Chapter 1 Relations and Functions (Concepts)

Relations on a Set

To understand a Relation on a Set, we must first understand the foundation upon which it is built: the Cartesian Product. A relation is not merely a connection but a precisely defined mathematical structure representing how elements interact within a system.

The Cartesian Product ($A \times A$)

If $A$ is a non-empty set, the Cartesian product $A \times A$ is the set of all possible ordered pairs $(a, b)$ where both $a$ and $b$ belong to set $A$. It represents the total universal space of all possible pairings within that set.

If the set $A$ contains $n$ elements, then the number of elements in the Cartesian product is given by:

Formal Mathematical Definition

A Relation $R$ on a set $A$ is defined as any subset of the Cartesian product $A \times A$. This means that $R$ consists of some (or all, or none) of the ordered pairs from the total possible pairings.

We say that an element $a$ is related to $b$ under the relation $R$ (written as $a R b$) if and only if the ordered pair $(a, b)$ belongs to the set $R$.

Total Number of Possible Relations

Since a relation is a subset of $A \times A$, the total number of distinct relations that can be defined on a set $A$ with $n$ elements is equal to the total number of subsets of $A \times A$ (the cardinality of the power set of $A \times A$).

Growth of Relations

The following table illustrates how the number of possible relations increases rapidly with the number of elements in set $A$:

Elements in $A$ ($n$)	Elements in $A \times A$ ($n^2$)	Total Relations ($2^{n^2}$)
1	1	$2^1 = 2$
2	4	$2^4 = 16$
3	9	$2^9 = 512$
4	16	$2^{16} = 65,536$

Domain, Range, and Codomain

When we define a relation $R$ on a set $A$, we can identify three specific components:

1. Domain: The set of all first elements of the ordered pairs belonging to $R$.

2. Range: The set of all second elements of the ordered pairs belonging to $R$.

3. Codomain: For a relation defined on a set $A$, the entire set $A$ is considered the codomain.

Methods of Representation

Relations are typically represented in three ways:

(i) Roster Form: Listing all the ordered pairs explicitly. Example: $R = \{ (1, 1), (2, 2), (3, 3) \}$

(ii) Set-Builder Form: Defining the relation using a rule or property. Example: $R = \{ (a, b) : a, b \in A, a = b \}$

(iii) Arrow Diagram: A visual representation using sets and directed arrows to show which element is related to which.

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Particular Relations on a Set

In mathematical logic, when we define a relation $R$ on a non-empty set $A$, we are essentially selecting a subset of the Cartesian product $A \times A$. There are three specific types of relations that represent the boundaries of this selection process.

1. Empty Relation (Void Relation)

A relation $R$ on a set $A$ is called an empty relation if no element of $A$ is related to any element of $A$. Since the empty set $\phi$ is a subset of every set, it is the smallest possible relation that can be defined on $A$.

Example: Suppose we have a set $A = \{1, 2, 3, 4\}$ and we define a relation $R$ such that $R = \{(a, b) : a + b = 15\}$. Since the maximum possible sum of any two elements in $A$ is $4 + 4 = 8$, no pair $(a, b)$ can satisfy the condition $a + b = 15$. Hence, $R$ contains no elements and is an empty relation.

2. Universal Relation

A relation $R$ on a set $A$ is called a universal relation if every element of $A$ is related to every element of $A$. This occurs when the relation $R$ is exactly equal to the Cartesian product $A \times A$. It is the largest possible relation on a set.

Example: Let $A$ be the set of all integers. Define a relation $R$ on $A$ by $R = \{(a, b) : |a - b| \geq 0\}$. Since the absolute difference between any two integers is always non-negative, every possible pair $(a, b)$ of integers satisfies this condition. Therefore, $R$ is a universal relation.

Note: Both the Empty Relation and the Universal Relation are collectively referred to as trivial relations.

3. Identity Relation

The relation $I_A$ is called the identity relation on set $A$ if every element of $A$ is related to itself only. This means that for every $a \in A$, the pair $(a, a)$ must be in the relation, and no other pairs of the form $(a, b)$ where $a \neq b$ are allowed.

Example: If we consider a set $A = \{x, y, z\}$, the identity relation is $I_A = \{(x, x), (y, y), (z, z)\}$. If we define another relation $R = \{(x, x), (y, y), (z, z), (x, z)\}$, this is not an identity relation because the element $x$ is related to $z$, violating the "only itself" rule.

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Comparison Summary

For a set $A$ containing $n$ elements, the number of ordered pairs in these particular relations varies as follows:

Type of Relation	Condition	Number of Pairs
Empty Relation	$R = \phi$	$0$
Universal Relation	$R = A \times A$	$n^2$
Identity Relation	$R = I_A$	$n$

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Classification of Relations

Relations are categorized based on how elements interact with themselves and each other. The three fundamental properties—Reflexivity, Symmetry, and Transitivity—form the backbone of higher mathematical analysis and are essential for defining equivalence relations.

(i) Reflexive Relation

A relation $R$ on a set $A$ is said to be Reflexive if every single element of the set is related to itself. If there is even one element $a \in A$ such that $(a, a) \notin R$, the relation fails to be reflexive.

Note: Unlike an identity relation, a reflexive relation allows elements to be related to other elements as well, provided the self-relation $(a, a)$ exists for all.

Examples of Reflexive Relations:

Example 1: Let $A$ be the set of all triangles in a plane. Let $R$ be a relation defined as "is congruent to". Since every triangle is congruent to itself ($\Delta \cong \Delta$), the relation is reflexive.

Example 2: Let $R$ be the relation "$\le$" (less than or equal to) on the set of Real numbers $\mathbb{R}$. Since $a \le a$ is mathematically true for any real number $a$, the relation is reflexive.

Example 3 (Counter-example): Let $A = \{1, 2, 3\}$ and $R = \{(1, 1), (2, 2)\}$. This relation is not reflexive because the element $3 \in A$ but $(3, 3) \notin R$.

(ii) Symmetric Relation

A relation $R$ on a set $A$ is Symmetric if the existence of a relationship between $a$ and $b$ implies the same relationship exists between $b$ and $a$. In simpler terms, if the "forward" pair is present, the "reverse" pair must also be present.

Examples of Symmetric Relations:

Example 1: Let $L$ be the set of all straight lines in a 2D plane. Let $R$ be the relation "is perpendicular to" ($\perp$). If line $L_1$ is perpendicular to line $L_2$, then $L_2$ is also perpendicular to $L_1$. Thus, the relation is symmetric.

Example 2: On the set of integers $Z$, let $R$ be defined as $a R b$ iff "$a + b$ is an even number". Since $a + b = b + a$, if $a + b$ is even, then $b + a$ is also even. Hence, $R$ is symmetric.

Example 3 (Counter-example): Let $R$ be the relation "is the father of". If $a$ is the father of $b$, then $b$ cannot be the father of $a$. This relation is not symmetric.

(iii) Transitive Relation

A relation $R$ on a set $A$ is Transitive if whenever an element $a$ is related to $b$, and $b$ is related to $c$, then $a$ must also be related to $c$. It establishes a "chain" of relationships.

Examples of Transitive Relations:

Example 1: Let $R$ be the relation "is a divisor of" on the set of natural numbers $N$. If $a$ divides $b$ and $b$ divides $c$, then $a$ must divide $c$. Thus, it is transitive.

Example 2: On the set of Real numbers, the relation "$<$" (less than) is transitive. If $x < y$ and $y < z$, then it logically follows that $x < z$.

Example 3 (Counter-example): On a set of lines, "is perpendicular to" ($\perp$) is not transitive. If line $L_1 \perp L_2$ and $L_2 \perp L_3$, then $L_1$ is actually parallel to $L_3$, not perpendicular to it.

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Summary of Properties

To identify the type of relation effectively, the following summary table can be used as a reference:

Property	Condition	Logical Meaning
Reflexive	$(a, a) \in R$	Self-relation for all elements.
Symmetric	$(a, b) \in R \implies (b, a) \in R$	Two-way relationship.
Transitive	$(a, b), (b, c) \in R \implies (a, c) \in R$	Chain-link relationship.

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Equivalence Relation and Equivalence Classes

An Equivalence Relation is a special type of relation that mimics the properties of "equality" in a set. It is widely used in mathematics to group elements that are "similar" in some specific way, effectively reducing the complexity of a set by treating similar elements as part of the same category.

Definition of an Equivalence Relation

A relation $R$ defined on a non-empty set $A$ is called an Equivalence Relation if and only if it satisfies the following three conditions simultaneously:

(i) Reflexivity

Every element in the set must be related to itself.

(ii) Symmetry

If an element $a$ is related to $b$, then $b$ must also be related to $a$.

(iii) Transitivity

If $a$ is related to $b$ and $b$ is related to $c$, then $a$ must be related to $c$.

Example: Let $A$ be the set of all triangles drawn in a plane. Let $R$ be the relation "is similar to" ($\sim$) on $A$.

1. Reflexive: Every triangle is similar to itself ($\Delta \sim \Delta$).

2. Symmetric: If $\Delta_1 \sim \Delta_2$, then clearly $\Delta_2 \sim \Delta_1$.

3. Transitive: If $\Delta_1 \sim \Delta_2$ and $\Delta_2 \sim \Delta_3$, then $\Delta_1 \sim \Delta_3$.

Since all three properties hold, the relation "is similar to" is an Equivalence Relation.

Equivalence Classes

When an equivalence relation $R$ is defined on a set $A$, it naturally groups elements into distinct subsets. Any element $a \in A$ belongs to a subset consisting of all elements $x \in A$ that are related to $a$. This set is called the Equivalence Class of $a$, denoted by $[a]$.

Example: Consider the set of integers $\mathbb{Z}$ and the relation $R$ defined by "$a-b$ is divisible by 3". This is an equivalence relation. The equivalence classes are:

$\bullet$ $[0] = \{ \dots, -6, -3, 0, 3, 6, \dots \}$ (Integers leaving remainder 0 when divided by 3)

$\bullet$ $[1] = \{ \dots, -5, -2, 1, 4, 7, \dots \}$ (Integers leaving remainder 1 when divided by 3)

$\bullet$ $[2] = \{ \dots, -4, -1, 2, 5, 8, \dots \}$ (Integers leaving remainder 2 when divided by 3)

Properties of Equivalence Classes

Equivalence classes have highly specific mathematical properties that lead to the concept of partitioning a set. Let $A_i$ and $A_j$ be two equivalence classes under a relation $R$ on set $A$:

1. Internal Consistency: All elements within a single equivalence class are related to each other.

2. Mutual Exclusivity: Two equivalence classes are either identical or completely disjoint. No two different equivalence classes can share a common element.

3. Exhaustive Union: The union of all possible equivalence classes equals the original set $A$.

Example: In the "divisibility by 3" example above, we can see that:

1. $[0], [1],$ and $[2]$ have no common elements (Disjoint).

2. $[0] \cup [1] \cup [2] = \mathbb{Z}$ (Union is the set of all integers).

3. Any two elements in $[1]$, say $4$ and $7$, are related because $4-7 = -3$, which is divisible by 3.

The Partitioning Concept

In general, an equivalence relation $R$ on set $A$ divides $A$ into mutually disjoint subsets $A_i$ (called partitions). This is often visualised through a Venn diagram where the set is divided into non-overlapping "tiles".

Fundamental Theorem of Equivalence Relations

The theorem states that for every equivalence relation on a set, there is a corresponding partition of the set, and conversely, for every partition of a set, there is a corresponding equivalence relation.

Property	Condition
Reflexive	$(a, a) \in R$
Symmetric	$(a, b) \in R \implies (b, a) \in R$
Transitive	$(a, b), (b, c) \in R \implies (a, c) \in R$
Equivalence Class	Set of all related elements

Important Derivation: Divisibility as an Equivalence Relation

Let us prove that the relation $R = \{ (a, b) : a, b \in \mathbb{Z}, n \text{ divides } (a-b) \}$ is an equivalence relation for any $n \in \mathbb{N}$.

Reflexive Proof:

Since $0$ is divisible by any integer $n$, $(a, a) \in R$. Thus, $R$ is reflexive.

Symmetric Proof:

Let $(a, b) \in R$. This means $(a - b)$ is divisible by $n$.

Since $-k$ is an integer, $(b - a)$ is also divisible by $n$. Thus, $(b, a) \in R$. $R$ is symmetric.

Transitive Proof:

Let $(a, b) \in R$ and $(b, c) \in R$.

Adding equations both the above equations, we get:

Since $(q_1 + q_2)$ is an integer, $(a - c)$ is divisible by $n$. Thus, $(a, c) \in R$. $R$ is transitive.

Hence, $R$ is an Equivalence Relation.

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Functions

In Mathematics, a function is a special case of a relation. To understand functions, we must first understand the concept of a Cartesian Product and Relations.

Cartesian Product and Relations

A relation $R$ from set $X$ to set $Y$ is defined as a subset of the Cartesian product $X \times Y$. The Cartesian product is the set of all possible ordered pairs $(x, y)$ where $x \in X$ and $y \in Y$.

Total Number of Relations

If $n(X) = m$ and $n(Y) = n$, then the total number of relations from $X$ to $Y$ is given by:

Definition of a Function

A function $f$ from a set $X$ to a set $Y$ is a rule that assigns to each element $x \in X$ exactly one element $y \in Y$. While every function is a relation, every relation is not necessarily a function.

For a relation to be a function, it must satisfy two conditions:

1. Every element in set $X$ must have an image in set $Y$.

2. No element in set $X$ can have more than one image in set $Y$.

Total Number of Functions

If $n(X) = m$ and $n(Y) = n$, the total number of functions possible is:

Domain, Codomain, and Range

Let $f: X \to Y$ be a function defined by $y = f(x)$.

1. Domain: The set $X$ is called the domain of the function. It represents all possible input values for which the function is defined.

2. Codomain: The set $Y$ is called the codomain. it represents the potential set of output values.

3. Range: The set of all actual images of elements of $X$ under $f$ is called the range. It is denoted by $f(X)$.

The following table distinguishes between the components of a mapping:

Term	Description	Visual Analogy
Domain	Input set (Set $X$)	The "Pre-images"
Codomain	The target set (Set $Y$)	The "Possibilities"
Range	Actual outputs $\{f(x)\}$	The "Images"

Vertical Line Test (Graphical Interpretation)

To determine if a graph represents a function, we use the Vertical Line Test. If any vertical line drawn parallel to the y-axis intersects the graph at more than one point, the graph does not represent a function.

Standard Real Functions

A function $f: A \to B$ is called a real-valued function if $B$ is a subset of $\mathbb{R}$ (the set of all real numbers). If $A$ is also a subset of $\mathbb{R}$, it is called a real function. Below are the precise definitions of various standard real functions.

1. Identity Function

The function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x$ for each $x \in \mathbb{R}$ is called the identity function. The domain and range of this function are both $\mathbb{R}$.

