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

Section 01: Mathematical Foundations & Algebra

Author

Dr. Nikolai Heinrichs & Dr. Tobias Vlćek

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: \(|x| = 5\)

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Ready? Let’s see how you did!

Learning Objectives

By the end of this session, you can:

Classify numbers within the hierarchy \(\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}\)
Read and write sets in roster, set-builder, and interval notation
Perform set operations and visualize them with Venn diagrams
Apply the commutative, associative, and distributive properties
Compute percentages and compound growth for business problems
Read basic logical statements and truth tables

Number Systems

The Number Hierarchy

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

\(\mathbb{N}\) = {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

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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 \(\mathbb{N}_0\). Note that the irrationals \(\mathbb{I}\) sit beside \(\mathbb{Q}\) inside \(\mathbb{R}\) — no number is both rational and irrational!

Picturing the Hierarchy

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Every natural number is an integer, every integer is rational — but \(\sqrt{2}\) and \(\pi\) live outside \(\mathbb{Q}\), in the irrationals.

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: \(C = [0, 1] = \{x \in \mathbb{R} : 0 \leq x \leq 1\}\)

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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: \([a, b] = \{x \in \mathbb{R} : a \leq x \leq b\}\) — both endpoints included
Open: \((a, b) = \{x \in \mathbb{R} : a < x < b\}\) — both endpoints excluded
Half-open: \([a, b)\) includes \(a\), excludes \(b\); \((a, b]\) excludes \(a\), includes \(b\)

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Square bracket \([\) means “included”, round parenthesis \((\) means “excluded”. Think of it as the bracket “grabbing” the endpoint!

Intervals on the Number Line

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A filled dot means the endpoint belongs to the interval, an open dot means it does not.

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

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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 for 5 minutes, 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

Working with Numbers & Sets

Venn Diagrams

Venn diagrams are visual representations of sets and their relationships: the rectangle is the universal set \(U\), each circle is a set.

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Venn diagrams help visualize complex set relationships!

Union

Union: \(A \cup B\) — all elements in A or B (or both)

Example: \(\{1,2\} \cup \{2,3\} = \{1,2,3\}\)

Intersection

Intersection: \(A \cap B\) — all elements in A and B

Example: \(\{1,2\} \cap \{2,3\} = \{2\}\)

Difference

Difference: \(A \setminus B\) — elements in A but not in B

Example: \(\{1,2,3\} \setminus \{2,3\} = \{1\}\)

Complement

Complement: \(\bar{A}\) — all elements of \(U\) not in A

Requires a 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

Work in pairs for 5 minutes

Using set notation and a Venn diagram, find:

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

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

Individual Exercise 02

Work individually for 5 minutes

Classify each (list ALL applicable sets):

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

Coffee 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!

Calculator: Entering Fractions

Two ways to enter fractions:

Mixed fraction: \(3\frac{1}{2}\)

Press SHIFT + fraction key (for mixed fraction template)
Enter: 3 → 1 → 2

Improper fraction: \(\frac{7}{2}\)

Press fraction key
Enter: 7 → 2

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Use the arrow keys to move between numerator and denominator fields.

Calculator: Fraction Calculations

Example: Calculate \(\frac{2}{3} + 1\frac{1}{2}\)

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Result: \(\frac{13}{6}\)

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To convert between improper and mixed fractions:

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Press S <-> D for conversion

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So far, nothing is really new, right?

Calculator: Calculation History

A time-saving feature:

Press ↑ (up arrow) to recall your previous calculation
Edit the expression and press = to recalculate
Navigate through multiple previous calculations with ↑ and ↓

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This is very useful when you made a small mistake or want to try different values!

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This only works as long as you don’t click the ON button, change the computation mode, or reset the data!

Calculator: Prime Factorization (FACT)

Factor integers into prime factors:

Enter a positive integer and press =
Press SHIFT + FACT

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Example: Find the prime factorization of 84

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Enter: 84 = then SHIFT + FACT
Result: \(2^2 \times 3 \times 7\)

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Works for integers up to 10 digits. Useful for simplifying fractions!

Calculator: GCD and LCM

Greatest Common Divisor and Least Common Multiple:

Access GCD and LCM functions via ALPHA + * or ALPHA + /.

Function	Syntax	Example
GCD	GCD(a; b)	GCD(24; 36) = 12
LCM	LCM(a; b)	LCM(24; 36) = 72

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Example: ALPHA + * then GCD(48; 180) = gives 12

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Useful for simplifying fractions: \(\frac{48}{180} = \frac{48 \div 12}{180 \div 12} = \frac{4}{15}\)

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	\(2^3 ≠ 3^2\)

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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: \(5000 \times (1.10)^3 = \text{€}6655\)

Compound Growth Visualized

€5000 growing at 10% per year:

Calculator: Percentage Calculations

Using the percent function:

Press SHIFT then ANS to access percentage calculations.

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Example: What is 15% of 240?

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Enter: 240 × 15 SHIFT ANS =

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Result: 36

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Example: Increase 200 by 8%

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Enter: 200 + 200 × 8 SHIFT ANS =

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Result: 216

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You can always just use 0.08 for 8% as well!

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

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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.

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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:

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

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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

Final Assessment

One quick check before you go:

To which number sets does \(-\frac{12}{4}\) belong?

And: what is \(\{1, 2, 3\} \setminus \{2, 3, 4\}\)?

Next Session Preview

Session 01-03: Core Algebra & Exponents

Order of operations and algebraic expressions
Laws of exponents and scientific notation
Absolute value and basic factorization

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Homework Assignment

Complete Tasks 01-02: set operations, number classification, proving/disproving properties — and prepare one presentation problem. Entry quiz next session on today’s material!