Chapter 1 Relations and Functions (Concepts)

Relations on a Set

In mathematics, a relation is a powerful concept used to describe a connection or relationship between elements. These can be elements within a single set or between two different sets. To formally define a relation, we must first understand the foundation on which it is built: the Cartesian Product.

Cartesian Product of Sets

The Cartesian Product is a way of creating a new set from two existing sets by forming all possible ordered pairs. It acts as a universe of all potential pairings between the elements of the sets.

Definition

Given two non-empty sets, $A$ and $B$, their Cartesian product, denoted as $A \times B$ (read as "A cross B"), is the set of all ordered pairs $(a, b)$ where the first element, $a$, is from set $A$ and the second element, $b$, is from set $B$.

The term "ordered pair" is crucial because the order matters. The pair $(a, b)$ is generally not the same as the pair $(b, a)$.

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Cardinality of Cartesian Product

If set $A$ has $m$ elements (denoted $|A| = m$) and set $B$ has $n$ elements (denoted $|B| = n$), then the number of pairs in their Cartesian product $A \times B$ is the product of their sizes.

In the example above, $|A|=2$ and $|B|=3$, so $|A \times B| = 2 \times 3 = 6$.

Definition of a Relation

A relation is a way to specify which of the possible pairings from a Cartesian product are meaningful according to a certain rule or condition. Think of the Cartesian product as a menu of all possible connections, and a relation as your specific order from that menu.

Formal Definition

A relation $R$ from a non-empty set $A$ to a non-empty set $B$ is formally defined as a subset of the Cartesian product $A \times B$.

If an ordered pair $(a, b)$ is in the set $R$, we say "$a$ is related to $b$ by $R$" and write $aRb$.

Most often in this topic, we discuss a relation on a single set $A$, which is a relation from $A$ to itself. In this case, $R \subseteq A \times A$.

Domain, Range, and Codomain

For any relation $R$ from a set $A$ to a set $B$:

• The Domain is the set of all the first elements of the pairs in $R$. It's a subset of $A$.
• The Range is the set of all the second elements of the pairs in $R$. It's a subset of $B$.
• The Codomain is the entire set $B$, which contains all possible second elements.

Total Number of Possible Relations

Since a relation is any subset of $A \times B$, we can count the total number of possible relations. A set with $N$ elements has $2^N$ subsets.

If $|A| = m$ and $|B| = n$, then $|A \times B| = mn$.

Therefore, the total number of possible relations from $A$ to $B$ is $2^{mn}$.

For a relation on a single set $A$ with $|A|=m$, the total number of relations is $2^{m^2}$.

Types of Relations on a Set $A$

When a relation is defined from a set A to itself (i.e., $R \subseteq A \times A$), we can classify it based on specific structural properties or "rules of behavior" it might possess. Understanding these properties—reflexivity, symmetry, and transitivity—is fundamental to advanced topics in mathematics, including order theory, graph theory, and abstract algebra.

Trivial Relations

There are two "extreme" relations that can be defined on any non-empty set A. They represent the minimum and maximum possible relations.

1. Empty Relation (or Void Relation)

This is the relation where no element is related to any element (including itself). The relation set contains no ordered pairs. It is the smallest possible relation.

Example: Let $A = \{1, 2, 3\}$. The relation $R$ on $A$ defined by $aRb$ if $a+b=10$ is an empty relation, as no two elements in A sum to 10.

2. Universal Relation

This is the relation where every element is related to every other element. The relation set contains all possible pairs from the Cartesian product $A \times A$. It is the largest possible relation.

Example: Let $A = \{1, 2\}$. The universal relation is $R = \{(1,1), (1,2), (2,1), (2,2)\}$.

Key Relational Properties

3. Identity Relation

The identity relation, denoted by $I_A$, is a very specific relation where every element is related only to itself. No other relationships are allowed.

For any given set A, there is only one unique identity relation.

Example: If $A=\{1,2,3\}$, then the identity relation on A is $I_A = \{(1,1), (2,2), (3,3)\}$. The relation $\{(1,1), (2,2)\}$ is not the identity relation on A because the pair $(3,3)$ is missing. The relation $\{(1,1), (2,2), (3,3), (1,2)\}$ is not the identity relation because it contains an extra pair, $(1,2)$, which is not a self-relation.