2. Constant Function

A function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = c$ for all $x \in \mathbb{R}$, where $c$ is a constant, is called a constant function.

The domain is $\mathbb{R}$ and the range is the singleton set $\{c\}$.

3. Polynomial Function

A function $f: \mathbb{R} \to \mathbb{R}$ is said to be a polynomial function if for every $x \in \mathbb{R}$, $y = f(x) = a_0 + a_1x + a_2x^2 + \dots + a_nx^n$, where $n$ is a non-negative integer and $a_0, a_1, \dots, a_n \in \mathbb{R}$.

4. Rational Function

Rational functions are functions of the form $\frac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomial functions of $x$ defined in a domain, where $q(x) \neq 0$.

Piecewise Defined Functions

5. Modulus Function

The function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = |x|$ for each $x \in \mathbb{R}$ is called the modulus function. It is defined as:

$f(x) = \begin{cases} x & , & x \geq 0 \\ -x & , & x < 0 \end{cases}$

The domain of the modulus function is $\mathbb{R}$ and the range is $[0, \infty)$.

6. Signum Function

The function $f: \mathbb{R} \to \mathbb{R}$ defined by:

$f(x) = \begin{cases} 1 & , & x > 0 \\ 0 & , & x = 0 \\ -1 & , & x < 0 \end{cases}$

is called the signum function. It can also be expressed as $f(x) = \frac{|x|}{x}$ for $x \neq 0$. The range is the set $\{-1, 0, 1\}$.

7. Greatest Integer Function (Step Function)

The function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = [x], x \in \mathbb{R}$ assumes the value of the greatest integer, less than or equal to $x$. Such a function is called the greatest integer function.

Transcendental Functions

8. Exponential Function

The function $f(x) = a^x$, where $a > 0$ and $a \neq 1$, is called an exponential function. The domain is $\mathbb{R}$ and the range is $(0, \infty)$.

9. Logarithmic Function

The inverse of the exponential function is the logarithmic function. For $a > 0, a \neq 1$:

The domain of the logarithmic function is $(0, \infty)$ and the range is $\mathbb{R}$.

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Types of Functions

In this section, we elaborate on the various types of mappings between sets. These classifications are fundamental for understanding invertible functions and calculus.

1. One-One Function (Injective Function)

A function $f : X \to Y$ is defined to be one-one (or injective), if the images of distinct elements of $X$ under $f$ are distinct. In other words, for any $x_1, x_2 \in X$, $f(x_1) = f(x_2)$ implies that $x_1 = x_2$. If a function is not one-one, it is called a many-one function.

Mathematical Condition for Injectivity

To establish that a function is one-one, we generally use one of the following two equivalent conditions:

Condition 1: Using Images

Condition 2: Using Elements

Visual Representation

In a mapping diagram, a function is one-one if every element in the codomain has at most one arrow pointing to it from the domain.

Graphical Interpretation (Horizontal Line Test)

A real function $y = f(x)$ is one-one if every horizontal line (line parallel to the x-axis) intersects the graph of the function in at most one point. If a horizontal line intersects the graph at more than one point, it indicates that multiple $x$ values share the same $y$ value, making it many-one.

Example. Show that the function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = 3x + 5$ is a one-one function.

To Prove: The function $f(x) = 3x + 5$ is one-one for all $x \in \mathbb{R}$.

Proof:

Let $x_1$ and $x_2$ be any two arbitrary elements in the domain $\mathbb{R}$ such that their images are equal.

Substituting the function definition:

Subtracting $5$ from both sides of the equation:

Dividing both sides by $3$:

Since we started with the assumption $f(x_1) = f(x_2)$ and successfully derived $x_1 = x_2$ as the only logical conclusion, the function satisfies the criteria for injectivity.

Conclusion: Hence, the function $f(x) = 3x + 5$ is one-one.

2. Many-One Function

A function $f : X \to Y$ is called a many-one function if two or more distinct elements in the domain $X$ have the same image in the codomain $Y$. In other words, a function is many-one if it is not one-one.

Mathematical Condition

A function $f$ is said to be many-one if there exist at least two elements $x_1, x_2 \in X$ such that:

Visual Representation

In a mapping diagram, a many-one function is identified when two or more arrows from the domain point to the same element in the codomain.

Graphical Test

According to the Horizontal Line Test, if any straight line drawn parallel to the x-axis intersects the graph of the function at two or more points, the function is many-one. This is because a single $y$-value corresponds to multiple $x$-values.

Example. Check whether the function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x^2$ is one-one or many-one.

To Find: The nature of the mapping for $f(x) = x^2$.

Solution:

To check if the function is many-one, we look for at least two different inputs that produce the same output.

Let us take two distinct real numbers from the domain $\mathbb{R}$:

Now, calculating their images under the given function:

From the above equations, we observe that:

Since $x_1 \neq x_2$ but $f(x_1) = f(x_2)$, the condition for a one-one function is not satisfied. Multiple elements of the domain are mapped to the same element in the codomain.

Conclusion: The function $f(x) = x^2$ is a many-one function.

3. Onto Function (Surjective Function)

A function $f : X \to Y$ is called an onto function (or surjective function) if every element of the codomain $Y$ is the image of at least one element of the domain $X$. In other words, for an onto function, there is no element left in set $Y$ without a pre-image in set $X$.

Mathematical Condition

A function $f : X \to Y$ is said to be onto if and only if for every $y \in Y$, there exists an element $x \in X$ such that:

Equivalently, a function is onto if the range of the function is equal to the codomain of the function.

Visual Representation

In a mapping diagram, a function is onto if every element in the codomain $Y$ has at least one arrow pointing to it. It does not matter if an element in $Y$ has more than one arrow (that would make it many-one), as long as none are left empty.

Graphical Test

For a real-valued function, the function is onto if the graph extends to cover the entire range specified in the codomain. If the codomain is $\mathbb{R}$, the graph must cover every point on the y-axis.

Example. Show that the function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = 2x - 3$ is an onto function.

To Prove: The function $f(x) = 2x - 3$ is surjective (onto).

Proof:

Let $y$ be any arbitrary element in the codomain $\mathbb{R}$. To prove $f$ is onto, we must find an element $x$ in the domain $\mathbb{R}$ such that $f(x) = y$.

Substituting the definition of the function:

Solving for $x$ in terms of $y$:

Since $y$ is a real number, the value $x = \frac{y + 3}{2}$ is also a real number. Therefore, for every $y$ in the codomain, there exists a corresponding $x$ in the domain such that:

Since every element $y$ in the codomain is an image of some $x$ in the domain, the range of the function is the entire set of real numbers $\mathbb{R}$.

Conclusion: Hence, the function $f(x) = 2x - 3$ is onto.

4. Into Function

A function $f : X \to Y$ is called an into function if there exists at least one element in the codomain $Y$ which is not the image of any element in the domain $X$. In other words, a function is described as "into" if it is not onto.

Mathematical Condition

For a function $f : X \to Y$ to be an into function, the range of the function must be a proper subset of the codomain $Y$. This is expressed mathematically as:

This implies that there is at least one $y \in Y$ for which there is no $x \in X$ such that $f(x) = y$.

Visual Representation

In a mapping diagram, a function is "into" if there is at least one element in set $Y$ that does not have an incoming arrow from set $X$.

Difference Between Onto and Into

While an onto function uses up every element in the codomain, an into function always leaves at least one element unused. If we consider the set of all real numbers $\mathbb{R}$ as the codomain, any function whose results are restricted (like only positive numbers) will be an into function.

