Sets and Set Operations

🎯 Learning Objectives

1. Classify numbers into $\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}$ and understand their hierarchy
2. Define a set using roster form and set-builder notation
3. Perform set operations — union, intersection, complement, difference, symmetric difference
4. Compute cardinalities using the inclusion–exclusion principle
5. Construct power sets and determine the number of subsets
6. Interpret set relationships using Venn diagrams

📋 Prerequisites

• Basic arithmetic — addition, subtraction, multiplication, division, modulo (remainder)
• Logical connectives — AND (∧), OR (∨), NOT (¬) — first encountered in primary school logic

📖 Core Content

1.1 Number Systems — The Foundation

1.1.1 Intuition: What Are Numbers?

• Counting → Natural numbers
• Debt / temperature → Integers
• Sharing a pizza → Rational numbers
• The diagonal of a square → Irrational numbers (Real numbers)
• Square roots of negatives → Complex numbers

💡 Why this matters: In data science, the type of number determines what operations are valid. You cannot average ZIP codes (they are labels, not quantities). You cannot take the log of a negative number (unless you use complex numbers). Understanding number systems prevents subtle bugs in data pipelines.

1.1.2 Formal Definitions

1.1.3 Important Properties

• Integers are discrete. Between 2 and 3 there is no other integer. You can list them in order: $\ldots, -2, -1, 0, 1, 2, \ldots$
• Rationals are dense. Between any two rational numbers, there is always another rational. For example, between $\frac{1}{3}$ and $\frac{1}{2}$, the average $\frac{1/3 + 1/2}{2} = \frac{5}{12}$ lies between them.
• Reals are dense too. Between any two reals, there is another real. But reals also include irrationals like $\sqrt{2}$ and $\pi$, which cannot be expressed as $\frac{p}{q}$. Prime Numbers: A prime $p$ is a natural number $> 1$ whose only factors are $1$ and $p$ itself. Examples: $2, 3, 5, 7, 11, 13, \ldots$ Every integer can be written uniquely as a product of primes (prime factorization). Example: $84 = 2^2 \times 3 \times 7$.

🔍 Edge case: $1$ is not prime — it has only one factor (itself). There is no largest prime (proved by Euclid).

1.1.4 Worked Examples

Proof

Proof by contradiction. Assume $\sqrt{2} = \frac{p}{q}$ where $p, q \in \mathbb{Z}$, $q \neq 0$, and $\gcd(p,q) = 1$ (fraction is in lowest terms).

Squaring both sides: $2 = \frac{p^2}{q^2} \implies p^2 = 2q^2$

This means $p^2$ is even, so $p$ must be even. Write $p = 2k$.

Then $(2k)^2 = 2q^2 \implies 4k^2 = 2q^2 \implies q^2 = 2k^2$

So $q^2$ is even, hence $q$ is even. But then $p$ and $q$ are both even — they share a factor of 2, contradicting $\gcd(p,q) = 1$.

Thus $\sqrt{2}$ cannot be expressed as $\frac{p}{q}$; it is irrational. ∎

1.2 The Concept of a Set

1.2.1 Intuition: What Is a Set?

💡 Why this matters: Sets are the mathematical language for "grouping things." When we talk about "all users in the database" or "all products whose price > $100," we're implicitly working with sets. SQL's UNION, INTERSECT, and EXCEPT are direct translations of set operations.

1.2.2 Formal Definition

• Roster form: List elements inside curly braces. $A = \{1, 2, 3, 4\}$
• Set-builder form: Describe the property that elements satisfy. $A = \{x \mid x \in \mathbb{N},\ x \geq 1,\ x \leq 4\}$ Read as: "the set of all $x$ such that $x$ is a natural number, $x \geq 1$, and $x \leq 4$"

• Unordered: $\{1,2,3\} = \{3,1,2\}$
• No duplicates: $\{1,1,2\} = \{1,2\}$
• Empty set $\emptyset = \{\}$ is a subset of every set.
• Cardinality $|A|$ is the number of distinct elements in $A$.

