(\exists\)	There exists	\(\exists x \in \mathbb{Z}: x < 0\)
\(\Rightarrow\)	Implies	\(x = 2 \Rightarrow x^2 = 4\)
\(\Leftrightarrow\)	If and only if	\(x^2 = 4 \Leftrightarrow x = \pm 2\)
\(\neg\)	Not	\(\neg (x > 0)\) means \(x \leq 0\)
\(\wedge\)	And	\(p \wedge q\)
\(\vee\)	Or	\(p \vee q\)

Let’s Practice Reading

Translate to English:

\(\forall x \in \mathbb{R}: x + 0 = x\)
- “For all real numbers x, x plus zero equals x”
\(\exists n \in \mathbb{N}: n > 1000000\)
- “There exists a natural number n greater than one million”
\(x \in A \cap B \Rightarrow x \in A\)
- “If x is in the intersection of A and B, then x is in A”

Individual Exercise 01

Work individually first, then compare with neighbors

Express the following in set notation (choose which notation to use):

The set of all even natural numbers
The set of all real numbers between -1 and 1 (inclusive)
The set of all integers divisible by 3

Break - 10 Minutes

Venn Diagrams

Venn diagrams are visual representations of sets and their relationships.

Components:

Rectangle: Universal set U
Circles: Individual sets
Overlaps: Intersections
Outside circles: Complements

Example:

U = {1, 2, 3, 4, 5, 6, 7, 8}
A = {1, 2, 3, 4}
B = {3, 4, 5, 6}
A ∩ B = {3, 4} (overlap)

Venn diagrams help visualize complex set relationships!

Working with Numbers & Sets

Set Operations

Union: \(A \cup B\)

All elements in A or B
Example: \(\{1,2\} \cup \{2,3\} = \{1,2,3\}\)

Intersection: \(A \cap B\)

All elements in A and B
Example: \(\{1,2\} \cap \{2,3\} = \{2\}\)

Difference: \(A \setminus B\)

Elements in A but not in B
Example: \(\{1,2,3\} \setminus \{2,3\} = \{1\}\)

Complement: \(\bar{A}\)

All elements not in A
Requires universal set U

Example: Employee Skills

A company tracks employee skills:

\(P\) = {Python, Java, SQL, R}
\(D\) = {SQL, Excel, Tableau, R}

Lets work together and find the following:

\(P \cup D\) (All skills available)
\(P \cap D\) (Versatile skills)
\(P \setminus D\) (Skills unique to Programmers)
\(D \setminus P\) (Skills unique to Data Analysts)

Classify \(\frac{22}{7}\)

Can be written as \(\frac{p}{q}\) ✓
Therefore: \(\frac{22}{7} \in \mathbb{Q}\)
Also: \(\frac{22}{7} \in \mathbb{R}\)
But: \(\frac{22}{7} \notin \mathbb{Z}\) (≈ 3.14…)

Classify \(\sqrt{9}\)

\(\sqrt{9} = 3\)
Therefore: \(3 \in \mathbb{N}\)
Also: \(3 \in \mathbb{Z}, \mathbb{Q}, \mathbb{R}\)

Be careful! \(\pi \approx \frac{22}{7}\) but \(\pi \neq \frac{22}{7}\)

\(\pi\) is irrational!
No fraction exactly equals \(\pi\)

What about the following?

Is \(0.\overline{45}\) rational?

Method
General Rule

Let \(x = 0.454545...\)

Multiply by 100: \(100x = 45.454545...\)
Subtract original: \(100x - x = 45.454545... - 0.454545...\)
Simplify: \(99x = 45\)
Solve: \(x = \frac{45}{99} = \frac{5}{11}\)

Yes! It’s rational.

For \(0.\overline{abc}\) with n repeating digits:

Multiply by \(10^n\)
Subtract original
Solve for x

Group Exercise 01

How is this class structured?

Using set notation and a Venn diagram, find:

How many study at least one subject?
How many study only Mathematics?

We are going to do this one together. Any suggestions on how to start?

