Algebra Properties and Inequalities

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Questions and Answers

If $r = ab$ is a root of the equation $x^n = m$, what can be concluded about $b$ given that $a$ and $b$ are relatively prime?

$b$ can be any positive integer.

$b$ must be coprime with $m$.

$b$ is equal to 1. (correct)

$b$ must always be greater than 1.

Why can we deduce that the only possible rational solutions of the equation $x^n = m$ are integers?

Because $m$ must be an even number.

Because $a$ and $b$ cannot be both even.

Because $b$ divides $an$ when $b = 1$. (correct)

Because $r$ can be expressed as a fraction of two integers.

Which of the following statements is true regarding prime numbers and irrationality?

Both prime numbers can never be even.

Any prime number $p$ is irrational. (correct)

Every prime number is rational.

The sum of two prime numbers is always rational.

What does Gauss’ lemma imply in the context of rational solutions to the polynomial equations?

If a prime divides a product, it must divide at least one factor. Signup and view all the answers

What can be inferred about the expression $p + q$ where $p$ and $q$ are both prime numbers?

$p + q$ is an irrational number. Signup and view all the answers

What is the solution set for the equation |x^2 + 4x - 5| = x^2 + 4x - 5?

]−∞, −5] ∪ [1, +∞[ Signup and view all the answers

For which condition does |a| < a hold true?

a < 0 Signup and view all the answers

What does the expression |x^2 + 4x - 5| ≤ x^2 + 4x - 5 imply about the solutions?

The solutions include some negative intervals. Signup and view all the answers

Which inequality describes the solutions for |x^2 + 4x - 5| > x^2 + 4x - 5?

]−5, 1[ Signup and view all the answers

What conclusion can be drawn from |a| ≥ a for any a ∈ R?

a is always non-negative. Signup and view all the answers

If x and y are both positive, which inequality relates x + y to 2(x + y)?

x + y ≤ 2(x + y) Signup and view all the answers

What is the correct interpretation of the set of solutions for the inequality |x^2 + 4x - 5| < x^2 + 4x - 5?

It has no solutions. Signup and view all the answers

What is the significance of M in relation to the sets A and B?

M is the least upper bound of A ∪ B. Signup and view all the answers

What is the outcome when solving the equation 3x - 1 + 2 = 1?

x = 3/2 Signup and view all the answers

What can be concluded about the element m regarding the lower bounds of sets A and B?

m is the greatest lower bound for A and B. Signup and view all the answers

If m0 is a lower bound of A ∪ B, what relation does it have with m?

m is greater than or equal to m0. Signup and view all the answers

What is the highest value that elements of set A can reach based on the provided content?

2 Signup and view all the answers

What is the established infimum of set A?

-2 Signup and view all the answers

What happens when elements of A and B are represented together?

Their upper and lower bounds merge. Signup and view all the answers

Which statement best describes sup A?

It is equal to the maximum of A. Signup and view all the answers

For values of m and n being non-zero integers, what bounds do they provide for set A?

Bounded above by 2 and below by -2. Signup and view all the answers

What is the infimum of the interval ]a, b[?

a Signup and view all the answers

If A = ]−∞, b], what can be said about the supremum of A?

It is equal to b. Signup and view all the answers

Which statement is true about the set A = {r ∈ Q, r < a}?

A admits a supremum. Signup and view all the answers

What can you conclude about the sets A and B if both are nonempty and bounded from above?

The product set AB could be unbounded. Signup and view all the answers

What must be true if A is one of the intervals [a, +∞[?

A has no upper bound. Signup and view all the answers

What does the supremum criterion axiom state?

Every nonempty bounded part of R has a supremum. Signup and view all the answers

Given the product set AB, how is sup(AB) related to sup A and sup B?

sup(AB) = (sup A)(sup B) Signup and view all the answers

Which of the following is true about an interval ]a, b]?

It is bounded below by a. Signup and view all the answers

What two properties must M satisfy to be considered the supremum of set A?

M is an upper bound of A and for any ε > 0, there exists a ∈ A such that M − ε < a ≤ M. Signup and view all the answers

If M is an upper bound of set A, what can we conclude about any ε > 0?

There will always be an a ∈ A such that M − ε < a ≤ M. Signup and view all the answers

What contradiction arises if M > M 0 where M 0 is another upper bound of A?

It leads to the conclusion that ε must be negative. Signup and view all the answers

What values are shown to be the supremum and infimum of set A in the exercise?

sup A = 1 and inf A = 0. Signup and view all the answers

What is the implication if ε is greater than 1/n0 in establishing the supremum of A?

It implies that un0 is an appropriate choice satisfying the conditions. