Step 1: Theoretical Definition of the Set

The set $C$ is defined by the set-builder notation: $C = \{x : x \in \mathbb{N}, 10 \le x \le 99, \text{sum of digits of } x = 8\}$.
Let the two-digit number be represented as $10a + b$, where $a \in \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ and $b \in \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$.
The constraint given is $a + b = 8$ [Constraint based on problem statement].

Step 2: Systematic Identification of Elements

To find all possible values for $x$, we iterate through all possible values of the tens digit ($a$) and determine the corresponding units digit ($b$) such that $b = 8 - a$. Since $x$ is a natural number, $a$ must be at least 1.

Step 3: Verification of Constraints

We check if any other values exist. If $a=9$, $b = 8 - 9 = -1$, which is not a natural digit [By definition of decimal numeral system]. Thus, the set is complete.

Step 4: Representation in Roster Form

The roster form requires listing all elements of the set separated by commas and enclosed within curly braces [Per the definition of Roster/Tabular form].

Final Solution: $C = \{17, 26, 35, 44, 53, 62, 71, 80\}$

Definition and Theoretical Framework

The set $A$ is defined using the set-builder notation: $A = \{x : x \text{ is an odd natural number}\}$. To list the elements of this set, we invoke the following definitions:

• Natural Numbers ($\mathbb{N}$): The set of counting numbers starting from 1, i.e., $\{1, 2, 3, 4, 5, \dots\}$.
• Odd Numbers: Integers not divisible by 2. Formally, an odd number is defined by the expression $2n - 1$, where $n \in \mathbb{N}$.

Step 1: Identifying the Sequence

We generate the elements of the set by substituting consecutive values of $n$ into the expression for odd numbers:

Step 2: Visual Representation of the Set Elements

The set $A$ constitutes an infinite sequence along the number line representing discrete odd points.

Step 3: Constructing the Roster Form

By the Roster Method (or Tabular Form), we list all elements within curly braces, separated by commas. [Per the Axiom of Extensionality, a set is defined solely by its elements].

$A = \{1, 3, 5, 7, 9, \dots\}$

Final Solution: The set of odd natural numbers is represented in roster form as $A = \{1, 3, 5, 7, 9, \dots\}$.

Step 1: Analyzing the Set-Builder Notation

The set $B$ is defined by the set-builder form $B = \{x : x \in \mathbb{N} \text{ and } x < 6\}$. This notation requires the identification of two specific constraints:

• Constraint 1: $x$ must be a member of the set of Natural Numbers ($\mathbb{N} = \{1, 2, 3, 4, 5, 6, ...\}$). [Per the definition of the set of natural numbers].
• Constraint 2: $x$ must satisfy the inequality $x < 6$. [Per the logical condition provided in the problem statement].

Step 2: Identifying Eligible Elements

We evaluate the natural numbers against the condition $x < 6$:

Step 3: Constructing the Roster Representation

In roster form, elements of a set are listed explicitly within curly braces, separated by commas. [Per the definition of the Roster/Tabular method]. Including all verified elements from Step 2:

$B = \{1, 2, 3, 4, 5\}$

Visual Representation of the Set on a Number Line

Final Solution: $B = \{1, 2, 3, 4, 5\}$

Step 1: Analyzing the Set Definition

The set $E$ is defined by the property of containing all unique letters found in the word "TRIGONOMETRY". In set theory, a set is defined as a collection of well-defined, distinct objects [Per the Zermelo-Fraenkel set theory axiom of extensionality].

Step 2: Listing the Characters

We extract each character from the word T-R-I-G-O-N-O-M-E-T-R-Y sequentially:

• T, R, I, G, O, N, O, M, E, T, R, Y

Step 3: Applying the Law of Distinctness

According to the standard definition of a set, each element must be unique; repeated elements are counted only once [Per the Fundamental Principle of Set Representation]. We identify and remove duplicates:

Step 4: Constructing the Roster Form

By collecting the unique letters identified in Step 3, we arrange them into the roster format, where elements are enclosed in curly braces $\{ \}$ and separated by commas:

$E = \{T, R, I, G, O, N, M, E, Y\}$

Visual Representation of Set Membership

Final Solution: $E = \{T, R, I, G, O, N, M, E, Y\}$

Step 1: Identifying the Universal Set

The universal set of concern is the English alphabet, $U = \{a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z\}$. [Per the definition of the English alphabet].

Step 2: Defining the Constraints

The set $F$ is defined by two conditions:

• Constraint A: The element $x$ must be a consonant. [By definition, consonants are alphabetic letters excluding the vowels $\{a, e, i, o, u\}$].
• Constraint B: The element $x$ must precede $k$ in the English alphabet. This implies the set of elements $S = \{a, b, c, d, e, f, g, h, i, j\}$.

Step 3: Evaluating the Intersection of Constraints

To determine the elements of $F$, we filter the set $S$ to exclude vowels.

Step 4: Final Set Enumeration

By collecting the included elements identified in Step 3, we construct the set $F$. [By the Roster Method of set representation].

Final Solution: $F = \{b, c, d, f, g, h, j\}$

Step 1: Analyzing the Set-Builder Notation

The set is defined as $A = \{x : x \in \mathbb{Z} \text{ and } -3 \le x < 7\}$. To convert this into roster form, we must identify all elements that satisfy the given conditions simultaneously [Per the definition of Set-Builder Notation].

• Constraint 1: $x$ must be an integer ($\mathbb{Z}$).
• Constraint 2: $x$ must be greater than or equal to $-3$ (inclusive, as indicated by $\le$).
• Constraint 3: $x$ must be strictly less than $7$ (exclusive, as indicated by $<$).

Step 2: Listing the Integer Interval

Based on the defined interval $[-3, 7)$, we list the integers sequentially starting from the lower bound ($-3$) and ending at the largest integer less than $7$ (which is $6$):

Step 3: Visual Representation on a Number Line

Step 4: Final Construction of the Set

By the Roster Method, elements of a set are listed within curly braces $\{ \}$, separated by commas [Per Set Theory conventions].

$A = \{-3, -2, -1, 0, 1, 2, 3, 4, 5, 6\}$

Final Solution: A = {-3, -2, -1, 0, 1, 2, 3, 4, 5, 6}

Definition of a Set

A set is defined as a well-defined collection of objects. In mathematics, "well-defined" implies that there is an objective, universal criterion that allows one to decide, without ambiguity, whether a specific object belongs to the collection or not [Per Cantor’s Set Theory].

Analysis of the Given Collection

The collection in question is: "A collection of most dangerous animals of the world."

Step 1: Evaluation of the Criterion

To determine if this collection constitutes a set, we must examine the term "most dangerous."

• Subjectivity: The perception of danger is highly subjective and varies based on individual experience, geographical context, and scientific framing (e.g., whether one measures danger by total human fatalities per year, venom toxicity, or predatory aggression).
• Lack of Universal Standard: There is no globally accepted, mathematically precise list that classifies animals by a single, undisputed metric of "most dangerous." An animal considered dangerous by one observer (e.g., a shark) may not be considered as dangerous as a different organism by another (e.g., a mosquito, which causes significantly more human deaths via disease transmission).

Step 2: Logical Application

Since the membership of this collection cannot be determined with absolute certainty due to the lack of a universal, objective, and well-defined rule, the collection fails the fundamental requirement of set theory.

Conclusion

Because the collection depends on the opinion of the observer rather than an objective mathematical or scientific standard, the rule defining the collection is not "well-defined."

Final Solution: The collection of the "most dangerous animals of the world" is not a set because the term "most dangerous" is subjective and lacks a universally accepted, precise definition.

Step 1: Analyzing the Set Definition

The set $A$ is given by the roster method as $A = \{1, 2, 3, 4, 5, 6\}$. This implies that the elements belonging to set $A$ are exclusively the integers from 1 to 6 inclusive [Definition of a Set].

