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

Section 01: Mathematical Foundations & Algebra

Author

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)

Classifying Numbers

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?

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

\((50 × 20) + (30 × 20) + (40 × 20)\)

\(20 × (50 + 30 + 40)\) 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

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!