4. Reflexive Relation

The core idea of a reflexive relation is that every element "is related to itself". It's a mirror property for every single element in the set.

Condition: For every element $a$ in set $A$, the ordered pair $(a, a)$ must be an element of the relation $R$.

A reflexive relation can contain other pairs like $(a, b)$ where $a \neq b$. This makes it a less strict condition than the identity relation.

Reflexive vs. Identity: Every identity relation is reflexive, but not every reflexive relation is an identity relation. An identity relation contains *only* self-related pairs, whereas a reflexive relation must contain *at least* all self-related pairs.

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5. Symmetric Relation

The core idea of a symmetric relation is that the relationship is a "two-way street". If an element $a$ is related to an element $b$, then $b$ must also be related to $a$ in the same way.

Condition: If an ordered pair $(a, b)$ is in the relation $R$, then its reverse pair $(b, a)$ must also be in $R$.

Note that pairs of the form $(a, a)$ do not violate this condition, because if $(a, a) \in R$, its reverse is also $(a, a)$, which is already in R.

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6. Transitive Relation

The core idea of a transitive relation is that relationships can form a "chain". If there is a path from $a$ to $b$, and another path from $b$ to $c$, then there must be a direct "shortcut" path from $a$ to $c$.

Condition: If a pair $(a, b)$ is in $R$ and another pair $(b, c)$ is in $R$, then the "bridging" pair $(a, c)$ must also be in $R$.

Important Note: If you cannot find any "chains" of the form $(a, b)$ and $(b, c)$ in the relation, the condition is never violated, and the relation is considered transitive by default (this is called being vacuously true). For example, $R = \{(1,2), (3,4)\}$ is transitive.

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Equivalence Relation

An equivalence relation is a special and very important type of relation that is used to group elements of a set that are "alike" or "equivalent" in some specific way. A relation $R$ on a set $A$ is an equivalence relation if and only if it satisfies all three of the key properties: it must be Reflexive, Symmetric, and Transitive (RST).

1. Reflexive: Every element is equivalent to itself.

   For all $a \in A, (a, a) \in R$.

2. Symmetric: If $a$ is equivalent to $b$, then $b$ must be equivalent to $a$.

   If $(a, b) \in R$, then $(b, a) \in R$.

3. Transitive: If $a$ is equivalent to $b$ and $b$ is equivalent to $c$, then $a$ must be equivalent to $c$.

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

Common examples include the "is equal to" (=) relation on numbers, the "is congruent to" relation on geometric shapes, and the "has the same birthday as" relation on a set of people.

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Equivalence Classes and Partitions

The most important consequence of an equivalence relation on a set is that it naturally partitions the set. A partition of a set is a way of splitting it into a collection of smaller, non-overlapping subsets such that every element of the original set belongs to exactly one of these subsets.

Equivalence Classes

Each of the subsets in the partition created by an equivalence relation is called an equivalence class. The equivalence class of an element $a$, denoted by $[a]$, is the set of all elements in the original set $A$ that are related to $a$ under the relation $R$.

In simple terms, an equivalence class is a "club" or "group" of all elements that are considered equivalent to each other under the rules of the relation.

Properties of Equivalence Classes

1. For any $a \in A$, $a$ is in its own equivalence class, so $[a]$ is never empty.
2. Any two equivalence classes are either identical or completely disjoint (they have no elements in common). $[a] = [b]$ or $[a] \cap [b] = \emptyset$.
3. The union of all distinct equivalence classes is the original set $A$.

Answer:

The relation $aRb$ means that $a$ and $b$ have the same parity (both are even or both are odd). This relation will partition the set of all integers $Z$ based on their parity.

Step 1: Find the equivalence class of an even number, like 0.

The equivalence class of 0, denoted $[0]$, is the set of all integers $b$ that are related to 0.

$[0] = \{b \in Z : (b, 0) \in R\} = \{b \in Z : b - 0 \text{ is even}\} = \{b \in Z : b \text{ is even}\}$

So, $[0]$ is the set of all even integers: $\{ \dots, -4, -2, 0, 2, 4, \dots \}$.

Step 2: Find the equivalence class of an odd number, like 1.

