30 Questions
What is the notation for the reciprocal of a real number?
1/A
What is the definition of a negative real number?
A real number is negative if A < 0
What is the set of all positive real numbers denoted by?
R⁺
What is the definition of the open interval?
The set of all points excluding a and b
What is the notation for the half-open interval from a to b, including a?
[a, b)
What is the real line sometimes referred to as?
An open interval
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
What is a positive integer according to the text?
A real number that belongs to every inductive set
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
What is a rational number according to the text?
A real number that is a quotient of integers
What is an irrational number according to the text?
A real number that is not a rational number
What is a neighborhood of a point in real analysis?
An open interval that contains the point
What is the characteristic of the set of all rational numbers?
Countable
Which of the following sets is known to be uncountable?
The set of all real numbers
What can be said about a subset of a countable set?
It is always countable
What is true about the set of all irrational numbers?
It is uncountable
What can be said about the Cantor set?
It is an uncountable set
What is true about the set of all sequences whose elements are the digits 0 and 1?
It is an uncountable set
What is the cardinal number of an empty set?
Zero
What is the condition for a function to be one-to-one?
If f(a) ≠ f(b) then a ≠ b
What is the notation for a one-to-one correspondence between two sets?
1-1
What can be said about two sets if there exists a 1-1 correspondence between them?
They are equivalent
What is the definition of a countable set?
A set that is equivalent to the set of all positive integers
What is the smallest size of infinite sets?
Countable
What is the least upper bound of a set?
The smallest number that is an upper bound for the set
What is the greatest lower bound of a set?
The largest number that is a lower bound for the set
Which of the following sets has a least upper bound and a greatest lower bound?
Set of all real numbers
What is the significance of the Least Upper Bound Axiom?
It states that the set of real numbers has no holes in it
What is the result when a set is a singleton set?
Its least upper bound and greatest lower bound are equal
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
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.
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