Step 2: Definition of the $\in$ and $\notin$ Symbols

• The symbol $\in$ (element of) is used to denote that an object is a member of a specific set. If $x \in A$, then $x$ is one of the elements listed within the braces of set $A$.
• The symbol $\notin$ (not an element of) is used to denote that an object is not contained within the specified set. If $x \notin A$, then $x$ does not satisfy the membership condition of set $A$.

Step 3: Verification of Membership

To determine the correct relation for the element $10$ with respect to set $A$, we perform an exhaustive comparison:

Since $10$ is not found in the collection $\{1, 2, 3, 4, 5, 6\}$, it fails the membership criteria [Set Membership Axiom].

Visual Representation of Set Membership

Final Solution: 10 $\notin$ A

Step 1: Theoretical Definition of a Set

In set theory, a set is defined as a well-defined collection of distinct objects. The term "well-defined" implies that for any given object, there must be an unambiguous criterion to determine whether that object belongs to the collection or not [Per Cantor’s Definition of a Set].

Step 2: Analysis of the Specific Collection

The collection in question is: "The collection of questions in this Chapter."

• Let $C$ be the collection of all questions contained within the specific textbook chapter.
• For any object $q$ (a question), one can verify if $q \in C$ by checking the contents of the physical or digital textbook chapter.
• Since the chapter content is finite, static, and explicitly documented, the membership of any question in this collection is not subject to personal opinion, bias, or fluctuation [Logical Criterion of Finitude and Objectivity].

Step 3: Verification Against Set Axioms

Step 4: Geometric Representation (Conceptual)

Step 5: Final Conclusion

Because the collection of questions in the chapter is fixed and possesses a clear, objective boundary that allows for definitive membership testing, it satisfies all axiomatic requirements of a mathematical set.

Final Solution: Yes, the collection of questions in this Chapter is a set.

Step 1: Analyzing Set (i) - {1, 2, 3, 6}

The set $\{1, 2, 3, 6\}$ contains positive integers that divide the number 6 without leaving a remainder. In set-builder notation, this is expressed as: $x \in \mathbb{N}$ such that $x$ is a divisor of 6.

Matching: Matches with (c) $\{x : x \text{ is a natural number and divisor of 6}\}$. [Per the definition of divisors in natural numbers].

Step 2: Analyzing Set (ii) - {2, 3}

The elements 2 and 3 are both prime numbers. Furthermore, both 2 and 3 are divisors of 6 (since $6 \div 2 = 3$ and $6 \div 3 = 2$).

Matching: Matches with (a) $\{x : x \text{ is a prime number and a divisor of 6}\}$. [Per the Fundamental Theorem of Arithmetic regarding prime factorization].

Step 3: Analyzing Set (iii) - {M, A, T, H, E, I, C, S}

This set represents the unique letters found in the word "MATHEMATICS". When writing a set in roster form, duplicate elements are listed only once. The letters are M, A, T, H, E, M (repeat), A (repeat), T (repeat), I, C, S.

Matching: Matches with (d) $\{x : x \text{ is a letter of the word MATHEMATICS}\}$. [Per the property of sets that elements are distinct and order is irrelevant].

Step 4: Analyzing Set (iv) - {1, 3, 5, 7, 9}

The elements are odd numbers ($2n-1$ for $n=1,2,3,4,5$) and all values are strictly less than 10.

Matching: Matches with (b) $\{x : x \text{ is an odd natural number less than 10}\}$. [Per the definition of odd natural numbers in the interval $(0, 10)$].

Final Correspondence Table

Final Solution: The correct matches are (i)-(c), (ii)-(a), (iii)-(d), and (iv)-(b).

