Number Theory Quiz

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Questions and Answers

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?

Integers

Whole numbers (correct)

Rational numbers

Natural numbers

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?

They cannot be expressed as a repeating decimal. Signup and view all the answers

What type of infinity is exhibited by the set of natural numbers?

Countable infinity Signup and view all the answers

What is a characteristic of the set of real numbers?

It consists of rational and irrational numbers. Signup and view all the answers

Which of the following sets is considered an uncountable infinity?

Irrational numbers Signup and view all the answers

How is the relationship between natural numbers, whole numbers, and integers expressed?

N ⊂ W ⊂ Z Signup and view all the answers

What property ensures that the result of the group operation does not leave the group?

Closure Property Signup and view all the answers

In a group G, what is the term for a subset that satisfies all the group axioms under the same operation?

Subgroup Signup and view all the answers

Which type of group allows for the order of operation to be switched without changing the result?

Abelian group Signup and view all the answers

What is the smallest subset which can generate all elements of a group called?

Generating set Signup and view all the answers

What is the term used for a representation that cannot be expressed as a direct sum of sub-representations?

Irreducible representation Signup and view all the answers

How is the degree of an irreducible representation related to the order of a finite group G?

It divides the order of G. Signup and view all the answers

What does the number of distinct irreducible representations in a finite group correspond to?

The number of conjugacy classes in the group. Signup and view all the answers

What is the equation representing the sum of the squares of the degrees of irreducible representations?

$ ext{(sum of } l_i^2 ext{)} = |G|$ Signup and view all the answers

If a finite group G has r conjugacy classes, what can be said about the degrees of the irreducible representations?

There are at most r distinct degrees. Signup and view all the answers

What does the least integer principle state about a nonempty subset of integers?

It contains a smallest integer. Signup and view all the answers

Which property does addition of integers not satisfy?

Notational Independence Signup and view all the answers

What is true regarding rational numbers compared to integers in the context of the largest integer principle?

Rational numbers can be bounded above without guaranteeing a largest number. Signup and view all the answers

Which of the following statements about integer multiplication is correct?

It is commutative and associative. Signup and view all the answers

How can any real number be approximated according to the content?

By a rational number. Signup and view all the answers

What is the relationship between integers and real numbers inferred from the content?

Integers are a sparse subset of real numbers. Signup and view all the answers

What can be said about the density of rational numbers in finite intervals?

There are infinite rational numbers in any finite interval. Signup and view all the answers

Which of the following properties is NOT satisfied by division of integers?

All listed properties Signup and view all the answers

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?

Order of the element Signup and view all the answers

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?

Ha is identical to H Signup and view all the answers

What must be true for two distinct elements a and b in G for them to belong to the same coset of H?

The element ab^{-1} must belong to H Signup and view all the answers

What is the implication of having two different cosets of H?

They are disjoint sets Signup and view all the answers

What will happen if the order of a coset Ha is less than the order of the subgroup H?

This is possible only if some elements are equal Signup and view all the answers

How do we generally refer to the left coset of H?

Coset Signup and view all the answers

In the context of groups, what can be inferred about the order of cosets compared to the order of their subgroup H?

The order of every coset of H is the same as that of H Signup and view all the answers

What does the notation Ha represent in group theory?

Set of all products $ha$ where $h$ is in H Signup and view all the answers

Which property ensures that the operation is unambiguously defined for multiple elements in a group?

Associative Property Signup and view all the answers

What is the order of a group?

The number of elements in a group Signup and view all the answers

Which of the following groups is NOT a finite group?

The set of integers under addition Signup and view all the answers

What defines an Abelian group?

Group product is commutative Signup and view all the answers

Which statement about a generating set is correct?

It always generates all elements of the group Signup and view all the answers

What is a cyclic group?

A group generated by a single element Signup and view all the answers

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?

Cancellation Law Signup and view all the answers

What type of subgroup is the identity set E = {e} classified as?

Trivial subgroup Signup and view all the answers

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 (Q̄): 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.
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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Description

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!