Sets, Relations, Functions: Definitions, Operations, and Graphs

TAKEAWAYS

• Sets are collections of distinct objects, and operations like union, intersection, and complement are visualized using Venn diagrams.
• Relations are mappings between two sets, classified as reflexive, symmetric, transitive, or equivalence relations.
• Functions are special relations where each input has exactly one output, with domain and range defining their scope.
• Cartesian products and ordered pairs form the basis for relations and functions, often represented as matrices or graphs.
• Real-world applications include database queries (sets), pathfinding algorithms (relations), and financial modeling (functions).
• Exam focus: Prove properties, solve domain/range problems, and match relations/functions to their graphical representations.

1. Sets: Definitions, Types, and Operations

1.1 What is a Set?

A set is a well-defined collection of distinct objects, called elements. Sets are denoted by capital letters (e.g., $A,B,U$), and elements by lowercase letters (e.g., $a,b,x$).

Example:

• $A=\{1,2,3,4\}$ (finite set)
• $B=\{x\mid x\text{ is an even prime number}\}=\{2\}$ (descriptive notation)
• $\mathbb{R}$ = set of all real numbers (infinite set)

1.2 Types of Sets

1.3 Set Operations

Worked Example 1: Given:

• $U=\{1,2,3,4,5,6\}$
• $A=\{1,2,3\}$
• $B=\{3,4,5\}$

Find:

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

Solution:

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

2. Relations: Ordered Pairs and Properties

2.1 Cartesian Product (A × 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 as Sets of Ordered Pairs

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

Example: Let $A=\{1,2\}$ and $B=\{3,4\}$. Define $R=\{(1,3),(2,4)\}$.

2.3 Properties of Relations

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

1. Reflexive
2. Symmetric
3. Transitive

Solution:

1. Reflexive: Yes, because $(1,1),(2,2),(3,3)\in R$.
2. Symmetric: Yes, because $(1,2)\in R$ implies $(2,1)\in R$, and vice versa.
3. Transitive: Yes, because there are no cases where $(a,b)$ and $(b,c)$ are in $R$ but $(a,c)$ is not.

graph LR
    A["(1,1)"] --> B["(1,2)"]
    B --> C["(2,1)"]
    C --> D["(2,2)"]
    D --> E["(3,3)"]
    title: "Relation R as a Directed Graph"

3. Functions: Definitions and Types

3.1 Definition of a Function

A function $f$ from set $A$ (domain) to set $B$ (codomain) assigns exactly one element in $B$ to each element in $A$. Notation: $f:A\to B$.

Example: $f(x)=2x+1$ is a function from $\mathbb{R}$ to $\mathbb{R}$.

3.2 Types of Functions

3.3 Domain and Range

• Domain: All possible input values ($x$-values).
• Range: All possible output values ($y$-values).

Worked Example 3: Find the domain and range of $f(x)=\sqrt{2-x-x^2}$.

Solution:

1. Domain: The expression under the square root must be non-negative: $$2-x-x^2\ge 0$$ $$x^2+x-2\le 0$$ Solve $x^2+x-2=0$: $$x=\frac{-1\pm \sqrt{1+8}}{2}=\frac{-1\pm 3}{2}$$ Roots: $x=1$ and $x=-2$. The parabola opens upwards, so the inequality holds between the roots: $$\text{Domain}=[-2,1]$$

2. Range: The maximum value of $\sqrt{2-x-x^2}$ occurs at the vertex of the parabola $y=-x^2-x+2$. Vertex $x$-coordinate: $x=-\frac{b}{2a}=-\frac{-1}{-2}=-0.5$. $$y=-(-0.5{)}^2-(-0.5)+2=-0.25+0.5+2=2.25$$ So, the maximum value of the square root is $\sqrt{2.25}=1.5$. At $x=-2$ and $x=1$, $y=0$. $$\text{Range}=[0,1.5]$$

## In the Real World

1. eSewa and Khalti (Digital Payments):

   • Sets: Transactions are processed as sets of unique IDs (e.g., user IDs, merchant IDs).
   • Relations: The "pays-to" relation maps users to merchants (e.g., $R=\{(user1,merchantA),(user2,merchantB)\}$).
   • Functions: The payment amount is a function of the user’s input (e.g., $f(user1)=500$).

2. Pathao (Ride-Hailing):

   • Sets: Drivers and passengers are two distinct sets.
   • Relations: A matching algorithm creates a relation $R$ where $(driver,passenger)$ pairs are formed based on location and time.
   • Functions: The fare is a function of distance and time (e.g., $fare=f(distance,time)$).

3. NTC (Network Traffic Routing):

   • Relations: Network nodes are connected via directed edges representing data flow (e.g., $R=\{(node1,node2),(node2,node3)\}$).
   • Functions: The delay in data transmission is a function of the path taken (e.g., $delay=f(path)$).

4. Bank Loan Interest (Geometric Progression):

   • Example: A loan of Rs. 10,000 at 10% annual interest compounded yearly.
     • After 1 year: $10,000\times 1.10=11,000$
     • After 2 years: $11,000\times 1.10=12,100$
     • This forms a geometric sequence where each term is multiplied by 1.10.

## Exam Tip

1. For Sets:

   • Always draw Venn diagrams for union, intersection, and complement questions.
   • Remember: $n(A\cup B)=n(A)+n(B)-n(A\cap B)$.

2. For Relations:

   • Check properties (reflexive, symmetric, transitive) systematically.
   • Use matrices to represent relations if asked to prove equivalence.

3. For Functions:

   • Domain: Solve inequalities for square roots, denominators, and logarithms.
   • Range: Find the maximum/minimum values of the function (e.g., vertex of a parabola).
   • Bijective Proofs: Show both injective (unique outputs) and surjective (covers codomain).

4. Common Mistakes:

   • Forgetting to check the domain for square roots or denominators.
   • Misapplying set operations (e.g., confusing $A\setminus B$ with $A\cap B^{\prime }$).
   • Not verifying all conditions for relation properties.

Visual Summary: