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Total Categories: 6
The set of integers includes all positive and negative whole numbers, but explicitly excludes zero.
Answer: False
The fundamental definition of an integer includes zero, along with positive and negative whole numbers.
The set of natural numbers (ℕ) is a subset of the set of integers (ℤ), and integers are a subset of rational numbers (ℚ).
Answer: True
The hierarchical relationship among number sets establishes natural numbers as a subset of integers, and integers as a subset of rational numbers.
Numbers like 9.75 and 5/4 are considered integers because they can be expressed as a ratio of two whole numbers.
Answer: False
Integers are whole numbers without fractional or decimal components. Numbers like 9.75 and 5/4 are not integers, although they are rational numbers.
According to the fundamental definition, which of the following is NOT considered an integer?
Answer: 1/2
Integers are whole numbers, including positive, negative, and zero, without any fractional or decimal components. Therefore, 1/2 is not an integer.
Which of the following statements accurately describes the relationship between integers and other number sets?
Answer: Natural numbers are a subset of integers.
The set of natural numbers is a subset of the set of integers, which in turn is a subset of rational numbers, and rational numbers are a subset of real numbers.
The symbol ℤ used to denote the set of all integers is derived from the Latin word 'Zahlen', meaning 'numbers'.
Answer: False
The symbol ℤ is derived from the German word 'Zahlen', not Latin.
The word 'integer' literally means 'whole' or 'untouched' and originates from the Latin word 'integer'.
Answer: True
The etymology of 'integer' traces back to the Latin 'integer', meaning 'whole' or 'untouched', derived from 'in' (not) and 'tangere' (to touch).
Historically, the term 'integer' always included negative numbers, even before the 18th century.
Answer: False
Historically, the term 'integer' initially referred only to positive whole numbers, and its definition expanded to include negative numbers later, notably by Leonhard Euler in 1765.
Georg Cantor's introduction of infinite sets and set theory influenced the adoption of the phrase 'the set of the integers' by the end of the 19th century.
Answer: True
The formal concept of 'the set of the integers' gained traction towards the end of the 19th century, directly influenced by Georg Cantor's foundational work in set theory.
The notation 'Z' for integers was first used in a textbook by David Hilbert in 1947.
Answer: False
While attributed to David Hilbert, the notation 'Z' for integers first appeared in a textbook titled 'Algèbre' by Nicolas Bourbaki in 1947, not a textbook by Hilbert himself.
The New Math movement in the 1950s clarified 'whole numbers' to include negative numbers, making it synonymous with 'integers'.
Answer: False
During the New Math movement, 'whole numbers' was redefined to refer specifically to natural numbers (excluding negative numbers), thus distinguishing it from 'integers' which included negative numbers.
The symbol ℤ used to denote the set of all integers originates from which language?
Answer: German
The symbol ℤ for integers is derived from the German word 'Zahlen', meaning 'numbers'.
When did the definition of 'integer' expand to include negative numbers?
Answer: As their usefulness became recognized, notably by Leonhard Euler in 1765.
The inclusion of negative numbers in the definition of integers became prominent with mathematicians like Leonhard Euler in the mid-18th century.
What mathematical concept influenced the adoption of the phrase 'the set of the integers' towards the end of the 19th century?
Answer: Infinite sets and set theory introduced by Georg Cantor.
Georg Cantor's work on infinite sets and set theory provided the formal framework that led to the widespread use of the phrase 'the set of the integers' by the late 19th century.
Who is credited with the use of the letter 'Z' to denote the set of integers?
Answer: David Hilbert
The use of the letter 'Z' to denote the set of integers is attributed to David Hilbert.
How was the term 'whole numbers' redefined in American elementary schools during the New Math movement of the late 1950s?
Answer: It referred specifically to natural numbers, excluding negative numbers.
During the New Math movement, 'whole numbers' was redefined to refer exclusively to natural numbers, thereby distinguishing it from 'integers' which include negative values.
The integers form the smallest group and the smallest ring that contain the natural numbers.
Answer: True
In abstract algebra, the integers are indeed recognized as the smallest group and the smallest ring that encompass the natural numbers.
In algebraic number theory, 'rational integers' are a broader category than 'algebraic integers'.
Answer: False
'Algebraic integers' represent a more general category, with 'rational integers' being a specific subset of algebraic integers that are also rational numbers.
The set of integers (ℤ) is closed under addition, subtraction, and multiplication.
Answer: True
The set of integers is closed under addition, subtraction, and multiplication, meaning that performing these operations on any two integers will always yield another integer.
The integers form a ring that is considered the 'initial object' in the category of rings, meaning a unique ring homomorphism exists from the integers into any other ring.
Answer: True
In ring theory, the integers are indeed the 'initial object' in the category of rings, implying a unique ring homomorphism from the integers to any other ring.
The unique homomorphism from integers to another ring is always injective, regardless of the target ring's characteristic.
Answer: False
The unique homomorphism from the integers to another ring is injective only if the characteristic of the target ring is zero; it is not always injective.
Integers are closed under division and exponentiation, as long as the result is a whole number.
Answer: False
Integers are not closed under division or exponentiation, as these operations can yield non-integer results (e.g., 1/2 or 2^-1).
The set of integers under addition forms an abelian group due to properties like closure, associativity, and the existence of identity and inverse elements.
Answer: True
The integers under addition satisfy all the axioms of an abelian group, including closure, associativity, commutativity, and the existence of an identity element (0) and inverse elements (negatives).
The set of integers under multiplication is considered a group because all integers have a multiplicative inverse within the set.
Answer: False
The set of integers under multiplication is not a group because most integers (e.g., 2) do not have a multiplicative inverse that is also an integer.
The lack of zero divisors in integers implies that if the product of two integers is zero, then at least one of the integers must be zero.
Answer: True
The property of having no zero divisors means that the product of two non-zero integers can never be zero, thus if a product is zero, at least one factor must be zero.
The smallest field that contains the integers as a subring is the field of rational numbers (ℚ).
Answer: True
The field of rational numbers is the smallest field that contains the integers, as it extends integers to include multiplicative inverses for all non-zero elements.
What basic algebraic structures do integers form that contain the natural numbers?
Answer: The smallest group and the smallest ring.
The integers are uniquely characterized as the smallest group and the smallest ring that contain the natural numbers.
Under which of the following operations is the set of integers NOT closed?
Answer: Division
The set of integers is not closed under division, as dividing one integer by another may result in a fraction, which is not an integer.
What unique property does the ring of integers possess in the context of ring theory?
Answer: It is considered the 'initial object' in the category of rings.
In ring theory, the ring of integers is uniquely characterized as the 'initial object', meaning there is a unique homomorphism from it to any other ring.
What is the smallest field that contains the integers as a subring?
Answer: The field of rational numbers (ℚ)
The field of rational numbers is the smallest field that contains the integers as a subring, as it provides multiplicative inverses for all non-zero integers.
The Euclidean algorithm is a method for computing the least common multiple (LCM) of two integers.
Answer: False
The Euclidean algorithm is used to compute the greatest common divisor (GCD) of two integers, not the least common multiple (LCM).
In Euclidean division for integers, if a = q × b + r, what is the condition for the remainder r?
Answer: 0 ≤ r < |b|
According to the definition of Euclidean division, the remainder r must be non-negative and strictly less than the absolute value of the divisor b.
The fact that integers form a Euclidean domain has what significant implication?
Answer: Any positive integer can be uniquely factored into prime numbers.
The property of integers forming a Euclidean domain directly implies the fundamental theorem of arithmetic, which states that every positive integer has a unique prime factorization.
Which of the following best describes the order-theoretic properties of the set of integers?
Answer: It is a totally ordered set with no upper or lower bound.
The set of integers is totally ordered, meaning any two elements can be compared, and it extends infinitely in both positive and negative directions, thus having no upper or lower bound.
In the formal set-theoretic construction, an integer represented as an ordered pair (a,b) intuitively stands for the sum a+b.
Answer: False
In the formal set-theoretic construction, an ordered pair (a,b) intuitively represents the difference a-b, not the sum a+b.
In the formal set-theoretic construction of integers using ordered pairs (a,b), what does the pair intuitively represent?
Answer: The difference a - b
Intuitively, an ordered pair (a,b) in the set-theoretic construction of integers represents the difference a - b.
How is the equivalence relation (a,b) ~ (c,d) defined for ordered pairs of natural numbers in the construction of integers?
Answer: a + d = b + c
The equivalence relation (a,b) ~ (c,d) is defined as a + d = b + c, ensuring that pairs representing the same integer are equivalent.
In the ordered pair construction of integers, how is the natural number 'n' identified?
Answer: [(n,0)]
In the ordered pair construction, a natural number 'n' is identified with the equivalence class [(n,0)].
The set of integers is countably infinite, meaning its elements can be put into a one-to-one correspondence with the natural numbers.
Answer: True
The set of integers is indeed countably infinite, a property demonstrated by the existence of a bijection between its elements and the natural numbers.
Why is the 'pair(x,y)' operation in the equivalence class construction of integers NOT considered a 'free constructor' in computer science?
Answer: The same integer can be represented by multiple algebraic terms.
The 'pair(x,y)' operation is not a free constructor because a single integer can be represented by multiple distinct ordered pairs (e.g., 0 can be (0,0), (1,1), etc.), violating the unique term representation property.
What is the cardinality of the set of integers?
Answer: Countably infinite
The set of integers is countably infinite, meaning its elements can be put into a one-to-one correspondence with the natural numbers.