Example. Let $f : \mathbb{N} \to \mathbb{N}$ be a function defined by $f(x) = 2x$. Check if the function is onto or into.

Given: $f(x) = 2x$ where the domain and codomain are the set of Natural Numbers $\mathbb{N} = \{1, 2, 3, 4, \dots\}$.

To Find: Whether the function is onto or into.

Solution:

Let us examine the elements of the range by substituting values from the domain $\mathbb{N}$:

For $x = 1, f(1) = 2(1) = 2$

For $x = 2, f(2) = 2(2) = 4$

For $x = 3, f(3) = 2(3) = 6$

The Range of the function is the set of even natural numbers:

However, the Codomain is the set of all natural numbers:

By comparing the above equations, we see that elements like $1, 3, 5, \dots$ (odd numbers) belong to the codomain but do not have any pre-image in the domain.

Since the range is not equal to the codomain, the function is not onto.

Conclusion: The function $f(x) = 2x$ over natural numbers is an into function.

5. Bijective Function (One-One Correspondence)

A function $f : X \to Y$ is defined to be bijective (or a one-one correspondence) if it is both one-one (injective) and onto (surjective). This means that every element in the domain is mapped to a unique element in the codomain, and every element in the codomain is the image of exactly one element in the domain.

Requirements for Bijectivity

For a function to be classified as bijective, it must satisfy two distinct mathematical conditions simultaneously:

Condition 1: Injectivity (One-one)

Each element of the domain must map to a unique element in the codomain. No two different inputs can have the same output.

Condition 2: Surjectivity (Onto)

The range of the function must be exactly equal to the codomain. Every element $y \in Y$ must have at least one pre-image $x \in X$.

Visual Representation

In a bijective mapping, the number of elements in the domain $X$ must be equal to the number of elements in the codomain $Y$ (for finite sets). There is a perfect pairing between the two sets.

Importance of Bijective Functions

Bijective functions are of great importance in mathematics because they are invertible. A function $f$ has an inverse $f^{-1}$ if and only if $f$ is a bijection. If a function is not bijective, its inverse relation will not satisfy the definition of a function.

Example. Show that the function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = 2x + 1$ is a bijective function.

To Prove: The function $f(x) = 2x + 1$ is bijective.

Proof Strategy: We must prove that $f$ is both one-one and onto.

Step 1: Proof of One-one (Injectivity)

Let $x_1, x_2 \in \mathbb{R}$ such that $f(x_1) = f(x_2)$.

Step 2: Proof of Onto (Surjectivity)

Let $y$ be an arbitrary element in the codomain $\mathbb{R}$. We seek $x \in \mathbb{R}$ such that $f(x) = y$.

Since for every $y \in \mathbb{R}$, there exists a corresponding $x = \frac{y-1}{2}$ in the domain $\mathbb{R}$, the function is onto.

Conclusion: Since the function $f$ is both one-one and onto, it is a bijective function.

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Combined Comparative Table

Mapping Characteristics	Type of Function	Key Condition
Distinct $x$ have distinct $y$; Range = Codomain	One-One Onto (Bijective)	$n(X) = n(Y)$ (for finite sets)
Distinct $x$ have distinct $y$; Range $\subset$ Codomain	One-One Into	$n(X) < n(Y)$ (for finite sets)
Multiple $x$ have same $y$; Range = Codomain	Many-One Onto	$n(X) > n(Y)$ (for finite sets)
Multiple $x$ have same $y$; Range $\subset$ Codomain	Many-One Into	At least one $y$ unused

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Combined Comprehensive Example

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Special Real Functions

Identity Function

Let $A$ be any non-empty set. A function $f : A \to A$ is said to be an identity function if it associates every element of set $A$ to itself. It is generally denoted by $I_A$.

Definition

Properties of Identity Function:

• The Domain and Codomain of the identity function are the same set $A$.
• The Range of the function is also $A$.
• It is always a bijective function (both one-one and onto).
• In a coordinate plane, the graph of the identity function $y = x$ is a straight line passing through the origin, making an angle of $45^\circ$ with the positive x-axis.

Constant Function

A function $f : A \to B$ is called a constant function if every element of the domain $A$ is mapped to the same fixed element $c$ in the codomain $B$.

Definition

Properties of Constant Function:

• The Range of a constant function is a singleton set $\{c\}$.
• It is a many-one function (unless the domain $A$ contains only one element).
• It is an into function (unless the codomain $B$ contains only the element $c$).
• The graph of a constant function $y = c$ is a straight line parallel to the x-axis.

Equal Functions

Two functions $f$ and $g$ are said to be equal ($f = g$) if they are identical in every respect. For two functions to be equal, they must satisfy the following three conditions simultaneously:

1. The domain of $f$ must be equal to the domain of $g$.

2. The codomain of $f$ must be equal to the codomain of $g$.

3. The value of the functions must be the same for every element in their common domain.

Zero Function

The zero function is a special type of constant function where every element of the domain is mapped to the additive identity, which is zero. It serves as the "zero element" in the vector space of functions.

Formal Definition

Let $D$ be the domain (such as the set of real numbers $\mathbb{R}$ or integers $\mathbb{Z}$). A function $f_0 : D \to \mathbb{R}$ is called a zero function if:

The zero function is often denoted simply by the symbol $0$. For instance, in the set of all real-valued functions, the zero function $f(x) = 0$ acts as the additive identity because $(f + f_0)(x) = f(x) + 0 = f(x)$.

Properties of the Zero Function

1. Domain and Range

The domain of a zero function can be any non-empty set (usually $\mathbb{R}$ in real analysis). However, the range is always a singleton set containing only zero.

2. Nature of Mapping

Since all elements of the domain map to the same image ($0$), the zero function is a many-one function (provided the domain has more than one element). It is also an into function if the codomain contains any element other than zero.

3. Parity (Even and Odd Nature)

The zero function is unique because it is the only function that is simultaneously an even function and an odd function.

Monotonic Functions

In real analysis, a monotonic function is a function between ordered sets that preserves or reverses the given order. These functions are categorized based on whether the value of the function increases, decreases, or stays the same as the input value increases.

Increasing and Strictly Increasing Functions

1. Increasing (Non-decreasing) Function

A real-valued function $f$ is called an increasing function if for all $x_1, x_2$ in its domain:

This means as we move from left to right on the x-axis, the graph of the function does not decrease; it either rises or remains horizontal.

2. Strictly Increasing Function

A function $f$ is strictly increasing if for all $x_1, x_2$ in its domain:

In this case, the graph must always rise as we move from left to right.

Decreasing and Strictly Decreasing Functions

3. Decreasing (Non-increasing) Function

A real-valued function $f$ is called a decreasing function if for all $x_1, x_2$ in its domain:

This implies that as the input increases, the output either falls or remains constant.

4. Strictly Decreasing Function

A function $f$ is strictly decreasing if for all $x_1, x_2$ in its domain:

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Monotonicity and Injectivity

There is a direct relationship between strict monotonicity and the one-one (injective) nature of a function. A strict monotonic function is always one-one.

Proof Sketch:

Let $f$ be a strictly increasing function. If we take two distinct points $x_1 \neq x_2$, then either $x_1 < x_2$ or $x_1 > x_2$.

If $x_1 < x_2$, then by definition $f(x_1) < f(x_2)$, which implies $f(x_1) \neq f(x_2)$.

If $x_1 > x_2$, then by definition $f(x_1) > f(x_2)$, which implies $f(x_1) \neq f(x_2)$.

In both cases, distinct inputs lead to distinct outputs, which is the definition of a one-one function.

Even and Odd Functions

Even Functions

A real-valued function $f(x)$ is said to be an even function if the value of the function remains unchanged when the sign of the independent variable is reversed. Mathematically, for a function $f$ with domain $D$:

Characteristics of Even Functions

1. Graphical Symmetry

The graph of an even function is always symmetrical about the y-axis. This means that the y-axis acts as a mirror; if you fold the graph along the y-axis, the two halves will coincide perfectly. Points $(x, y)$ and $(-x, y)$ both lie on the graph.

2. Examples of Even Functions

Standard examples include polynomial functions with only even powers and certain trigonometric functions:

• $f(x) = x^2, x^4, x^6, \dots$
• $f(x) = \cos x$
• $f(x) = |x|$ (Modulus function)
• $f(x) = \text{sec } x$

Odd Functions

A real-valued function $f(x)$ is said to be an odd function if the sign of the function value is reversed when the sign of the independent variable is reversed. Mathematically, for a function $f$ with domain $D$:

Characteristics of Odd Functions

1. Graphical Symmetry

The graph of an odd function is symmetrical about the origin. This is also known as rotational symmetry of $180^\circ$. If a point $(x, y)$ exists on the graph, then the point $(-x, -y)$ must also exist on the graph. For an odd function defined at $x=0$, it must pass through the origin ($f(0)=0$).

2. Examples of Odd Functions

Standard examples include polynomial functions with only odd powers and specific trigonometric functions:

• $f(x) = x, x^3, x^5, \dots$
• $f(x) = \sin x$
• $f(x) = \tan x$
• $f(x) = \text{cosec } x$

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Composition of Functions

In many mathematical applications, we encounter situations where the output of one function becomes the input for another. This process of combining two or more functions to form a new function is known as the Composition of Functions. It is also referred to as the resultant of functions or function of a function.

The Mechanism of Composition

Consider three sets $A, B,$ and $C$. Let $f : A \to B$ be a function such that for every $x \in A$, there is a unique $y \in B$. Let $g : B \to C$ be another function such that for every $y \in B$, there is a unique $z \in C$.

By substituting $y = f(x)$ into the second function, we obtain:

This resultant mapping is the composite function $g \circ f$. It is often referred to as a function of a function because the value of $g$ depends on the value of $f(x)$.

Condition for Existence (Domain-Range Constraint)

A composition is not always possible. For $g(f(x))$ to be defined, every output produced by $f$ must be a valid input for $g$. If $f$ produces an element that does not belong to the domain of $g$, then $g$ cannot process that element, and the composition fails.

The Essential Criterion

The composition $g \circ f$ exists if and only if:

Similarly, for the composition $f \circ g$ to be defined, the following condition must be met:

Comparison of Mapping Components

The following table illustrates the roles of different sets in the composition $g \circ f : A \to C$.

Component	Role in $f$	Role in $g$	Role in $g \circ f$
Set $A$	Domain	-	Domain
Set $B$	Codomain	Domain	Intermediate Link
Set $C$	-	Codomain	Codomain

Visual Illustration

The arrow diagram below represents the flow of elements. Note that $f$ acts first (rightmost in notation), and $g$ follows.

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Properties of Composition of Functions

The composition of functions is not merely a way to combine rules; it possesses specific algebraic properties that govern how these functions behave during operations. Understanding these properties is crucial for solving complex functional equations in higher mathematics.

1. Associativity of Composition

While the composition of functions is generally not commutative, it is always associative. This fundamental property ensures that when combining three or more functions, the grouping of the functions does not affect the final mapping, as long as the sequential order of the functions remains unchanged.

Mathematical Statement and Derivation

Let $f: A \to B$, $g: B \to C$, and $h: C \to D$ be three functions. To prove that composition is associative, we must show that $h \circ (g \circ f) = (h \circ g) \circ f$.

Consider an arbitrary element $x$ in the domain $A$. We apply the left-hand side and right-hand side operations to this element:

Left-Hand Side (LHS) Analysis:

Right-Hand Side (RHS) Analysis:

Comparing the above results, we see that for every $x \in A$:

Visual Illustration of Associativity

The following diagram depicts four sets $A, B, C,$ and $D$ with mappings $f, g,$ and $h$. It illustrates that whether we first combine $f$ and $g$ into $(g \circ f)$ or combine $g$ and $h$ into $(h \circ g)$, the path from $A$ to $D$ remains identical.

2. Non-Commutativity

In algebra, an operation is called commutative if changing the order of the operands does not change the result. However, the composition of functions is generally not commutative. This means that the resultant function depends heavily on which function is applied first and which is applied second.

Mathematical Definition

For two functions $f$ and $g$, the composition $g \circ f$ and $f \circ g$ are usually distinct entities. Mathematically, this is expressed as:

Even in cases where both $g \circ f$ and $f \circ g$ are defined and have the same domain and codomain, their output values for the same input $x$ are typically different.

Visualizing the Order of Operations

The non-commutative nature arises because $g(f(x))$ means we first apply the rule of $f$ and then use that result as an input for $g$, whereas $f(g(x))$ means we apply $g$ first. The sequence of logical operations is reversed.

Example. If $f: \mathbb{R} \to \mathbb{R}$ is defined by $f(x) = x^2$ and $g: \mathbb{R} \to \mathbb{R}$ is defined by $g(x) = x + 1$, show that the composition is not commutative.

To Prove: $(g \circ f)(x) \neq (f \circ g)(x)$

Step 1: Calculate $(g \circ f)(x)$

By the definition of composition, we apply $f$ first:

Substituting $f(x) = x^2$ into the expression:

Applying the rule of $g$ (which adds $1$ to the input):

Step 2: Calculate $(f \circ g)(x)$

Here, we apply $g$ first:

Substituting $g(x) = x + 1$ into the expression:

Applying the rule of $f$ (which squares the input):

Expanding the identity $(a+b)^2 = a^2 + 2ab + b^2$:

Step 3: Comparison

Comparing equation (i) and equation (ii), we observe that:

Since the functions are not identical for all $x$ in the domain, we conclude that the order of composition matters.

Conclusion: Hence, $g \circ f \neq f \circ g$.

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Special Exception

The only common case where composition is commutative ($g \circ f = f \circ g$) is when one function is the Inverse of the other, or when one of the functions is the Identity function.

3. Preservation of Mapping Types

One of the most significant features of the composition of functions is that it preserves the structural properties of the individual functions involved. If the constituent functions are one-one or onto, the resultant composite function will inherit these characteristics.

(i) Preservation of Injectivity (One-One Nature)

If two functions $f$ and $g$ are both injective, their composition $g \circ f$ is also guaranteed to be injective. This means that if distinct elements are processed through a chain of one-one mappings, the final results will also remain distinct.

Formal Proof

Let $f: A \to B$ and $g: B \to C$ be two one-one functions. To prove $g \circ f: A \to C$ is one-one, we must show that $(g \circ f)(x_1) = (g \circ f)(x_2)$ implies $x_1 = x_2$.

Given:

Proof:

Assume that for $x_1, x_2 \in A$:

Conclusion: Since $(g \circ f)(x_1) = (g \circ f)(x_2) \implies x_1 = x_2$, the function $g \circ f$ is one-one.

(ii) Preservation of Surjectivity (Onto Nature)

If two functions $f$ and $g$ are both surjective, their composition $g \circ f$ is also surjective. This ensures that every element in the final codomain $C$ has at least one pre-image in the initial domain $A$.

