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Questions and Answers
What is the set of natural numbers defined as?
What is the set of natural numbers defined as?
- {0, 1, 2,..., ∞}
- {−∞,..., ∞}
- {−1, 0, 1, 2,..., ∞}
- {1, 2, 3,..., ∞} (correct)
Which of the following sets includes zero?
Which of the following sets includes zero?
- Integers
- Whole numbers (correct)
- Rational numbers
- Natural numbers
What defines a rational number?
What defines a rational number?
- A number that can be expressed as a ratio of two integers (correct)
- A number that can be written as a non-repeating decimal
- Any integer number including zero
- A number without a decimal representation
Which statement about irrational numbers is true?
Which statement about irrational numbers is true?
What type of infinity is exhibited by the set of natural numbers?
What type of infinity is exhibited by the set of natural numbers?
What is a characteristic of the set of real numbers?
What is a characteristic of the set of real numbers?
Which of the following sets is considered an uncountable infinity?
Which of the following sets is considered an uncountable infinity?
How is the relationship between natural numbers, whole numbers, and integers expressed?
How is the relationship between natural numbers, whole numbers, and integers expressed?
What property ensures that the result of the group operation does not leave the group?
What property ensures that the result of the group operation does not leave the group?
In a group G, what is the term for a subset that satisfies all the group axioms under the same operation?
In a group G, what is the term for a subset that satisfies all the group axioms under the same operation?
Which type of group allows for the order of operation to be switched without changing the result?
Which type of group allows for the order of operation to be switched without changing the result?
What is the smallest subset which can generate all elements of a group called?
What is the smallest subset which can generate all elements of a group called?
What is the term used for a representation that cannot be expressed as a direct sum of sub-representations?
What is the term used for a representation that cannot be expressed as a direct sum of sub-representations?
How is the degree of an irreducible representation related to the order of a finite group G?
How is the degree of an irreducible representation related to the order of a finite group G?
What does the number of distinct irreducible representations in a finite group correspond to?
What does the number of distinct irreducible representations in a finite group correspond to?
What is the equation representing the sum of the squares of the degrees of irreducible representations?
What is the equation representing the sum of the squares of the degrees of irreducible representations?
If a finite group G has r conjugacy classes, what can be said about the degrees of the irreducible representations?
If a finite group G has r conjugacy classes, what can be said about the degrees of the irreducible representations?
What does the least integer principle state about a nonempty subset of integers?
What does the least integer principle state about a nonempty subset of integers?
Which property does addition of integers not satisfy?
Which property does addition of integers not satisfy?
What is true regarding rational numbers compared to integers in the context of the largest integer principle?
What is true regarding rational numbers compared to integers in the context of the largest integer principle?
Which of the following statements about integer multiplication is correct?
Which of the following statements about integer multiplication is correct?
How can any real number be approximated according to the content?
How can any real number be approximated according to the content?
What is the relationship between integers and real numbers inferred from the content?
What is the relationship between integers and real numbers inferred from the content?
What can be said about the density of rational numbers in finite intervals?
What can be said about the density of rational numbers in finite intervals?
Which of the following properties is NOT satisfied by division of integers?
Which of the following properties is NOT satisfied by division of integers?
What is the term used for the smallest positive integer k such that $a^k = e$ for an element a in a finite group G?
What is the term used for the smallest positive integer k such that $a^k = e$ for an element a in a finite group G?
If H is a subgroup of G, what is the relationship between the left coset Ha and the subgroup H for elements a that are already in H?
If H is a subgroup of G, what is the relationship between the left coset Ha and the subgroup H for elements a that are already in H?
What must be true for two distinct elements a and b in G for them to belong to the same coset of H?
What must be true for two distinct elements a and b in G for them to belong to the same coset of H?
What is the implication of having two different cosets of H?
What is the implication of having two different cosets of H?
What will happen if the order of a coset Ha is less than the order of the subgroup H?
What will happen if the order of a coset Ha is less than the order of the subgroup H?
How do we generally refer to the left coset of H?
How do we generally refer to the left coset of H?
In the context of groups, what can be inferred about the order of cosets compared to the order of their subgroup H?
In the context of groups, what can be inferred about the order of cosets compared to the order of their subgroup H?
