What is a Set?
- Notation of Sets
- Types of Sets
- Set Operations
What is a Relation?
- Cartesian Product
- Representation of Relations
- Types of Relations
What is a Function?
- Definition of a Function
- Examples of Functions
- Types of Functions
- Graph of a Function
Differences Between Relations and Functions
Applications
FAQs on Sets, Relations And Functions

Sets, Relations And Functions

Sets, relations, and functions form the foundational concepts of mathematics. These concepts are not just theoretical; they have practical applications in computer science, physics, economics, and many other fields. This article provides a detailed explanation of these topics using simple language and examples to help you grasp them easily.

What is a Set?

A set is a collection of well-defined and distinct objects. These objects are called elements or members of the set. Sets are usually represented by capital letters such as $A$ $A$ , $B$ $B$ , or $C$ $C$ .

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Notation of Sets

Sets are written in curly braces, $\{\}$ ${$ $}$ , and the elements are separated by commas. For example:

$A = \{1, 2, 3, 4, 5\}$ $A$ $=$ ${$ $1$ $,$ $2$ $,$ $3$ $,$ $4$ $,$ $5$ $}$ represents a set of numbers.
$B = \{\text{apple}, \text{banana}, \text{cherry}\}$ $B$ $=$ ${$ $apple$ $,$ $banana$ $,$ $cherry$ $}$ represents a set of fruits.

Types of Sets

Empty Set (Null Set): A set with no elements.
Example: $\emptyset = \{\}$ $\emptyset$ $=$ ${$ $}$
Finite Set: A set with a limited number of elements.
Example: $A = \{1, 2, 3\}$ $A$ $=$ ${$ $1$ $,$ $2$ $,$ $3$ $}$
Infinite Set: A set with an infinite number of elements.
Example: $B = \{1, 2, 3, 4, \dots\}$ $B$ $=$ ${$ $1$ $,$ $2$ $,$ $3$ $,$ $4$ $,$ $\dots$ $}$
Subset: If every element of set $A$ $A$ is also an element of set $B$ $B$ , then $A$ $A$ is a subset of $B$ $B$ .
Example: $A = \{1, 2\}$ $A$ $=$ ${$ $1$ $,$ $2$ $}$ is a subset of $B = \{1, 2, 3\}$ $B$ $=$ ${$ $1$ $,$ $2$ $,$ $3$ $}$ .
Universal Set: A set that contains all possible elements under consideration.
Example: For all integers, the universal set can be $U = \{\dots, -2, -1, 0, 1, 2, \dots\}$ $U$ $=$ ${$ $\dots$ $,$ $-2$ $,$ $-1$ $,$ $0$ $,$ $1$ $,$ $2$ $,$ $\dots$ $}$ .
Power Set: A set of all subsets of a given set.
Example: If $A = \{1, 2\}$ $A$ $=$ ${$ $1$ $,$ $2$ $}$ , then $\text{Power Set of A} = \{\emptyset, \{1\}, \{2\}, \{1, 2\}\}$ $Power Set of A$ $=$ ${$ $\emptyset$ $,$ ${$ $1$ $}$ $,$ ${$ $2$ $}$ $,$ ${$ $1$ $,$ $2$ $}}$ .

Set Operations

Union ( $\cup$ $\cup$ ): Combines all elements of two sets without repetition.
Example: If $A = \{1, 2\}$ $A$ $=$ ${$ $1$ $,$ $2$ $}$ and $B = \{2, 3\}$ $B$ $=$ ${$ $2$ $,$ $3$ $}$ , then $A \cup B = \{1, 2, 3\}$ $A$ $\cup$ $B$ $=$ ${$ $1$ $,$ $2$ $,$ $3$ $}$ .
Intersection ( $\cap$ $\cap$ ): Includes only the common elements of two sets.
Example: $A \cap B = \{2\}$ $A$ $\cap$ $B$ $=$ ${$ $2$ $}$ .
Difference ( $A - B$ $A$ $-$ $B$ ): Contains elements of $A$ $A$ that are not in $B$ $B$ .
Example: $A - B = \{1\}$ $A$ $-$ $B$ $=$ ${$ $1$ $}$ .
Complement: Contains all elements not in the given set.
Example: If $U = \{1, 2, 3, 4\}$ $U$ $=$ ${$ $1$ $,$ $2$ $,$ $3$ $,$ $4$ $}$ and $A = \{1, 2\}$ $A$ $=$ ${$ $1$ $,$ $2$ $}$ , then $A' = \{3, 4\}$ $A$ $'$ $=$ ${$ $3$ $,$ $4$ $}$ .

What is a Relation?

A relation is a way to describe a relationship between two sets. It is defined as a subset of the Cartesian product of two sets.

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Cartesian Product

If $A$ $A$ and $B$ $B$ are two sets, the Cartesian product $A \times B$ $A$ $\times$ $B$ is the set of all ordered pairs where the first element comes from $A$ $A$ and the second from $B$ $B$ .

Example:

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

$A$ $\times$ $B$ $=$ ${($ $1$ $,$ $x$ $)$ $,$ $($ $1$ $,$ $y$ $)$ $,$ $($ $2$ $,$ $x$ $)$ $,$ $($ $2$ $,$ $y$ $)}$ .

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Representation of Relations

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

Example:

If $A = \{1, 2\}$ $A$ $=$ ${$ $1$ $,$ $2$ $}$ and $B = \{3, 4\}$ $B$ $=$ ${$ $3$ $,$ $4$ $}$ , a relation $R$ $R$ could be $R = \{(1, 3), (2, 4)\}$ $R$ $=$ ${($ $1$ $,$ $3$ $)$ $,$ $($ $2$ $,$ $4$ $)}$ .

Types of Relations

Empty Relation: No pair in the relation.
Example: $R = \emptyset$ $R$ $=$ $\emptyset$
Universal Relation: Includes all possible pairs in $A \times B$ $A$ $\times$ $B$ .
Reflexive Relation: Every element is related to itself.
Example: $(a, a) \in R$ $($ $a$ $,$ $a$ $)$ $\in$ $R$ for all $a \in A$ $a$ $\in$ $A$ .
Symmetric Relation: If $(a, b) \in R$ $($ $a$ $,$ $b$ $)$ $\in$ $R$ , then $(b, a) \in R$ $($ $b$ $,$ $a$ $)$ $\in$ $R$ .
Example: $R = \{(1, 2), (2, 1)\}$ $R$ $=$ ${($ $1$ $,$ $2$ $)$ $,$ $($ $2$ $,$ $1$ $)}$ .
Transitive Relation: If $(a, b) \in R$ $($ $a$ $,$ $b$ $)$ $\in$ $R$ and $(b, c) \in R$ $($ $b$ $,$ $c$ $)$ $\in$ $R$ , then $(a, c) \in R$ $($ $a$ $,$ $c$ $)$ $\in$ $R$ .
Example: $R = \{(1, 2), (2, 3), (1, 3)\}$ $R$ $=$ ${($ $1$ $,$ $2$ $)$ $,$ $($ $2$ $,$ $3$ $)$ $,$ $($ $1$ $,$ $3$ $)}$ .
Equivalence Relation: A relation that is reflexive, symmetric, and transitive.

What is a Function?

A function is a special type of relation where each element in the first set (domain) is related to exactly one element in the second set (codomain).

Definition of a Function

A function $f$ $f$ from set $A$ $A$ to set $B$ $B$ is written as $f: A \to B$ $f$ $:$ $A$ $\to$ $B$ . For every $a \in A$ $a$ $\in$ $A$ , there is exactly one $b \in B$ $b$ $\in$ $B$ such that $f(a) = b$ $f$ $($ $a$ $)$ $=$ $b$ .

Examples of Functions

If $A = \{1, 2, 3\}$ $A$ $=$ ${$ $1$ $,$ $2$ $,$ $3$ $}$ and $B = \{4, 5\}$ $B$ $=$ ${$ $4$ $,$ $5$ $}$ , a function $f$ $f$ could be $f = \{(1, 4), (2, 4), (3, 5)\}$ $f$ $=$ ${($ $1$ $,$ $4$ $)$ $,$ $($ $2$ $,$ $4$ $)$ $,$ $($ $3$ $,$ $5$ $)}$ .

Types of Functions

One-to-One (Injective): Every element in the domain maps to a unique element in the codomain.
Example: $f(x) = x + 1$ $f$ $($ $x$ $)$ $=$ $x$ $+$ $1$ .
Onto (Surjective): Every element in the codomain has at least one pre-image in the domain.
Example: $f(x) = x^2$ $f$ $($ $x$ $)$ $=$ $x$ $2$ (for $x \in \mathbb{R}^+$ $x$ $\in$ $R$ $+$ ).
Bijective (One-to-One and Onto): A function that is both injective and surjective.
Example: $f(x) = x$ $f$ $($ $x$ $)$ $=$ $x$ (for $x \in \mathbb{R}$ $x$ $\in$ $R$ ).
Constant Function: Maps every element of the domain to the same element in the codomain.
Example: $f(x) = c$ $f$ $($ $x$ $)$ $=$ $c$ , where $c$ $c$ is constant.

Graph of a Function

The graph of a function represents the relationship between the domain and codomain. For example, the graph of $y = x^2$ $y$ $=$ $x$ $2$ is a parabola.

Differences Between Relations and Functions

Aspect	Relation	Function
Definition	Any subset of $A \times B$ $A$ $\times$ $B$ .	A relation where each element of $A$ $A$ has exactly one image in $B$ $B$ .
Mapping	Can have multiple mappings for one element.	Only one mapping for each element in $A$ $A$ .
Example	$R = \{(1, 2), (1, 3)\}$ $R$ $=$ ${($ $1$ $,$ $2$ $)$ $,$ $($ $1$ $,$ $3$ $)}$ .	$f = \{(1, 2), (2, 3)\}$ $f$ $=$ ${($ $1$ $,$ $2$ $)$ $,$ $($ $2$ $,$ $3$ $)}$ .

Applications

Sets: Used in database management, Venn diagrams, and probability.
Relations: Used in graph theory and social networks.
Functions: Used in calculus, physics, and computer algorithms.

FAQs on Sets, Relations And Functions

What exactly is a set relation?

A relation between two sets is a collection of ordered pairs that each contains one object from the other set. If object x belongs to the first set and object y belongs to the second set, the objects are said to be related if the ordered pair (x,y) is in the relation.

What is the fundamental relationship between sets?

Sets are well-defined collections that are entirely defined by their elements. As a result, two sets are equal if and only if they have the same elements. The fundamental relationship in set theory is that of element hood, also known as membership.