Formal Proof

Let $f: A \to B$ and $g: B \to C$ be two onto functions. To prove $g \circ f: A \to C$ is onto, we must show that for every $z \in C$, there exists $x \in A$ such that $(g \circ f)(x) = z$.

Proof:

1. Let $z$ be an arbitrary element in set $C$.

2. Since $g: B \to C$ is onto, there must exist an element $y \in B$ such that:

3. Now, for this $y \in B$, since $f: A \to B$ is onto, there must exist an element $x \in A$ such that:

4. Substituting (ii) into (i):

Conclusion: Since every $z \in C$ has a pre-image $x \in A$ under $g \circ f$, the function is onto.

3. Preservation of Bijectivity

A function is bijective if it is both one-one and onto. Based on the two properties derived above, we can conclude:

Theorem: If $f: A \to B$ and $g: B \to C$ are both bijective functions, then their composition $g \circ f: A \to C$ is also a bijective function.

Example. Let $f: \{1, 2\} \to \{3, 4\}$ be defined as $f = \{(1, 3), (2, 4)\}$ and $g: \{3, 4\} \to \{5, 6\}$ be defined as $g = \{(3, 5), (4, 6)\}$. Verify if $g \circ f$ is bijective.

Step 1: Check $f$ and $g$ individually.

For $f$: Distinct elements map to distinct images, and the range $\{3, 4\}$ equals the codomain. So, $f$ is bijective.

For $g$: Distinct elements map to distinct images, and the range $\{5, 6\}$ equals the codomain. So, $g$ is bijective.

Step 2: Find $g \circ f$.

$g \circ f = \{(1, 5), (2, 6)\}$.

Conclusion: Since $g \circ f$ is both one-one and onto, it is bijective. This verifies the property that the composition of two bijections is a bijection.

4. Identity Property

In the algebra of functions, the Identity Function acts as the identity element for the operation of composition, much like the number $1$ acts as the identity element for multiplication. Composing any function $f$ with an identity function results in the function $f$ itself.

Definitions of Identity Functions

Let $A$ and $B$ be two non-empty sets. We define two specific identity functions based on these sets:

1. Identity on Set A ($I_A$): A function $I_A: A \to A$ defined by $I_A(x) = x$ for all $x \in A$.

2. Identity on Set B ($I_B$): A function $I_B: B \to B$ defined by $I_B(y) = y$ for all $y \in B$.

Statement and Derivation

For any function $f: A \to B$, the following identity properties hold true:

1. Right Identity Property ($f \circ I_A = f$)

To prove this, we apply the composition to an arbitrary element $x \in A$:

Since $(f \circ I_A)(x) = f(x)$ for all $x \in A$, we conclude $f \circ I_A = f$.

2. Left Identity Property ($I_B \circ f = f$)

To prove this, we apply the composition to an arbitrary element $x \in A$:

Since $f(x)$ is an element of the codomain $B$, and $I_B$ maps every element of $B$ to itself:

Since $(I_B \circ f)(x) = f(x)$ for all $x \in A$, we conclude $I_B \circ f = f$.

Visual Interpretation

In a mapping diagram, the identity function is represented by arrows that connect each element to itself. Composing with such a mapping does not change the destination of any element starting from set $A$.

Comparison with Numerical Identity

The following table draws a parallel between identity in real numbers and identity in function composition.

Feature	Real Numbers (Multiplication)	Functions (Composition)
Identity Element	$1$	$I(x) = x$
Property	$a \times 1 = a$	$f \circ I = f$
Operation	Multiplication	Composition ($\circ$)

Example. Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = x^3 + 2x$. Verify that $(f \circ I)(x) = f(x)$, where $I$ is the identity function on $\mathbb{R}$.

To Verify: $(f \circ I)(x) = f(x)$

Given:

Proof:

By the definition of composition:

Substituting $I(x) = x$:

Substituting the actual expression of $f(x)$:

Conclusion: Since the result in (vi) is identical to the original function given in (v), the identity property is verified for the given function.

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Real Functions and Domains

When we define a composite function, it is a common misconception to assume that its domain is simply the domain of the first function applied. In reality, real functions require a "Double-Filtering" process. For the composition $g \circ f$ to be valid, an input $x$ must not only be acceptable for $f$, but the resulting output $f(x)$ must also be acceptable for $g$.

Mathematical Definition of Composite Domains

The domain of a composite function is the set of all inputs $x$ that satisfy two simultaneous conditions. Let $D_f$ and $R_f$ be the domain and range of $f$, and $D_g$ and $R_g$ be the domain and range of $g$.

1. Domain of $g \circ f$

For $g(f(x))$ to exist, $x$ must first be in the domain of $f$, and then the value $f(x)$ must fall within the domain of $g$.

2. Domain of $f \circ g$

Similarly, for $f(g(x))$ to exist, $x$ must first be in the domain of $g$, and then the value $g(x)$ must fall within the domain of $f$.

Visualizing the Domain Restriction

Think of function composition as a conveyor belt with two processing machines. If an object passes through the first machine ($f$) but comes out in a shape that the second machine ($g$) cannot handle, that object is rejected from the domain of the composite function.

Comparison Table: Domain and Range Interaction

Composition	Primary Constraint	Impact on Domain
$g \circ f$	$\text{Range}(f) \cap \text{Domain}(g) \neq \emptyset$	Usually a subset of $D_f$
$f \circ g$	$\text{Range}(g) \cap \text{Domain}(f) \neq \emptyset$	Usually a subset of $D_g$

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Invertible Functions

The concept of invertibility is central to understanding how mathematical operations can be undone. An invertible function, often called a bijective mapping, provides a perfect one-to-one link between two sets, allowing for a unique reverse path from every output back to its original input.

The Necessity of Bijectivity

For a function $f: X \to Y$ to have an inverse $f^{-1}: Y \to X$, it must strictly satisfy two conditions. If either is missing, the reverse rule fails to qualify as a "function."

1. Requirement of One-one (Injectivity)

If $f$ were many-one, two different inputs (say $x_1$ and $x_2$) would map to the same output $y$. When trying to reverse the process, the inverse rule would not know whether to send $y$ back to $x_1$ or $x_2$. This violation of the "single output" rule prevents it from being a function.

2. Requirement of Onto (Surjectivity)

If $f$ were an into function, there would be some elements in the codomain $Y$ that are not images of any $x \in X$. For the inverse function $f^{-1}$ (which has $Y$ as its domain), these "leftover" elements would have no destination, violating the rule that every element in a function's domain must have a mapping.

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Formal Definition via Composition

A function $f: X \to Y$ is said to be invertible if there exists another function $g: Y \to X$ such that the "round trip" from $X$ to $Y$ and back to $X$ (or vice versa) brings you exactly to the starting point. This is expressed using identity functions $I_X$ and $I_Y$.

Here, $g$ is uniquely identified as $f^{-1}$.

Domain and Range Relationships

The inverse function essentially swaps the roles of the domain and the range. This is a crucial property used in solving complex inverse problems.

Property	Function $f$	Inverse Function $f^{-1}$
Domain	$X$	$Y$
Range	$Y$	$X$
Point Format	$(x, y)$	$(y, x)$

Graphical Symmetry (Reflection Property)

One of the most powerful visual tools for invertible functions is their symmetry. If we plot the graphs of $f$ and $f^{-1}$ on the same coordinate plane, they appear as mirror images of each other.

The line of symmetry is always the identity line $y = x$. This is because the inverse operation simply swaps the $x$ and $y$ coordinates of every point on the original graph.

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Properties of Invertible Functions

Invertible functions possess several key properties that allow us to manipulate functional equations and understand the deep relationships between domains and ranges. A function $f: X \to Y$ is called invertible only if it is a bijective function (both one-one and onto).

1. Uniqueness of the Inverse

Every bijective function has exactly one inverse. If a function is invertible, its inverse is unique. This ensures that the reverse mapping remains consistent across all mathematical operations.

Proof of Uniqueness:

Suppose $g_1$ and $g_2$ are two inverses of a function $f: X \to Y$. Then, for any $y \in Y$:

From the above equations, we have $f(g_1(y)) = f(g_2(y))$. Since $f$ is one-one (a requirement for invertibility):

Thus, $g_1 = g_2$, proving the inverse is unique.

2. Double Inverse Property

The inverse of an inverse function is the original function itself. If we reverse a mapping and then reverse it again, we return to the starting rule.

This property also implies that the inverse of a bijective function is itself bijective. In terms of ordered pairs, if $(x, y) \in f$, then $(y, x) \in f^{-1}$, and reversing the latter yields $(x, y)$, which is back in $f$.

3. Composition with the Inverse (Identity Result)

When an invertible function $f: A \to B$ is composed with its inverse $f^{-1}$, the result is an identity function. This operation effectively "cancels out" the functional transformation.

The Left and Right Identities:

1. Identity on Domain A: If we apply $f$ first and then $f^{-1}$:

2. Identity on Codomain B: If we apply $f^{-1}$ first and then $f$:

4. The Reversal Law of Composition

The Reversal Law of Composition is a fundamental theorem in functional algebra. It states that the inverse of a composite function is the composition of the inverses of the individual functions, but in the reverse order. This principle is not only mathematically significant but also intuitively logical when viewed through step-by-step processes.

Formal Mathematical Statement

Let $f : A \to B$ and $g : B \to C$ be two bijective (one-one and onto) functions. Since $f$ and $g$ are bijections, their composition $g \circ f : A \to C$ is also a bijection. Consequently, the inverse function $(g \circ f)^{-1}$ exists and maps elements from set $C$ back to set $A$.

The law is defined as:

Proof of the Reversal Law

To prove that $f^{-1} \circ g^{-1}$ is indeed the inverse of $g \circ f$, we must show that their composition results in the identity function.

Part 1: Composition from the Right

By applying the Associative Property of composition:

Since $f \circ f^{-1} = I_B$ (Identity function on set $B$):

Part 2: Composition from the Left

Similarly, we can show the reverse order:

Conclusion: Since both compositions yield the identity function, $f^{-1} \circ g^{-1}$ is the unique inverse of $g \circ f$.

The "Shoes and Socks" Analogy

To understand why the order must be reversed, consider the daily routine of dressing your feet:

1. The Forward Process ($g \circ f$): You first put on your Socks ($f$), then you put on your Shoes ($g$). Note that in notation, $f$ is applied first, so it is written on the right.

2. The Inverse Process ($(g \circ f)^{-1}$): To return to the original state of bare feet, you cannot remove your socks first. You must first take off your Shoes ($g^{-1}$) and then take off your Socks ($f^{-1}$).

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Binary Operations

The term "Binary" is derived from the Latin word 'binarius', which means "consisting of two." In mathematics, a binary operation is a calculation that combines two elements (called operands) to produce another element. It is the fundamental building block of abstract algebra, providing the framework for structures like groups, rings, and fields.

Detailed Mathematical Definition

A binary operation $*$ on a non-empty set $A$ is formally defined as a function whose domain is the Cartesian product $A \times A$ and whose codomain is $A$ itself.

This definition implies three critical requirements for any rule to be considered a binary operation:

(a) Requirement of Two Inputs (Ordered Pairs)

The operation acts on an ordered pair $(a, b)$ where $a, b \in A$. Since the domain is $A \times A$, the operation must be defined for every possible pair of elements from the set.

(b) Uniqueness of Result

Since a binary operation is a function, for every ordered pair $(a, b)$, there must exist one and only one image in $A$. If a rule produces two different results for the same two inputs, it is not a binary operation.

(c) Closure Property

The result of the operation, denoted by $a \ast b$, must belong to the same set $A$. In mathematical terms:

Importance of the Word "Ordered"

In the definition, we refer to the ordered pair $(a, b)$. This signifies that the position of the elements matters. A rule might assign a different value to $(a, b)$ than it does to $(b, a)$.

If $a \ast b = b \ast a$ for all elements, the operation is called commutative. However, if there exists even a single pair where $a \ast b \neq b \ast a$, the order becomes the deciding factor of the outcome.

Example 1: Addition on the set of Integers ($\mathbb{Z}$)

Let $A = \mathbb{Z} = \{..., -2, -1, 0, 1, 2, ...\}$. Let the operation be addition ($+$).

Since the sum of any two integers is always a unique integer, addition is a binary operation on $\mathbb{Z}$.

Example 2: Subtraction on the set of Natural Numbers ($\mathbb{N}$)

Let $A = \mathbb{N} = \{1, 2, 3, ...\}$. Let the operation be subtraction ($-$).

Because the result $-2$ does not belong to the set of natural numbers, the "Closure Property" fails. Therefore, subtraction is not a binary operation on $\mathbb{N}$.

General Representation of Operations

While $+$ and $\times$ are common, a binary operation can be any arbitrary rule. For instance, we can define an operation $\oplus$ on the set of Real Numbers $\mathbb{R}$ as:

In this case, since the average of two real numbers is always a unique real number, $\oplus$ is a valid binary operation on $\mathbb{R}$.

Example. Show that the operation $\ast$ defined on $\mathbb{R}$ by $a \ast b = ab^2$ is a binary operation.

To prove that $\ast$ is a binary operation on $\mathbb{R}$, we must check if the result is always a unique real number for any $a, b \in \mathbb{R}$.

Step 1: Let $a, b$ be any two real numbers.

Step 2: By the properties of real numbers, the square of a real number ($b^2$) is always a unique real number.

Step 3: The product of two real numbers ($a$ and $b^2$) is also always a unique real number.

Since the rule assigns a unique real number to every ordered pair $(a, b)$, $\ast$ is a binary operation on $\mathbb{R}$.

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Binary Operations on Different Sets

The applicability of a rule as a binary operation depends entirely on the Set upon which it is defined. For a rule to be a binary operation, it must satisfy the Closure Property, meaning the result of the operation for any two elements in the set must also be an element of that same set.

Binary Operations on the Set of Natural Numbers ($\mathbb{N}$)

The set of Natural Numbers is defined as $\mathbb{N} = \{1, 2, 3, 4, ...\}$. We examine the four fundamental arithmetic operations on this set:

(i) Addition ($+$)

Addition is a binary operation on $\mathbb{N}$. If we take any two natural numbers $a$ and $b$, their sum $a + b$ is always a unique natural number.

(ii) Multiplication ($\times$)

Multiplication is also a binary operation on $\mathbb{N}$ because the product of any two natural numbers is always a unique natural number.

(iii) Subtraction ($-$)

Subtraction is not a binary operation on $\mathbb{N}$. While $5 - 3 = 2$ (a natural number), the reverse order or certain pairs result in values outside the set.

Since the result is not always a natural number, subtraction fails the closure property on $\mathbb{N}$.

(iv) Division ($\div$)

Division is not a binary operation on $\mathbb{N}$ because the quotient of two natural numbers is often a fraction or a decimal.

Binary Operations on the Set of Real Numbers ($\mathbb{R}$)

The set of Real Numbers ($\mathbb{R}$) includes all rational and irrational numbers. The behavior of operations changes significantly here.

(i) Addition, Subtraction, and Multiplication

These three operations are all binary operations on $\mathbb{R}$. For any $a, b \in \mathbb{R}$, the values $a+b$, $a-b$, and $ab$ are always unique real numbers.

(ii) Division ($\div$)

Surprisingly, division is not a binary operation on the complete set of real numbers $\mathbb{R}$. This is because of the existence of zero ($0 \in \mathbb{R}$).

Since the operation does not produce a real number when the second element of the ordered pair $(a, b)$ is zero, the rule is not defined for all pairs in $\mathbb{R} \times \mathbb{R}$.

Binary Operations on Non-Zero Real Numbers ($\mathbb{R}_0$)

If we exclude zero from the set of real numbers, let $\mathbb{R}_0 = \mathbb{R} - \{0\}$.

In this restricted set, Division becomes a binary operation. For every ordered pair $(a, b) \in \mathbb{R}_0 \times \mathbb{R}_0$, the image $a/b$ is a unique non-zero real number.

Summary Table of Closure Property

The following table summarizes whether the basic arithmetic operations are binary operations on common number sets:

Operation	Natural Numbers ($\mathbb{N}$)	Integers ($\mathbb{Z}$)	Real Numbers ($\mathbb{R}$)	Non-zero Real ($\mathbb{R}_0$)
Addition ($+$)	Yes	Yes	Yes	No (e.g., $5+(-5)=0$)
Subtraction ($-$)	No	Yes	Yes	No (e.g., $5-5=0$)
Multiplication ($\times$)	Yes	Yes	Yes	Yes
Division ($\div$)	No	No	No	Yes

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Types and Properties of Binary Operations

A binary operation can possess several algebraic properties that define the structure of the set it operates on. These properties—Commutativity, Associativity, Identity, and Inverse—are essential for solving algebraic equations and understanding advanced mathematical systems.

1. Commutative Property (Abelian Property)

A binary operation $\ast$ on a set $A$ is said to be commutative (or Abelian) if the order of the operands does not change the result.

Observations:

1. Addition and Multiplication are commutative on $\mathbb{R}$ because $a + b = b + a$ and $a \times b = b \times a$.

2. Subtraction and Division are not commutative on $\mathbb{R}$. For example:

2. Associative Property

A binary operation $\ast$ on a set $A$ is associative if the grouping of elements does not affect the final outcome when performing the operation on three or more elements.

Observations:

1. Addition and Multiplication on $\mathbb{R}$ are associative.

2. Subtraction is not associative. Consider $a=7, b=4, c=3$:

From the above equations, we see $(a - b) - c \neq a - (b - c)$.

3. Division is not associative on the set of non-zero real numbers $\mathbb{R}_0$. Changing the grouping changes the divisor in the subsequent step.

Example: Let $a = 8, b = 4, c = 2$.

Comparing the above equations, we find $1 \neq 4$, hence division is not associative.

3. Identity Element

An element $e \in A$ is called the identity element for the binary operation $\ast$ if every element $a \in A$ remains unchanged when operated with $e$.

Key Identities in $\mathbb{R}$:

1. Additive Identity: $0$ is the identity for addition because $a + 0 = a = 0 + a$.

2. Multiplicative Identity: $1$ is the identity for multiplication because $a \times 1 = a = 1 \times a$.

3. Uniqueness: If an identity element exists for a binary operation, it is always unique.

4. Invertible Element (Inverse)

If an identity element $e$ exists for an operation $\ast$, then an element $a \in A$ is said to be invertible if there exists another element $b \in A$ such that their composition results in the identity element.

The element $b$ is called the inverse of $a$ and is denoted as $a^{-1}$.

Key Inverses in $\mathbb{R}$:

1. Additive Inverse: For every $a \in \mathbb{R}$, there exists $-a \in \mathbb{R}$ such that $a + (-a) = 0$.

2. Multiplicative Inverse: For every non-zero $a \in \mathbb{R}$, there exists $\frac{1}{a} \in \mathbb{R}$ such that $a \times \frac{1}{a} = 1$.

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Answer:

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Operation Table

For a finite non-empty set $A$, we can represent a binary operation $\ast$ using a square array known as an Operation Table (or Cayley Table). This table serves as a comprehensive visual representation of all possible results obtained by applying the operation to any two elements of the set.

The structure of the table is such that the elements of set $A$ are listed in the header row (top) and the header column (left). The entry located at the intersection of the $i^{th}$ row and $j^{th}$ column represents the result of the operation: $(i^{th} \text{ entry on the left}) \ast (j^{th} \text{ entry at the top})$.

Symmetry and Commutativity

One of the most significant advantages of an operation table is that it allows us to determine if an operation is commutative at a glance. A binary operation is commutative if and only if the operation table is symmetric across its main diagonal (the diagonal line running from the top-left to the bottom-right).

In mathematical terms, for an operation to be commutative, the following condition must hold for all entries:

Example. Let $A = \{1, -1\}$ be a finite set. Construct the operation table for $A$ under the binary operation of multiplication ($\times$).

To construct the table, we multiply each element in the left column with each element in the top row as follows:

$\bullet$ $1 \times 1 = 1$

$\bullet$ $1 \times (-1) = -1$

$\bullet$ $(-1) \times 1 = -1$

$\bullet$ $(-1) \times (-1) = 1$

$\times$	1	-1
1	1	-1
-1	-1	1

Observation:

In the table above, the elements across the main diagonal (1, 1) are symmetric. Specifically:

Since the table is symmetric about the main diagonal, we can conclude that multiplication is commutative on the set $A = \{1, -1\}$.

Properties Identifiable via Table

1. Closure: If all the results within the table (excluding the headers) are elements of the set $A$, the operation is closed.

2. Identity Element: If a row is exactly the same as the top header row, and the corresponding column is exactly the same as the leftmost header column, that element is the identity element.

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Important Formulas

In the study of Binary Operations and Functions, it is essential to quantify the number of possible operations and understand the procedure for finding the inverse of a given function. This section provides the mathematical derivations and detailed examples for these concepts.

1. Total Number of Binary Operations

A binary operation on a finite set $A$ is essentially a function that maps an ordered pair from the Cartesian product $A \times A$ to an element in $A$. To find the total number of such operations, we use the principles of functional counting.

Derivation:

Let $A$ be a finite set containing $n$ elements. We denote the number of elements in $A$ (cardinality) as $O(A) = n$.

1. The Domain: A binary operation acts on the set $A \times A$. The number of elements in the Cartesian product is calculated as:

2. The Codomain: The result of the operation must belong to the set $A$, which has $n$ elements.

3. Functional Count: From the theory of sets and functions, the total number of functions from a set with $x$ elements to a set with $y$ elements is given by $y^x$.

In this context, we have:

$\bullet$ Number of elements in domain ($x$) = $n^2$

$\bullet$ Number of elements in codomain ($y$) = $n$

Substituting these values into the formula, we get the total number of binary operations:

2. Commutative Binary Operations

While the total number of operations is $n^{n^2}$, the number of commutative operations is fewer because the result of $a \ast b$ must be the same as $b \ast a$.

Formula:

For a set with $n$ elements, the number of commutative binary operations is:

3. Finding the Inverse of a Function

The inverse of a function $f$, denoted by $f^{-1}$, is a rule that "reverses" the action of $f$. For $f^{-1}$ to exist, the function must be one-to-one (injective) and onto (surjective).

Answer: