Sets and Field Axioms for Real Numbers

A set must have well-defined and unambiguous criteria for its elements. The term 'cute' is subjective, varying from person to person, which makes it difficult to determine which puppies qualify as 'cute.' This lack of clear definition prevents the grouping of puppies into a set, as individuals might disagree on which puppies meet this criterion. Therefore, the vagueness of the term undermines the ability to create a precise set based on the given description.

A valid set is defined as a collection of distinct objects that can be clearly identified and defined. "The presidents of the United States of America" represents a specific, well-defined group with a clear membership criterion—individuals who have held the office of the presidency. In contrast, the other options are more subjective and less precisely defined, as terms like "tall," "nice," and "fast" can vary greatly in interpretation and may not yield a consistent membership list.

The mathematical statement ∃ k ∈ K translates to "there exists a number k belonging to the set K." This means that at least one element, denoted as k, can be found within the specified set K. It emphasizes the existence of such an element without specifying which one, contrasting with statements that imply all elements or none at all.

The statement ∀ w > 1, w² > w asserts that for every value of w that is greater than 1, the square of w (w²) will always exceed the value of w itself. This can be understood mathematically, as squaring a number greater than 1 results in a larger number, confirming that w² will indeed be greater than w for all such values. This is a fundamental property of numbers in the context of inequalities.

Natural numbers (1, 2, 3, ...) are a subset of integers (which include negative numbers and zero). Integers are in turn a subset of rational numbers (which can be expressed as fractions of integers). Finally, rational numbers are included within the real numbers, which encompass all rational and irrational numbers. This hierarchical structure demonstrates that the set of natural numbers is indeed contained within the set of integers, which is then contained within the set of rational numbers, and all of these are part of the broader set of real numbers.

-7 is a negative integer, which means it belongs to the set of integers. Since all integers are also rational numbers (as they can be expressed as a fraction), -7 is part of the rational numbers. Additionally, all rational numbers are included in the real numbers. Finally, the set of complex numbers encompasses all real numbers, including integers and rational numbers. Therefore, -7 is classified within the sets of complex numbers, real numbers, rational numbers, and integers.

A number is classified as rational if it can be expressed as a fraction of two integers, where the denominator is not zero. The given number, 0.013378966667772845, is a terminating decimal, meaning it has a finite number of decimal places. This allows it to be represented as a fraction, specifically 13378966667772845/1000000000000000, where both the numerator and denominator are integers. Therefore, it is rational, as it meets the criteria for being expressible as a ratio of two integers.

A number is classified as irrational if it cannot be expressed as a fraction of two integers. The square root of 10 does not fit this criterion, as there are no integers \( a \) and \( b \) such that \( \sqrt{10} = \frac{a}{b} \). Additionally, its decimal expansion is non-terminating and non-repeating, which further confirms its irrationality. In contrast, rational numbers can be expressed as fractions and have either terminating or repeating decimal forms.

4/36 simplifies to 1/9, which is a rational number because it can be expressed as a fraction of two integers. Since all rational numbers are also real numbers, 1/9 belongs to the set of real numbers. Additionally, complex numbers include all real numbers, so 1/9 is also classified as a complex number. Thus, 4/36 fits into the sets of complex, real, and rational numbers.

The phrase "There exists a number b belonging to the set B" indicates the existence of at least one element within the set B. In mathematical notation, this is represented by the existential quantifier "∃," which signifies "there exists." Therefore, "∃ b ∈ B" correctly captures the meaning that there is at least one element b that is a member of the set B. Other options either suggest universal membership or relationships that do not convey the intended existence of an element.

To find \( i^3 \), we start with the definition of \( i \), where \( i = \sqrt{-1} \). First, we calculate \( i^2 \), which equals \(-1\). Then, to find \( i^3 \), we multiply \( i^2 \) by \( i \):

\[
i^3 = i^2 \cdot i = (-1) \cdot i = -i.
\]

Thus, \( i^3 \) simplifies to \(-i\), making it the correct answer.

To solve for \( i^4 \), we start with the definition of \( i \), where \( i = \sqrt{-1} \). Calculating the powers of \( i \) gives us: \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), and \( i^4 = 1 \). Notably, \( i^4 \) returns to 1, demonstrating that the powers of \( i \) cycle every four terms. Thus, \( i^4 \) equals 1, confirming the result.

To find \( i^6 \), we can use the property of powers of \( i \). The powers of \( i \) cycle every four terms: \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), and \( i^4 = 1 \). Since \( i^6 \) can be expressed as \( i^{4+2} = i^4 \cdot i^2 \), we substitute \( i^4 = 1 \) and \( i^2 = -1 \). Therefore, \( i^6 = 1 \cdot (-1) = -1 \).

-7 is classified as an integer because it is a whole number that can be positive, negative, or zero. While it is also a rational number (as it can be expressed as -7/1), the question asks for the most restrictive set. Natural numbers are only positive whole numbers, excluding -7. Therefore, among the options provided, integers represent the most specific category that includes -7, making it the correct answer.

The statement "Every negative integer j is less than or equal to its inverse" asserts that for all integers j that are less than zero (negative integers), the relationship j ≤ -j holds true. The chosen option correctly captures this universal quantification over negative integers, indicating that for every such j, the inequality is satisfied. The other options either misrepresent the quantifiers or apply to integers that are not strictly negative, thus failing to accurately reflect the intended meaning of the original statement.