Step 1: Identifying the Pattern of Elements

Given the set $A = \{5, 25, 125, 625\}$. We observe that each element is a positive integral power of the base number $5$:

• $5^1 = 5$
• $5^2 = 25$
• $5^3 = 125$
• $5^4 = 625$

Generalizing this, every element $x$ in the set follows the rule $x = 5^n$, where $n$ is a natural number [Definition of Natural Numbers, $\mathbb{N}$].

Step 2: Determining Constraints on the Variable

The set consists of exactly four elements, corresponding to the values of $n = 1, 2, 3, 4$. Therefore, the variable $n$ must satisfy the following inequality constraint:

$1 \le n \le 4$ where $n \in \mathbb{N}$

Step 3: Constructing the Set-Builder Form

The set-builder notation requires defining a variable, assigning the rule for the variable, and stating the domain constraint. We define the set as follows:

$A = \{x : x = 5^n, n \in \mathbb{N} \text{ and } 1 \le n \le 4\}$

Visual Representation of Power Growth

Final Solution: \{x : x = 5^n, n \in \mathbb{N}, 1 \le n \le 4\}

Step 1: Identifying the Pattern of the Set

Given the set $A = \{2, 4, 6, . . .\}$, we analyze the constituent elements:

• $2 = 2 \times 1$
• $4 = 2 \times 2$
• $6 = 2 \times 3$

Step 2: Defining the Variable and Domain

Let $x$ represent an arbitrary element of the set. Each element $x$ can be expressed in the form $x = 2n$, where $n$ is a natural number. The set of all natural numbers is denoted by $\mathbb{N} = \{1, 2, 3, . . .\}$. Since the sequence $\{2, 4, 6, . . .\}$ is infinite, $n$ must span the entire set of natural numbers $\mathbb{N}$. [By Definition of Natural Numbers and Even Integers]

Step 3: Constructing the Set-Builder Notation

The set-builder form is structured as $\{x : P(x)\}$, where $P(x)$ is the property that defines the membership of the set.
The property here is that $x$ is an even natural number, or algebraically, $x = 2n$ such that $n \in \mathbb{N}$.

Visual Representation of the Set Construction

Step 4: Formalizing the Output

Combining the variable definition and the property, the set-builder form is written as:
$A = \{x : x = 2n, n \in \mathbb{N}\}$
Alternatively, this can also be expressed as:
$A = \{x \in \mathbb{N} : x \text{ is an even number}\}$

Final Solution: \{x : x = 2n, n \in \mathbb{N}\}

Step 1: Definition of a Set

According to Cantor’s set theory, a set is defined as a well-defined collection of objects [Definition per Set Theory Axioms]. The term "well-defined" implies that for any given object $x$ and a collection $A$, there exists an unambiguous rule to determine whether $x \in A$ or $x \notin A$.

Step 2: Analysis of the Provided Collection

The collection in question is: "The collection of all boys in your class."

• Let $C$ represent the specific classroom of interest.
• Let $U$ represent the universal set of all humans.
• The collection $B$ is defined by the predicate $P(x)$, where $P(x)$ is true if $x$ is a student in class $C$ AND $x$ is of the male gender.

Step 3: Verification of Well-Definedness

To determine if the collection is a set, we test the criterion of objectivity:

Since the criteria for membership are based on factual attributes (enrollment status and gender identification as recorded in institutional documentation), the collection is not subject to individual opinion, ambiguity, or variable interpretation [Ref: Axiom of Specification].

Visual Representation of Set Membership

Step 4: Conclusion

Because the collection is well-defined and the boundary for membership is clear, fixed, and objective, it satisfies the fundamental criteria of a mathematical set.

Final Solution: The collection of all boys in your class is a set, because it is a well-defined collection of objects where membership is objective and verifiable.

Step 1: Analyzing the Inequality

The set $B$ is defined by the condition $x \in \mathbb{Z}$ (where $\mathbb{Z}$ is the set of integers) subject to the interval constraints:

$-\frac{1}{2} < x < \frac{9}{2}$

[By the property of real numbers, we evaluate the decimal equivalents to identify the boundaries of the interval].