1.2.3 Set Comprehension Pattern

• Generate: Draw elements from an existing set
• Filter: Keep only those satisfying a condition
• Transform: (Optional) Apply a function to each kept element Examples:
• Even integers: $\{x \mid x \in \mathbb{Z},\ x \bmod 2 = 0\}$
• Perfect squares: $\{n^2 \mid n \in \mathbb{N}\}$
• Rational numbers in lowest terms: $\{p/q \mid p,q \in \mathbb{Z},\ q \neq 0,\ \gcd(p,q)=1\}$

1.2.4 Power Set

1.2.5 Worked Examples

⚠️ Edge case: $\{3,4\}$ is a single element of $A$, so its cardinality counts as 1, not 2. This is a common mistake.

1.3 Set Operations

1.3.1 Intuition: Combining Sets

• Union ($A \cup B$): Users who liked X OR Y (or both)
• Intersection ($A \cap B$): Users who liked both X AND Y
• Difference ($A \setminus B$): Users who liked X but NOT Y
• Complement ($A^c$ or $\overline{A}$): Users who did NOT like X (Diagram)

🔍 Key relationship: $A \triangle B = (A \cup B) \setminus (A \cap B)$

1.3.2 Properties of Set Operations

• $(A \cup B)^c = A^c \cap B^c$
• $(A \cap B)^c = A^c \cup B^c$

💡 Memory Aid: "Break the parenthesis, flip the operator." Union becomes intersection, intersection becomes union, complement each term.

1.3.3 Cardinality and Inclusion–Exclusion

1.3.4 Worked Examples

⚠️ Edge case: Inclusion–exclusion can exceed the total population, revealing measurement error.

📐 Key Formulas — Summary Table

⚠️ Common Pitfalls

Pitfall 1: Confusing $\emptyset$ and $\{\emptyset\}$

• $\emptyset \subseteq \{\emptyset\}$ (true — empty set is subset of every set)
• $\emptyset \in \{\emptyset\}$ (true — the empty set is an element of $\{\emptyset\}$)
• $\{\emptyset\} \subseteq \emptyset$ (false — $\{\emptyset\}$ has an element but $\emptyset$ has none)

Pitfall 2: Double-Counting in Set Cardinality Problems

Pitfall 3: Confusing $\subset$ (proper subset) with $\subseteq$ (subset)

• $A \subseteq B$ means every element of $A$ is also in $B$ (possibly equal sets)
• $A \subset B$ means $A \subseteq B$ and $A \neq B$ (strict containment) Example: $\{1,2\} \subseteq \{1,2\}$ is true; $\{1,2\} \subset \{1,2\}$ is false.

Pitfall 4: Misapplying De Morgan's Laws

📝 Practice Questions

Q1: Let $A = \{x \mid x \in \mathbb{Z},\ x^2 \leq 16\}$. List $A$ in roster form.

Strategy Hint: Find all integers whose square is ≤ 16.

Step 1: Integers whose square ≤ 16: $(-4)^2 = 16$, $(-3)^2 = 9$, $(-2)^2 = 4$, $(-1)^2 = 1$, $0^2 = 0$, $1^2 = 1$, $2^2 = 4$, $3^2 = 9$, $4^2 = 16$.

Step 2: These are $-4, -3, -2, -1, 0, 1, 2, 3, 4$.

$\boxed{A = \{-4, -3, -2, -1, 0, 1, 2, 3, 4\}}$ Q2: If $|A| = 5$, how many proper subsets does $A$ have?

Strategy Hint: Proper subsets exclude the set itself.

Step 1: Total subsets = $2^5 = 32$ Step 2: Proper subsets = $32 - 1 = 31$

$\boxed{31}$ Q3: Let $U = \{1,2,3,4,5,6\}$, $A = \{1,2,3\}$, $B = \{2,4,6\}$. Find $(A \cup B)^c$.

Strategy Hint: Find the union first, then complement relative to U.

Step 1: $A \cup B = \{1,2,3,4,6\}$ Step 2: $(A \cup B)^c = U \setminus (A \cup B) = \{5\}$

$\boxed{\{5\}}$ Q4: In a group of 60 people, 25 like tea, 30 like coffee, and 10 like both. How many like neither?

Strategy Hint: Use inclusion–exclusion.

Step 1: $|T| = 25$, $|C| = 30$, $|T \cap C| = 10$ Step 2: $|T \cup C| = 25 + 30 - 10 = 45$ Step 3: $|(T \cup C)^c| = 60 - 45 = 15$

$\boxed{15 \text{ people}}$ Q5: Write the set of all perfect squares less than 50 in set-builder notation.

Strategy Hint: The transform can be applied to the generated variable.

$\boxed{\{n^2 \mid n \in \mathbb{N},\ n^2 < 50\}}$

Or in roster form: $\{0, 1, 4, 9, 16, 25, 36, 49\}$. Q6: How many subsets of $\{a, b, c, d\}$ contain the element $a$?

Strategy Hint: $a$ is fixed as "in". The remaining 3 elements each have 2 choices.

Step 1: $a$ is forced to be in every such subset. Step 2: For each of $b, c, d$: in or out → $2^3 = 8$ choices.

$\boxed{8}$ Q7: If $A \subseteq B$ and $B \subseteq C$, prove $A \subseteq C$.

Strategy Hint: Use the definition of subset directly.

Proof: Let $x \in A$. Since $A \subseteq B$, $x \in B$. Since $B \subseteq C$, $x \in C$. Therefore every element of $A$ is in $C$, so $A \subseteq C$. ∎ Q8: Verify De Morgan's law: $(A \cap B)^c = A^c \cup B^c$ for $U = \{1,\ldots,10\}$, $A = \{2,4,6,8\}$, $B = \{1,2,3,4,5\}$.

Strategy Hint: Compute both sides independently and compare.

Step 1: $A \cap B = \{2,4\}$ Step 2: $(A \cap B)^c = \{1,3,5,6,7,8,9,10\}$ Step 3: $A^c = \{1,3,5,7,9,10\}$, $B^c = \{6,7,8,9,10\}$ Step 4: $A^c \cup B^c = \{1,3,5,6,7,8,9,10\}$ Step 5: Both sides equal $\{1,3,5,6,7,8,9,10\}$. ✓

$\boxed{\text{Verified}}$ Q9: 70 students: 40 play cricket, 35 play football, 20 play both. How many play exactly one sport?

Strategy Hint: Exactly one = (Cricket only) + (Football only).

Step 1: Cricket only = $40 - 20 = 20$ Step 2: Football only = $35 - 20 = 15$ Step 3: Exactly one = $20 + 15 = 35$

$\boxed{35 \text{ students}}$ Q10: Show that $\sqrt{3}$ is irrational.

Strategy Hint: Follow the same contradiction proof as $\sqrt{2}$.

Proof: Assume $\sqrt{3} = p/q$ in lowest terms. Then $p^2 = 3q^2$. So $p^2$ is divisible by 3 ⇒ $p$ is divisible by 3 (since if a prime divides $p^2$, it divides $p$). Write $p = 3k$. Then $9k^2 = 3q^2 \Rightarrow q^2 = 3k^2$, so $q$ is also divisible by 3. Contradiction (gcd would be at least 3). ∎ Q11: Let $A = \{\emptyset, \{\emptyset\}\}$. Find $\mathcal{P}(A)$.

Strategy Hint: $A$ has two elements: $\emptyset$ and $\{\emptyset\}$.

Step 1: $|A| = 2$, so $|\mathcal{P}(A)| = 4$. Step 2: $\mathcal{P}(A) = \{\emptyset,\ \{\emptyset\},\ \{\{\emptyset\}\},\ \{\emptyset, \{\emptyset\}\}\}$

$\boxed{\{\emptyset,\ \{\emptyset\},\ \{\{\emptyset\}\},\ \{\emptyset, \{\emptyset\}\}\}}$

Note: Don't confuse $\emptyset$ (empty set), $\{\emptyset\}$ (set containing empty set), and $\{\{\emptyset\}\}$ (set containing the set containing empty set)! Q12: Among 120 students: 65 enrolled in ML, 55 in AI, 45 in DS, 25 in ML & AI, 20 in ML & DS, 15 in AI & DS, 10 in all three. How many enrolled in at least one?

Strategy Hint: Three-set inclusion–exclusion.

Step 1: $|ML \cup AI \cup DS| = 65 + 55 + 45 - 25 - 20 - 15 + 10$ $= 165 - 60 + 10 = 115$

$\boxed{115 \text{ students}}$

🔗 Cross-References

• Next topic: Relations and Their Properties
• Related: Functions and Their Types — functions are special relations
• Across courses: BSMA1003 Maths 2 (sets form the foundation of linear algebra); BSMA1002 Stats 1 (sample spaces are sets) Join Discord Next1.2 Relations