The equivalence class of 1, denoted $[1]$, is the set of all integers $b$ that are related to 1.

$[1] = \{b \in Z : (b, 1) \in R\} = \{b \in Z : b - 1 \text{ is even}\}$

If $b-1$ is an even number, then $b$ itself must be an odd number.

So, $[1]$ is the set of all odd integers: $\{ \dots, -3, -1, 1, 3, 5, \dots \}$.

Step 3: Conclusion.

If we pick any other integer (e.g., 4), its equivalence class $[4]$ will be the set of all even integers, which is the same as $[0]$. If we pick another odd integer (e.g., -5), its class $[-5]$ will be the set of all odd integers, which is the same as $[1]$.

Thus, there are only two distinct equivalence classes. These classes form a partition of the set of integers $Z$:

• Disjoint: The sets are disjoint as no integer is both even and odd: $[0] \cap [1] = \emptyset$.
• Exhaustive: Their union covers all integers: $[0] \cup [1] = Z$.

Functions

A function is one of the most fundamental concepts in mathematics. It acts like a rule or a machine that takes an input and produces a single, predictable output. It is a special, more structured type of relation where every input is associated with exactly one output.

Definition of a Function

A function $f$ from a non-empty set $A$ to a non-empty set $B$ is a specific type of relation where every element in set $A$ is mapped to exactly one element in set $B$.

We denote a function $f$ from set $A$ to set $B$ as $f: A \to B$. This is read as "$f$ is a function from $A$ to $B$".

The Two Rules of a Function

For a relation $f \subseteq A \times B$ to be a function, it must obey two strict rules:

1. Every element in the domain must be used. Every element in set $A$ must have an output in set $B$. No input can be left behind. (Formally, the domain of the relation must be the entire set $A$).
2. Each element in the domain must have only one output. An element in set $A$ cannot be mapped to two or more different elements in set $B$. (Formally, if $(a, b_1) \in f$ and $(a, b_2) \in f$, then it must be that $b_1 = b_2$).

In simple terms, every input has exactly one output.

Key Terminology

If $(a, b)$ is a pair in the function $f$, we write it in the familiar notation $f(a) = b$.

• $b$ is the image of $a$ under $f$. (The output)
• $a$ is a pre-image of $b$ under $f$. (The input)

For a function $f: A \to B$:

• Domain: The set $A$ of all possible inputs.
• Codomain: The set $B$ of all possible outputs.
• Range: The set of all actual outputs that the function produces. The range is always a subset of the codomain.

Types of Functions

Functions can be classified into different types based on how they map elements from the domain to the codomain. The properties of a function, such as whether it is one-to-one or onto, are critically dependent on the specified domain and codomain. Let's consider a function $f: A \to B$.

1. One-to-One Function (Injective)

A function is one-to-one (or injective) if every distinct element in the domain maps to a distinct element in the codomain. In simple terms, different inputs always produce different outputs. No two different inputs can share the same output.

Formal Definition 1:

Formal Definition 2 (more useful for proofs):

Many-to-One Function

A function that is not one-to-one is called a many-to-one function. This means there exist at least two distinct elements in the domain that map to the same element in the codomain.

The Horizontal Line Test (for graphical functions)

A function is one-to-one if and only if no horizontal line intersects its graph more than once. If you can draw a horizontal line that cuts the graph in two or more places, the function is many-to-one.

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2. Onto Function (Surjective)

A function is onto (or surjective) if every element in the codomain has at least one corresponding input in the domain. In other words, every element in the codomain is "hit" or mapped to. There are no elements left over in the codomain that are not an output of the function.

This is equivalent to saying the Range is equal to the Codomain.

Formal Definition:

Into Function

A function that is not onto is called an into function. In an into function, the range is a proper subset of the codomain. This means there is at least one element in the codomain that is not the image of any element in the domain.

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3. One-to-One and Onto Function (Bijective)

A function is bijective if it is both one-to-one AND onto. A bijection creates a perfect pairing between the elements of two sets. Every input has a unique output, and every possible output is mapped to by exactly one input.

For finite sets, a bijection can only exist if the domain and codomain have the same number of elements ($|A| = |B|$).

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Answer:

The function is $f(x) = x^2$ with domain $R$ and codomain $R$.

Check for One-to-One:

Let's test two different inputs. Consider $x_1 = 2$ and $x_2 = -2$.

$f(2) = 2^2 = 4$

$f(-2) = (-2)^2 = 4$

We have different inputs ($2 \neq -2$) but they produce the same output (4). Therefore, the function is not one-to-one. It is a many-to-one function.

Check for Onto:

The codomain is all real numbers, $R$. We need to check if the range equals the codomain.

The function is $f(x) = x^2$. The square of any real number is always non-negative ($x^2 \geq 0$).

This means the range of the function is the set of non-negative real numbers, $[0, \infty)$.

The range $[0, \infty)$ is not equal to the codomain $R$. For instance, the number $-1$ is in the codomain but there is no real number $x$ such that $x^2 = -1$. Therefore, the function is not onto. It is an into function.

Check for Bijective:

Since the function is neither one-to-one nor onto, it is not bijective.

Note on Domain and Codomain

The properties of a function heavily depend on its specified domain and codomain. For instance, if we define a new function $g: [0, \infty) \to [0, \infty)$ by $g(x) = x^2$, this new function is bijective.

Other Specific Types of Functions

Identity Function

An identity function, denoted $I_A$, is a function from a set A to itself that maps every element to itself. The rule is $f(x) = x$. The domain and codomain must be the same set. An identity function is always bijective.

Example: The function $f: R \to R$ defined by $f(x) = x$ is the identity function on the real numbers.

Constant Function

A constant function is a function where every element of the domain maps to the exact same, single element in the codomain. The rule is $f(x) = c$, where $c$ is a constant.

Example: The function $f: R \to R$ defined by $f(x) = 5$ is a constant function. Its range is just the set $\{5\}$.

Zero Function

The zero function is a specific type of constant function where the constant is 0. The rule is $f(x) = 0$.

Equal Functions

Two functions, $f$ and $g$, are said to be equal if and only if they satisfy two conditions:

1. They have the exact same domain.
2. For every element $x$ in their common domain, their rules produce the same output, i.e., $f(x) = g(x)$.

Monotonic Functions

These functions describe a consistent trend in their values.

• Increasing: For all $x_1 < x_2$, $f(x_1) \le f(x_2)$. (The function never goes down).
• Decreasing: For all $x_1 < x_2$, $f(x_1) \ge f(x_2)$. (The function never goes up).
• Strictly Increasing: For all $x_1 < x_2$, $f(x_1) < f(x_2)$. (The function always goes up).
• Strictly Decreasing: For all $x_1 < x_2$, $f(x_1) > f(x_2)$. (The function always goes down).

Even and Odd Functions

These classifications are based on the function's symmetry with respect to the y-axis and the origin.

Even Function: A function $f$ is even if its graph is symmetric with respect to the y-axis.
Condition: $f(-x) = f(x)$ for all $x$ in the domain.
Example: $f(x) = x^2$ is even because $f(-x) = (-x)^2 = x^2 = f(x)$. Also, $f(x) = \cos(x)$ is even.

Odd Function: A function $f$ is odd if its graph is symmetric with respect to the origin.
Condition: $f(-x) = -f(x)$ for all $x$ in the domain.
Example: $f(x) = x^3$ is odd because $f(-x) = (-x)^3 = -x^3 = -f(x)$. Also, $f(x) = \sin(x)$ is odd.

A function can be even, odd, or neither. For example, $f(x) = x+1$ is neither even nor odd.

Composition of Functions

Just as we combine numbers using operations like addition and multiplication to create new numbers, we can combine functions to create new, more complex functions. The most important method for combining functions is function composition. This powerful concept allows us to build a "chain reaction" or a "pipeline" where the output of one function becomes the input for the next function.

Definition of Composition of Functions

Let's say we have two functions, $f$ and $g$. The composition of $g$ with $f$, written as $g \circ f$, represents the function that results from first applying function $f$ to an input $x$, and then applying function $g$ to the output of $f$. It is read as "$g$ composed with $f$", "$g$ of $f$", or "$g$ after $f$".

Condition for Composition

For the composition $g \circ f$ to be defined, there's a crucial compatibility condition: the output of the first function ($f$) must be a valid input for the second function ($g$). This means that every value in the range of $f$ must be included in the domain of $g$.

Formally, the range of $f$ must be a subset of the domain of $g$.

The Composition Rule

If the condition is met, the composite function $(g \circ f)(x)$ is defined by the following rule:

This formula is the core of function composition. It instructs us to evaluate the expression from the inside out:

1. Take an input $x$.
2. Calculate the output of the inner function, $f(x)$.
3. Use this entire result, $f(x)$, as the input for the outer function, $g$.

Domain and Codomain of a Composite Function

If $f: A \to B$ and $g: B \to C$, the composite function $g \circ f$ acts as a direct shortcut, taking an input from the domain of $f$ (set A) and producing an output in the codomain of $g$ (set C).

Order Matters!

It is very important to remember that function composition is not commutative. In general, the order in which you compose functions drastically changes the result. The function $g \circ f$ ("g after f") is different from the function $f \circ g$ ("f after g").

$g \circ f \neq f \circ g$

Composition of Real Functions and Their Domains

When dealing with real functions (functions whose inputs and outputs are subsets of the real numbers), the composition involves substituting one function's formula into the other. While finding the formula for a composite function is often straightforward, the main challenge is correctly determining its domain.

The domain of a composite function $(g \circ f)(x)$ is not simply the intersection of the domains of $f$ and $g$. It's a more careful, two-step process. Think of it like a two-stage security check: an input $x$ is only "valid" for the composite function if it can successfully pass through both stages.

The domain of $(g \circ f)(x)$ is the set of all $x$ values that satisfy two conditions simultaneously:

1. Stage 1 (Inner Function): The input $x$ must be in the domain of the inner function, $f$.
2. Stage 2 (Outer Function): The output of the inner function, $f(x)$, must be a valid input for the outer function, $g$. That is, $f(x)$ must be in the domain of $g$.

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Answer:

Properties of Composition of Functions

Function composition follows several important algebraic properties that are similar to properties of operations on numbers.

1. Associativity of Composition

The associative property means that when you compose three or more functions, the grouping of the operations doesn't matter. The order of application ($f$, then $g$, then $h$) is fixed, but you can compute the intermediate steps in different ways and get the same result.

For three functions $f: A \to B$, $g: B \to C$, and $h: C \to D$:

Proof: We show both sides give the same output for any input $x \in A$.

Left-Hand Side: Calculate $(g \circ f)$ first.

$(h \circ (g \circ f))(x) = h((g \circ f)(x)) = h(g(f(x)))$

Right-Hand Side: Calculate $(h \circ g)$ first.

$((h \circ g) \circ f)(x) = (h \circ g)(f(x)) = h(g(f(x)))$

Since both sides simplify to the same expression, the property is proven.

2. Composition with an Identity Function

The identity function is a special function that simply returns its input. For a set $A$, the identity function $I_A: A \to A$ is defined as $I_A(x) = x$. It's the functional equivalent of the number 1 in multiplication or 0 in addition.

Composing any function $f$ with an appropriate identity function leaves $f$ unchanged.

If $f: A \to B$, then:

• $f \circ I_A = f$ (Composing with identity on the domain)
• $I_B \circ f = f$ (Composing with identity on the codomain)

Proof for $f \circ I_A = f$:

For any $x \in A$, $(f \circ I_A)(x) = f(I_A(x)) = f(x)$. Since the outputs are the same for all inputs, the functions are identical.

3. Composition of Bijective Functions

An important theorem states that if you compose two bijective functions, the resulting composite function is also bijective.

If $f: A \to B$ is bijective and $g: B \to C$ is bijective, then their composition $g \circ f: A \to C$ is also bijective.

Proof Outline:

(i) Proving $g \circ f$ is One-to-One (Injective):

Assume $(g \circ f)(x_1) = (g \circ f)(x_2)$ for some $x_1, x_2 \in A$. This means $g(f(x_1)) = g(f(x_2))$. Since $g$ is injective, its inputs must be equal: $f(x_1) = f(x_2)$. And since $f$ is also injective, its inputs must be equal: $x_1 = x_2$. This proves $g \circ f$ is injective.

(ii) Proving $g \circ f$ is Onto (Surjective):

Take any element $z \in C$. Since $g$ is surjective, there exists a $y \in B$ such that $g(y) = z$. Since $f$ is surjective, there exists an $x \in A$ such that $f(x) = y$. Therefore, we can write $(g \circ f)(x) = g(f(x)) = g(y) = z$. We have found a pre-image $x$ for any element $z$, so $g \circ f$ is surjective.

Since $g \circ f$ is both injective and surjective, it is bijective.

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Invertible Functions

An invertible function is a function for which we can define a "reverse" or "undo" operation. This reverse operation is itself a function, called the inverse function. If a function $f$ takes an input $x$ from set A to an output $y$ in set B, its inverse function, denoted $f^{-1}$, takes that output $y$ and returns the original input $x$. This concept is fundamental for solving equations and understanding many mathematical structures.

Definition of an Invertible Function

A function $f: A \to B$ is called invertible if there exists another function, $g: B \to A$, that perfectly reverses its effect.

Specifically, $f$ is invertible if we can find a function $g$ such that composing the two functions in either order results in the identity function (the "do nothing" function).

1. Applying $f$ and then $g$ gets you back to the start in set A.

   This means the composition $g \circ f$ must be the identity function on set A ($I_A$).

   $(g \circ f)(x) = g(f(x)) = x$ for all $x \in A$.

2. Applying $g$ and then $f$ gets you back to the start in set B.

   This means the composition $f \circ g$ must be the identity function on set B ($I_B$).

   $(f \circ g)(y) = f(g(y)) = y$ for all $y \in B$.

If such a function $g$ exists, it is called the inverse of $f$ and is uniquely denoted by $f^{-1}$.

Important Note: The notation $f^{-1}$ is a special symbol for the inverse function and does not mean the reciprocal $\frac{1}{f(x)}$.

The Condition for Invertibility: Bijectivity

Not all functions have an inverse. For a function to be invertible, it must create a perfect, unambiguous pairing between the elements of its domain and codomain. This leads to a crucial theorem:

A function $f: A \to B$ is invertible if and only if it is bijective (both one-to-one and onto).

Why Bijectivity is Necessary

• Why must it be one-to-one?
  Consider a function that is not one-to-one (it's many-to-one). This means two different inputs, say $a_1$ and $a_2$, map to the same output $b$. If we tried to define an inverse, what should $f^{-1}(b)$ be? Should it be $a_1$ or $a_2$? The inverse would not be a function because one input ($b$) would have multiple outputs, which violates the definition of a function.
  
• Why must it be onto?
  Consider a function that is not onto. This means there is some element $b$ in the codomain that is never an output (it's "unmapped"). If we tried to define an inverse, what is the value of $f^{-1}(b)$? There is no input in A that maps to $b$, so the inverse would not be defined for all elements in its domain (which is B). This also violates the definition of a function.
  

Therefore, a function must be both one-to-one and onto (bijective) to guarantee that its inverse is a well-defined function.

Proof of the Condition

The proof shows that the concepts of invertibility and bijectivity are logically equivalent.

Part 1: If a function is invertible, it must be bijective.

• One-to-one: Assume $f(a_1) = f(a_2)$. Apply the inverse function $g$ to both sides: $g(f(a_1)) = g(f(a_2))$. Since $g(f(x))=x$, this simplifies to $a_1=a_2$. This proves $f$ is one-to-one.
• Onto: Take any element $y$ in the codomain B. We need to find an input $x$ that maps to it. Let $x = g(y)$. Then $f(x) = f(g(y))$. Since $f(g(y))=y$, we have $f(x)=y$. We found an input for every output, so $f$ is onto.

Part 2: If a function is bijective, it must be invertible.

• Since $f$ is bijective, for every $y$ in B, there is a unique $x$ in A such that $f(x)=y$. This allows us to define an inverse function $g: B \to A$ by the rule: $g(y) = x$. This $g$ will perfectly undo $f$, satisfying the conditions for an inverse.

How to Find the Inverse of a Function Algebraically

If a function is defined by a formula and is known to be bijective, you can find the formula for its inverse by following these steps.

1. Replace $f(x)$ with $y$: Start with the function's equation, $y = f(x)$.
2. Swap the variables $x$ and $y$: This is the crucial step that represents the "reversal" of inputs and outputs. The equation becomes $x = f(y)$.
3. Solve for the new $y$: Use algebraic manipulation to isolate $y$ in the new equation.
4. Replace $y$ with $f^{-1}(x)$: The resulting expression for $y$ is the formula for the inverse function.

The domain of $f^{-1}$ is the range of $f$, and the range of $f^{-1}$ is the domain of $f$.

Answer:

Answer:

Inverse of a Composite Function

There is a special rule for finding the inverse of a composite function, sometimes called the "socks and shoes rule". To undo a sequence of operations, you must undo them in the reverse order.

If $f$ and $g$ are two invertible functions, then their composition $g \circ f$ is also invertible, and its inverse is given by:

Notice that the order of the inverse functions is reversed. To undo "apply $f$, then apply $g$", you must first undo $g$ (with $g^{-1}$), and then undo $f$ (with $f^{-1}$).

Proof Outline:

To prove this, we must show that composing $(f^{-1} \circ g^{-1})$ with $(g \circ f)$ gives the identity function.

Consider $(f^{-1} \circ g^{-1}) \circ (g \circ f)$:

$= f^{-1} \circ (g^{-1} \circ g) \circ f$ (by associativity)

$= f^{-1} \circ I_B \circ f$ (since $g^{-1} \circ g$ is the identity)

$= f^{-1} \circ f$ (since composing with identity does nothing)

$= I_A$ (since $f^{-1} \circ f$ is the identity)

A similar check for $(g \circ f) \circ (f^{-1} \circ g^{-1})$ confirms the result.

Binary Operations

Operations like addition, subtraction, and multiplication are fundamental in mathematics. They are all examples of a general concept called a binary operation. A binary operation is a rule for combining two elements of a set to produce a third element, which must also belong to the same set.

Definition of a Binary Operation

A binary operation $*$ on a non-empty set $A$ is a function that takes any two elements from $A$, in a specific order, and maps them to a single element that is also in $A$.

Formally, $*$ is a function from the Cartesian product $A \times A$ to the set $A$.

This means for any pair $(a, b)$ where $a \in A$ and $b \in A$, the result of the operation, written as $a*b$, is a unique element that also belongs to set $A$.

Closure Property

The most important part of this definition is that the result $a*b$ must be an element of $A$. This is known as the closure property. If a set is "closed" under an operation, it means performing that operation will never take you outside of the set.

Answer:

Properties of Binary Operations

Binary operations can have several important properties that define their behavior. Let $*$ be a binary operation on a set $A$.

1. Commutativity

An operation is commutative if the order in which you combine two elements does not change the result.

For example, on the set of integers $Z$, addition is commutative ($3+5=5+3$) but subtraction is not ($3-5 \neq 5-3$).

Answer:

2. Associativity

An operation is associative if, when combining three or more elements, the grouping (or order of calculation) does not change the result.

For example, on the set of real numbers $R$, multiplication is associative ($(2 \times 3) \times 4 = 2 \times (3 \times 4)$), but division is not ($(8 \div 4) \div 2 \neq 8 \div (4 \div 2)$).

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3. Existence of an Identity Element

An identity element, usually denoted by $e$, is a special element in the set that, when combined with any other element, leaves it unchanged.

• For addition on $R$, the identity element is 0 ($a+0=a$).
• For multiplication on $R$, the identity element is 1 ($a \times 1=a$).

If an identity element exists, it is unique.

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4. Existence of Inverse Elements

If a binary operation has an identity element $e$, then an element $a$ is invertible if there exists another element $b$ (called the inverse of $a$) in the set, such that combining them gives the identity element.

The inverse of $a$ is often denoted by $a^{-1}$.

• For addition on $R$, the inverse of $a$ is $-a$ (since $a + (-a) = 0$).
• For multiplication on $R$, the inverse of a non-zero $a$ is $\frac{1}{a}$ (since $a \times \frac{1}{a} = 1$).

Answer:

Answer:

5. Operation Table (Cayley Table)

For a binary operation on a finite set, we can represent all possible outcomes in a square grid called an operation table or a Cayley table. The rows and columns are headed by the elements of the set. The entry at the intersection of row 'a' and column 'b' is the result of the operation $a*b$.

Answer: