Real Numbers Axiomatics Quiz

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Questions and Answers

Which of the following is NOT an axiom for addition in real numbers?

For all real numbers $a$, $b$, and $c$, $(a + b) + c = a + (b + c)$

For all real numbers $a$ and $b$, $a + b = b + a$

There exists a real number $0$ such that for all real numbers $a$, $a + 0 = 0 + a = a$

For all real number $a$ exists $b$ real number such that $a * b= 1$ (correct)

Which axiom guarantees the existence of an opposite element for every real number under addition?

Associativity

Existence of a neutral element

Commutativity

Existence of an opposite element (correct)

What does the axiom of transitivity in the order axioms state?

For any two real numbers, either one is less than or equal to the other

If $a$ is less than or equal to $b$ and $b$ is less than or equal to $c$, then $a$ is less than or equal to $c$. (correct)

If $a$ is less than or equal to $b$ and $b$ is less than or equal to $a$, then $a$ equals $b$.

Every real number is less than or equal to itself.

According to the axioms of communication, what does distributivity apply to?

Multiplication over addition (A) Signup and view all the answers

Which of the following is the axiom of completeness intended to guarantee?

That there is no 'gap' in the real numbers. If two sets of reals $X$ and $Y$ are such that every element in $X$ is less than or equal to every element in $Y$, there exists a real number $c$ such that all elements in $X$ are less or equal to $c$, and $c$ is less or equal to all elements in $Y$. (B) Signup and view all the answers

What is a difference between two real numbers $a$ and $b$?

A number $x$ such that $x + b = a$. (B) Signup and view all the answers

What is considered a countable set?

Any set that can be placed in a one-to-one correspondence with the set of natural numbers. (B) Signup and view all the answers

Which operation does the axiom of distributivity connect?

Addition and Multiplication (D) Signup and view all the answers

Given a non-empty set X that is bounded above, what can be said about its exact upper bound?

It exists and is unique. (A) Signup and view all the answers

What does Archimedes' principle imply about the set of natural numbers (N)?

N is bounded below, but not above. (A) Signup and view all the answers

If $x$ is a positive real number and $x < \frac{1}{n}$ for all natural numbers $n$, what must be the value of $x$?

x = 0 (C) Signup and view all the answers

What does Cantor's principle state about a system of nested segments $I_1 \supset I_2 \supset ... \supset I_n \supset ...$?

There is at least one point common to all segments. (C) Signup and view all the answers

Given the set $A = [2; 8]$, what are the values of $inf A$ and $sup A$ respectively?

inf A = 2, sup A = 8 (D) Signup and view all the answers

Which of the following describes a system of nested sets?

A sequence of sets where each set is a subset of the previous set. (B) Signup and view all the answers

What is the intersection of the intervals $(0, \frac{1}{n})$ for all natural numbers $n$, denoted as $\bigcap_{n=1}^{\infty} (0, \frac{1}{n})$?

The empty set, $\emptyset$ (B) Signup and view all the answers

Given $a, b \in R$ such that $a < b$, what can be said about the existence of a rational number $r$ between $a$ and $b$?

There are infinitely many rational numbers between $a$ and $b$. (C) Signup and view all the answers

Which of the following correctly describes a finite set?

A set whose cardinality can be assigned a natural number. (D) Signup and view all the answers

Which set includes numbers that can be expressed as a ratio of two integers, where the denominator is a natural number?

The set of rational numbers. (C) Signup and view all the answers

Which of the following numbers is classified as an irrational number?

$\sqrt{7}$ (A) Signup and view all the answers

What condition defines a set X as bounded above?

There exists some number $c$ in the real numbers where all elements of X are less than or equal to $c$. (C) Signup and view all the answers

What does it mean for a set to be unbounded below?

For every real number $b$, there exists an element in the set that is less than $b$. (D) Signup and view all the answers

What defines the exact upper bound (supremum) of a set?

The smallest of the set's upper bounds. (A) Signup and view all the answers

Which statement correctly describes the exact lower bound (infimum) of a set?

It is the largest of all lower bounds of the set. (D) Signup and view all the answers

Can a set have an exact upper bound without having a maximum element?

Yes, a set may have an exact upper bound without a maximum element. (D) Signup and view all the answers

Study Notes