• Lower bound: $-\frac{1}{2} = -0.5$
• Upper bound: $\frac{9}{2} = 4.5$

Step 2: Identifying Integers within the Interval

We must select all integers $x$ such that $-0.5 < x < 4.5$. We test consecutive integers:

[Per the definition of the set of integers $\mathbb{Z} = \{..., -2, -1, 0, 1, 2, ...\}$, we include only discrete values falling strictly between the calculated bounds].

Step 3: Visual Representation on a Number Line

Step 4: Final Set Construction

Collecting the valid integers identified in Step 2, we list the elements within the curly braces according to the roster method [Definition of Roster/Tabular Form of Sets].

Final Solution: $B = \{0, 1, 2, 3, 4\}$

1. Definitions and Theoretical Foundation

The symbol $\in$ denotes that an element belongs to a set (is a member of the set), while the symbol $\notin$ denotes that an element does not belong to a set (is not a member of the set). [Per the definition of Set Membership in Zermelo-Fraenkel Set Theory].

2. Analysis of the Given Set

The set $A$ is defined by the roster method as:

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

The elements contained within set $A$ are exclusively the integers $\{1, 2, 3, 4, 5, 6\}$.

3. Verification of Element Membership

To determine the correct symbol for $0 \dots A$, we examine if the element $0$ exists within the list of elements defining set $A$:

• The set $A$ contains only elements $x$ such that $x \in \{1, 2, 3, 4, 5, 6\}$.
• Comparing the target element $0$ against the members of $A$: $0 \neq 1, 0 \neq 2, 0 \neq 3, 0 \neq 4, 0 \neq 5, 0 \neq 6$.

4. Mathematical Justification

Since the integer $0$ is not listed among the defined members of set $A$, it fails the condition of membership [By the Principle of Extensionality, a set is determined by its elements].

Final Solution: 0 $\notin$ A

Step 1: Formal Definition of the Set Condition

The set $C$ is defined by the set-builder notation: $C = \{x : x \in \mathbb{Z}, x^2 \le 4\}$.

• The universal set for the variable $x$ is the set of integers, denoted by $\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}$. [By Definition of Integers]
• The constraint provided is the quadratic inequality $x^2 \le 4$.

Step 2: Analytical Solution of the Inequality

To find the elements of $C$, we must solve for $x$ in the inequality $x^2 - 4 \le 0$.

Factoring the difference of squares [By Algebraic Identity $a^2 - b^2 = (a-b)(a+b)$]:

$(x - 2)(x + 2) \le 0$

The roots of the corresponding equation $(x - 2)(x + 2) = 0$ are $x = 2$ and $x = -2$. These values divide the number line into three intervals: $(-\infty, -2]$, $[-2, 2]$, and $[2, \infty)$.

• Testing $x = -3$: $(-3)^2 = 9 > 4$ (Does not satisfy condition)
• Testing $x = -2$: $(-2)^2 = 4 \le 4$ (Satisfies condition)
• Testing $x = 0$: $0^2 = 0 \le 4$ (Satisfies condition)
• Testing $x = 2$: $2^2 = 4 \le 4$ (Satisfies condition)
• Testing $x = 3$: $3^2 = 9 > 4$ (Does not satisfy condition)

Step 3: Verification of Integer Elements

Since $x$ must be an integer, we identify all integers $x$ such that $x^2 \le 4$ within the closed interval $[-2, 2]$.

Step 4: Visual Representation

Final Solution: $C = \{-2, -1, 0, 1, 2\}$

Step 1: Theoretical Foundation of Set Membership

To determine the correct relation between an element $x$ and a set $A$, we apply the fundamental definition of set membership [Per Zermelo-Fraenkel Set Theory]:

• The symbol $\in$ denotes that an element "belongs to" or is a "member of" the specified set.
• The symbol $\notin$ denotes that an element "does not belong to" or is "not a member of" the specified set.

Step 2: Analysis of the Given Set

The set is defined by the roster method as:

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

The elements contained within set $A$ are explicitly listed as the natural numbers from $1$ to $6$. Any integer $n$ such that $1 \le n \le 6$ satisfies the condition $n \in A$.

Step 3: Verification of the Element

We evaluate the given value $4$ against the contents of set $A$:

1. Compare the element $4$ to the members of set $A = \{1, 2, 3, 4, 5, 6\}$.
2. Since $4$ is the fourth listed element in the set $A$, it holds that $4$ is an element of $A$.
3. [By the Definition of Set Inclusion], the statement $4 \in A$ is true.

Step 4: Final Symbolic Representation

Based on the analysis that the digit $4$ is explicitly present within the set $A$, we use the membership symbol $\in$.

Final Solution: 4 $\in$ A

Step 1: Analyzing the Set Definition

The set $A$ is defined by the roster method as $A = \{1, 2, 3, 4, 5, 6\}$. This implies that the set contains exactly six distinct elements: $1, 2, 3, 4, 5, \text{and } 6$. Any value not present within these curly braces is strictly excluded from the set [Per the definition of set membership].

Step 2: Verification of Element Inclusion

We test the candidate element $x = 8$ against the set $A$:

• The set $A$ contains integers $n$ such that $1 \le n \le 6$ and $n \in \mathbb{Z}^+$.
• Comparing $8$ to the elements of $A$:
• $8 \neq 1, 8 \neq 2, 8 \neq 3, 8 \neq 4, 8 \neq 5, 8 \neq 6$.

Step 3: Logical Application of Symbols

The symbol $\in$ denotes "is an element of," implying a truth value of $True$ if the element is contained within the set. The symbol $\notin$ denotes "is not an element of," implying a truth value of $True$ if the element is not found within the set [By Axiom of Extensionality].

Step 4: Conclusion

Since the number $8$ is not included in the listing of elements for set $A$, the correct relation is expressed as $8 \notin A$.

Final Solution: 8 $\notin$ A

Step 1: Definition of a Set in Roster Form

In set theory, a set is defined as a well-defined collection of distinct objects. When representing a set in Roster (or Tabular) form, every unique element of the set is listed exactly once, separated by commas, and enclosed within curly brackets $\{ \}$. [Per the principles of Zermelo-Fraenkel set theory regarding the extensionality of sets].

Step 2: Analyzing the Given Word

The given condition defines set $D$ as containing the letters of the word "LOYAL". We must perform a lexicographical breakdown of the word to identify individual letters:

• Letter 1: L
• Letter 2: O
• Letter 3: Y
• Letter 4: A
• Letter 5: L

Step 3: Applying the Law of Uniqueness

A fundamental property of sets is that elements must be distinct. If an element is repeated in the descriptive criteria, it appears only once in the roster representation [By the Axiom of Specification].

Step 4: Construction of the Set

By collecting the unique identifiers identified in Step 3, we construct the set $D$:

$D = \{L, O, Y, A\}$

Final Solution: $D = \{L, O, Y, A\}$

Step 1: Theoretical Foundation of Set Membership

The symbol $\in$ denotes that an object is an element of a specific set (membership), while the symbol $\notin$ denotes that an object is not an element of a specific set (non-membership) [Per Zermelo-Fraenkel set theory definition of the element relation].

Step 2: Analysis of the Given Set

The set $A$ is defined by the roster method as:

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

This implies the collection of elements contained within $A$ is limited to the integers $1$ through $6$ inclusive.

Step 3: Verification of the Element

We examine the given element $x = 2$ against the set $A$.

• The set $A$ explicitly contains the integer $2$.
• By the Law of Identity and Set Membership criteria, since $2$ is listed as a member in the roster notation of $A$, the relationship $2 \in A$ must hold true.

Step 4: Visual Representation of Set Membership

Step 5: Final Evaluation

Final Solution: 2 $\in$ A