Individual Exercise 02

Number Classification

Classify each (list ALL applicable sets):

\(-\frac{8}{2}\)
\(\sqrt{7}\)
\(0.\overline{55}\)
\(\pi + 1\)

Break - 15 Minutes

Properties of Operations

The Big Three Properties

Commutative: Order doesn’t matter
- \(a + b = b + a\)
- \(a \times b = b \times a\)
Associative: Grouping doesn’t matter
- \((a + b) + c = a + (b + c)\)
- \((a \times b) \times c = a \times (b \times c)\)
Distributive: Multiplication distributes over addition
- \(a(b + c) = ab + ac\)

Business Application

Revenue Calculation

A store sells 3 products:

Product A: 50 units at €20 each
Product B: 30 units at €20 each
Product C: 40 units at €20 each

Method 1
Method 2
Common Mistakes

\((50 × 20) + (30 × 20) + (40 × 20)\)

\(20 × (50 + 30 + 40)\) Distributive property!

Remember: multiplication before addition!

Which Operations Commute?

Operation	Commutative?	Example
Addition	✓ Yes	\(3 + 5 = 5 + 3 = 8\)
Multiplication	✓ Yes	\(3 × 5 = 5 × 3 = 15\)
Subtraction	✗ No	\(5 − 3 ≠ 3 − 5\)
Division	✗ No	\(6 ÷ 2 ≠ 2 ÷ 6\)
Exponentiation	✗ No	\(2^3 ≠ 3^2\)

Common Mistake

Students somtimes assume all operations commute!

Percentage Calculations

Basic Percentage
Percentage Change
Compound Growth

Finding x% of a number:

\(\text{Result} = \frac{x}{100} \times \text{Base}\)

Example: 15% of 240
Solution: 15% of 240 = \(\frac{15}{100} \times 240 = 36\)

Finding the change:

\(\text{Change \%} = \frac{\text{New} - \text{Old}}{\text{Old}} \times 100\%\)

Example: From €5000 to €5500
Solution: \(\frac{5500-5000}{5000} \times 100\% = 10\%\)

Multiple periods:

\(\text{Final} = \text{Initial} \times (1 + r)^n\)

Where r = rate (as decimal), n = periods
Example: €5000 at 10% for 3 years
Solution: \(5000 \times (1.10)^3 = €6655\)

Mathematical Logic Basics

Truth Tables for propositions \(p\) and \(q\)

\(p\)	\(q\)	\(p \wedge q\) (and)	\(p \vee q\) (or)	\(p \Rightarrow q\) (imp.)
T	T	T	T	T
T	F	F	T	F
F	T	F	T	T
F	F	F	F	T

Understanding Implication (\(p \Rightarrow q\))

Think of it as a promise: “If it is raining (\(p\)), then I will carry an umbrella (\(q\)).” The only way the promise is broken (the statement is False) is if it’s raining (\(p\)=T) but I don’t have my umbrella (\(q\)=F).

Soft Introduction to Proofs

A proof is a logical argument that shows a statement is true.

The goal is to move from what we know (assumptions) to what we want to show (conclusion) using small, logical steps.
A direct proof is the most common form:
Assume \(p\) is true and show that \(q\) must logically follow.

No need to worry about this topic too much! We just cover the absolute basics here just for you to know what a proof is.

Practice Time

Group Exercise 02

Working in pairs, determine if these statements are true or false:

\(\mathbb{Z} \subset \mathbb{Q}\)
\(\sqrt{4} \in \mathbb{N}\)
\(0.333... \in \mathbb{Q}\)
\(\{1, 2\} \subset \{1, 2, 3\}\)
\(\emptyset \subset \mathbb{N}\)

Take 5 minutes, then we’ll discuss!

Wrap-up

Key Takeaways

Mathematical notation is precise and universal
Venn diagrams visualize set relationships
Number systems form a hierarchy: \(\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}\)
Repeating decimals are rational numbers
Operations have specific properties we can exploit
Percentages and compound growth are essential for business
Logic helps us reason systematically

For Next Time

Homework: Complete Tasks 01-02

Focus on:

Set operations practice
Number classification
Proving/disproving properties
One presentation problem

Entry quiz next session on today’s material!