Sets, Venn Diagrams & Probability Rules

TAKEAWAYS:

• Sets are collections of distinct objects; operations like union, intersection, and complement are visualized using Venn diagrams.
• Probability measures likelihood using classical ($P(E)=\frac{\text{favorable outcomes}}{\text{total outcomes}}$), empirical ($P(E)=\frac{\text{frequency}}{\text{trials}}$), and conditional ($P(A|B)=\frac{P(A\cap B)}{P(B)}$) rules.
• Cartesian products ($A\times B$) and relations (reflexive, symmetric, transitive) model real-world pairings like user-password systems.
• Probability applications include risk assessment (e.g., loan defaults in banks) and optimization (e.g., Pathao driver routing).
• Exam focus: Solve problems using Venn diagrams for set operations and probability trees for conditional events.

1. Sets and Set Operations

1.1 Definitions and Notations

A set is a well-defined collection of distinct objects, called elements. Notation:

• $A=\{1,2,3\}$ (roster method)
• $B=\{x\mid x\text{ is a prime number}\}$ (set-builder notation)

Universal set (U): Contains all elements under consideration. Empty set ($\varnothing$): Contains no elements.

1.2 Set Operations

Operation	Symbol	Venn Diagram	Example (if $A=\{1,2,3\}$, $B=\{3,4,5\}$)
Union	$A\cup B$		$\{1,2,3,4,5\}$
Intersection	$A\cap B$		$\{3\}$
Complement	$A^{\prime }$ or $A^c$		If $U=\{1,2,3,4,5\}$, $A^{\prime }=\{4,5\}$
Difference	$A-B$		$\{1,2\}$

Worked Example 1: If $U=\{1,2,3,4,5,6\}$, $A=\{1,2,3\}$, $B=\{3,4,5\}$, find:

1. $A\cup B$
2. $A\cap B$
3. $A^{\prime }\cap B$

Solution:

1. $A\cup B=\{1,2,3,4,5\}$
2. $A\cap B=\{3\}$
3. $A^{\prime }=\{4,5,6\}$, so $A^{\prime }\cap B=\{4,5\}$

2. Cartesian Product and Relations

2.1 Cartesian Product ($A\times B$)

The Cartesian product of sets $A$ and $B$ is the set of all ordered pairs $(a,b)$ where $a\in A$ and $b\in B$.

Example: If $A=\{1,2\}$ and $B=\{x,y\}$, then: $$A\times B=\{(1,x),(1,y),(2,x),(2,y)\}$$

2.2 Relations on Sets

A relation $R$ from set $A$ to set $B$ is a subset of $A\times B$. Types of Relations:

• Reflexive: $(a,a)\in R$ for all $a\in A$.
• Symmetric: If $(a,b)\in R$, then $(b,a)\in R$.
• Transitive: If $(a,b)\in R$ and $(b,c)\in R$, then $(a,c)\in R$.

Worked Example 2: Let $A=\{1,2,3\}$ and define $R=\{(1,1),(2,2),(3,3),(1,2)\}$. Check if $R$ is reflexive, symmetric, or transitive.

Solution:

• Reflexive? Yes, because $(1,1),(2,2),(3,3)\in R$.
• Symmetric? No, because $(1,2)\in R$ but $(2,1)\notin R$.
• Transitive? No, because $(1,2)\in R$ and $(2,2)\in R$, but $(1,2)\notin R$ (transitivity fails here).

3. Probability Basics

3.1 Types of Probability

3.2 Probability Rules

1. Addition Rule: $$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$
2. Multiplication Rule (Independent Events): $$P(A\cap B)=P(A)\times P(B)$$
3. Complement Rule: $$P(A^{\prime })=1-P(A)$$

Worked Example 3: A bag contains 3 red and 2 blue balls. Two balls are drawn without replacement. Find:

1. Probability both are red.
2. Probability the first is red and the second is blue.

Solution:

1. $P(\text{Red then Red})=\frac{3}{5}\times \frac{2}{4}=\frac{6}{20}=0.3$.
2. $P(\text{Red then Blue})=\frac{3}{5}\times \frac{2}{4}=0.3$.

## In the Real World

1. eSewa Transactions:

   • Set Theory: Transactions are processed based on sets of valid users and payment methods. For example, $U=\{\text{all users}\}$, $A=\{\text{users with eSewa wallet}\}$, $B=\{\text{users with credit card}\}$. The union $A\cup B$ represents all users who can complete a payment.
   • Probability: The probability of a transaction failing due to insufficient funds is calculated empirically: $P(\text{failure})=\frac{\text{failed transactions}}{\text{total transactions}}$.

2. Pathao Driver Routing:

   • Cartesian Product: Pathao matches drivers ($D$) with passengers ($P$) using $D\times P$ to generate possible routes. A relation $R$ (e.g., "driver $d$ can pick up passenger $p$") is defined based on location and time.
   • Probability: The probability a driver accepts a ride is modeled using conditional probability: $P(\text{accept}|\text{high demand})=\frac{P(\text{accept and high demand})}{P(\text{high demand})}$.

3. Nepal Rastra Bank Loan Approvals:

   • Set Operations: Loan applicants are categorized into sets like $A=\{\text{applicants with collateral}\}$, $B=\{\text{applicants with high credit score}\}$. The intersection $A\cap B$ represents applicants most likely to get approved.
   • Probability: The probability of default is calculated using historical data: $P(\text{default})=\frac{\text{number of defaults}}{\text{total loans}}$.

4. Probability Trees and Conditional Probability

Worked Example 4 (Real-World Tie-In): In Kathmandu traffic, suppose:

• 60% of drivers ignore red lights.
• 30% of those who ignore red lights cause an accident.
• 5% of those who obey red lights cause an accident.

Find the probability a randomly selected driver causes an accident.

Solution:

graph TD
    A["Start"] --> B["Ignore Red Light (60%)"]
    A --> C["Obey Red Light (40%)"]
    B --> D["Accident (30%)"]
    B --> E["No Accident (70%)"]
    C --> F["Accident (5%)"]
    C --> G["No Accident (95%)"]
    D --> H["P(Accident|Ignore) = 0.3"]
    F --> I["P(Accident|Obey) = 0.05"]

Using the Law of Total Probability: $$P(\text{Accident})=P(\text{Accident}|\text{Ignore})\times P(\text{Ignore})+P(\text{Accident}|\text{Obey})\times P(\text{Obey})$$ $$P(\text{Accident})=0.3\times 0.6+0.05\times 0.4=0.18+0.02=0.20\text{ or }20\%.$$

5. Comparing Set Theory and Probability

## Exam Tip

1. For Set Theory:

   • Always draw Venn diagrams for union, intersection, and complement questions.
   • Memorize the number of elements in operations like $|A\cup B|=|A|+|B|-|A\cap B|$.
   • Common Pitfalls: Forgetting to subtract $|A\cap B|$ in union questions.

2. For Probability:

   • Classical vs. Empirical: Use classical for theoretical problems (e.g., dice), empirical for real-world data (e.g., Daraz delays).
   • Conditional Probability: Draw probability trees to visualize dependencies.
   • Key Formula: $P(A|B)=\frac{P(A\cap B)}{P(B)}$ is often tested with word problems.

3. Application Questions:

   • Nepal Context: Expect questions on eSewa transactions, bank loan risks, or traffic accidents (as in Worked Example 4).
   • Global Context: WhatsApp message delivery success rates or YouTube video recommendations (using probability of user engagement).

Final Note: Master Venn diagrams and probability trees—they save time and reduce errors in exams. Practice with real-world data (e.g., Ncell call drop rates or NEPSE stock volatility) to build intuition.