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 $A$, $B$, or $C$.

Notation of Sets

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

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

Types of Sets

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

Set Operations

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

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.

Cartesian Product

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

Example:

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

A×B={(1,x),(1,y),(2,x),(2,y)}.

Representation of Relations

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

Example:

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

Types of Relations

1. Empty Relation: No pair in the relation.
   Example: $R = \emptyset$
2. Universal Relation: Includes all possible pairs in $A \times B$.
3. Reflexive Relation: Every element is related to itself.
   Example: $(a, a) \in R$ for all $a \in A$.
4. Symmetric Relation: If $(a, b) \in R$, then $(b, a) \in R$.
   Example: $R = \{(1, 2), (2, 1)\}$.
5. Transitive Relation: If $(a, b) \in R$ and $(b, c) \in R$, then $(a, c) \in R$.
   Example: $R = \{(1, 2), (2, 3), (1, 3)\}$.
6. 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 $f$ from set $A$ to set $B$ is written as $f: A \to B$. For every $a \in A$, there is exactly one $b \in B$ such that $f(a) = b$.

Examples of Functions

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

Types of Functions

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

Differences Between Relations and Functions

Applications

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