Real Numbers Axioms

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Questions and Answers

What is the notation for the reciprocal of a real number?

A²

A⁻¹

1/A (correct)

A³

What is the definition of a negative real number?

A real number is negative if A < 0 (correct)

A real number is negative if A ≠ 0

A real number is negative if A = 0

A real number is negative if A > 0

What is the set of all positive real numbers denoted by?

R⁺ (correct)

What is the definition of the open interval?

The set of all points excluding a and b Signup and view all the answers

What is the notation for the half-open interval from a to b, including a?

[a, b) Signup and view all the answers

What is the real line sometimes referred to as?

An open interval Signup and view all the answers

What is an inductive set in the context of real numbers?

A set of real numbers that contains 1 and the successor of every element in the set Signup and view all the answers

What is a positive integer according to the text?

A real number that belongs to every inductive set Signup and view all the answers

What is a fundamental theorem in the context of integers?

Every integer can be represented as a product of prime factors in only one way Signup and view all the answers

What is a rational number according to the text?

A real number that is a quotient of integers Signup and view all the answers

What is an irrational number according to the text?

A real number that is not a rational number Signup and view all the answers

What is a neighborhood of a point in real analysis?

An open interval that contains the point Signup and view all the answers

What is the characteristic of the set of all rational numbers?

Countable Signup and view all the answers

Which of the following sets is known to be uncountable?

The set of all real numbers Signup and view all the answers

What can be said about a subset of a countable set?

It is always countable Signup and view all the answers

What is true about the set of all irrational numbers?

It is uncountable Signup and view all the answers

What can be said about the Cantor set?

It is an uncountable set Signup and view all the answers

What is true about the set of all sequences whose elements are the digits 0 and 1?

It is an uncountable set Signup and view all the answers

What is the cardinal number of an empty set?

Zero Signup and view all the answers

What is the condition for a function to be one-to-one?

If f(a) ≠ f(b) then a ≠ b Signup and view all the answers

What is the notation for a one-to-one correspondence between two sets?

1-1 Signup and view all the answers

What can be said about two sets if there exists a 1-1 correspondence between them?

They are equivalent Signup and view all the answers

What is the definition of a countable set?

A set that is equivalent to the set of all positive integers Signup and view all the answers

What is the smallest size of infinite sets?

Countable Signup and view all the answers

What is the least upper bound of a set?

The smallest number that is an upper bound for the set Signup and view all the answers

What is the greatest lower bound of a set?

The largest number that is a lower bound for the set Signup and view all the answers

Which of the following sets has a least upper bound and a greatest lower bound?

Set of all real numbers Signup and view all the answers

What is the significance of the Least Upper Bound Axiom?

It states that the set of real numbers has no holes in it Signup and view all the answers

What is the result when a set is a singleton set?

Its least upper bound and greatest lower bound are equal Signup and view all the answers

What is true about the least upper bound and greatest lower bound of a set?

A set can have at most one least upper bound and at most one greatest lower bound Signup and view all the answers

Study Notes

Notation and Definitions

The reciprocal of a real number ( x ) is denoted as ( \frac{1}{x} ).
A negative real number is any real number less than zero.
The set of all positive real numbers is denoted by ( \mathbb{R}^+ ).

Intervals

An open interval is defined as the set of all real numbers between two endpoints, excluding the endpoints themselves.
The notation for a half-open interval from ( a ) to ( b ), including ( a ) but excluding ( b ), is written as ([a, b)).

Real Numbers and Sets

The real line is sometimes referred to as the continuum.
An inductive set consists of a non-empty set of real numbers closed under successor operations.

Integers and Rationality

A positive integer is defined as any integer greater than zero.
The fundamental theorem in the context of integers pertains to the unique factorization of integers into prime numbers.

Rational and Irrational Numbers

A rational number is defined as a number that can be expressed as the quotient of two integers, where the denominator is not zero.
An irrational number cannot be expressed as a simple fraction or quotient of integers.

Neighborhoods and Characteristics

A neighborhood of a point in real analysis is an interval surrounding that point.
The set of all rational numbers is dense in the real numbers, meaning between any two real numbers, there is at least one rational number.

Countability and Cardinality

The set of all irrational numbers is known to be uncountable.
A subset of a countable set may be finite or countably infinite.
The Cantor set is an example of a set that is uncountable while having measure zero.
The set of all sequences whose elements are the digits 0 and 1 is also uncountable.

Function Properties

The cardinal number of an empty set is zero.
For a function to be one-to-one (injective), it must map distinct elements of the domain to distinct elements of the codomain.
Notation for a one-to-one correspondence between two sets is typically depicted by a bijection between the sets.

Set Comparisons and Definitions

If there exists a one-to-one correspondence between two sets, they are considered to be of the same cardinality.
A countable set is defined as a set that is either finite or can be put in a one-to-one correspondence with the natural numbers.
The smallest size of infinite sets is the cardinality of the natural numbers.

Bounds and Axioms

The least upper bound of a set is the smallest number that is greater than or equal to all numbers in the set.
The greatest lower bound of a set is the largest number that is less than or equal to all numbers in the set.
A set with a least upper bound and a greatest lower bound may be bounded or unbounded.
The Least Upper Bound Axiom asserts that every non-empty set of real numbers that is bounded above has a least upper bound.
For a singleton set, the least upper bound and greatest lower bound are both equal to the single element of the set.

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Description

Test your understanding of real number axioms, including the existence of negative numbers, reciprocals, and the order axioms. Understand how these axioms relate to each other and real number properties.