What does the notation Ha represent in group theory?
What does the notation Ha represent in group theory?
Which property ensures that the operation is unambiguously defined for multiple elements in a group?
Which property ensures that the operation is unambiguously defined for multiple elements in a group?
What is the order of a group?
What is the order of a group?
Which of the following groups is NOT a finite group?
Which of the following groups is NOT a finite group?
What defines an Abelian group?
What defines an Abelian group?
Which statement about a generating set is correct?
Which statement about a generating set is correct?
What is a cyclic group?
What is a cyclic group?
Which consequence of group axioms states that if two elements result in the same product with a third element, then those two elements must be equal?
Which consequence of group axioms states that if two elements result in the same product with a third element, then those two elements must be equal?
What type of subgroup is the identity set E = {e} classified as?
What type of subgroup is the identity set E = {e} classified as?
Study Notes
Number Theory
-
Types of Numbers:
- Natural Numbers (N): Counting numbers {1, 2, 3,..., ∞}.
- Whole Numbers (W): Includes zero, {0, 1, 2,..., ∞}, also called non-negative integers.
- Integers (Z): Positive and negative whole numbers and zero, Z = {..., −2, −1, 0, 1, 2,..., ∞}.
- Real Numbers (R): Continuous and uncountable set, R = {−∞,..., ∞}.
- Rational Numbers (Q): Real numbers expressible as a ratio of integers, often finite or repeating decimals.
- Irrational Numbers (QÌ„): Non-rational real numbers, cannot be expressed as finite or repeating decimals.
-
Infinities:
- Integers, whole numbers, and natural numbers are countably infinite; same level of infinity.
- The set of rational numbers is also countably infinite, proven by Cantor's diagonalization method.
- Real numbers form a continuous, uncountable infinity.
- Rational numbers are dense; any real number can be approximated by rationals within any interval.
-
Integer Principles:
- Least Integer Principle: Any non-empty integer subset bounded below contains a smallest integer.
- Largest Integer Principle: Any non-empty integer subset bounded above contains a largest integer.
-
Arithmetic Operations on Integers:
- Commutativity: a + b = b + a; a · b = b · a.
- Associativity: a + (b + c) = (a + b) + c; a · (b · c) = (a · b) · c.
- Distributivity: a · (b + c) = a · b + a · c.
- Subtraction is defined through addition of negative integers; division does not share these properties.
Group Theory
-
Group Definition and Axioms:
- A group G includes elements with a binary operation (group product) that satisfies:
- Closure Property: If a, b ∈ G, then a · b ∈ G.
- Associative Property: a · (b · c) = (a · b) · c.
- Unique Identity: There exists a unique element e ∈ G such that a · e = a.
- Unique Inverse: For every a ∈ G, a unique b exists such that a · b = e.
- A group G includes elements with a binary operation (group product) that satisfies:
-
Group Order:
- The number of elements in a group is its order; can be finite, countably infinite, or uncountably infinite.
- An Abelian group is one where the group product is commutative: ab = ba.
-
Subgroups:
- A subset S of G is a subgroup if it satisfies all group axioms under the same operation.
- Trivial subgroups: E = {e} is a subgroup of every group.
-
Generating Sets and Cyclic Groups:
- A generating set is the smallest subset whose elements can generate every element of G.
- A cyclic group is formed by a single generator; every finite group has a cyclic subgroup.
-
Cosets:
- For a subgroup H of G and a ∈ G, Ha is the left coset; aH is the right coset.
- The order of cosets is equal to the order of H. Different cosets of H are disjoint.
-
Coset Properties:
- If a, b belong to the same coset of H, then there exists some h ∈ H such that ab−1 ∈ H.
- Each element of G belongs to at least one coset of H.
Representations of Groups
- Irreducible Representations:
- A representation is irreducible if it cannot be decomposed into simpler representations.
- The degree of an irreducible representation divides the order of a finite group G.
- The number of distinct irreducible representations is equal to the number of conjugacy classes in G.
- The sum of the squares of the dimensions of distinct irreducible representations equals the order of the group.
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Test your knowledge on various types of numbers including natural, whole, integers, and more. This quiz will also explore concepts of infinity and the relationships between different number sets. Challenge yourself with questions that delve into the intricacies of number theory!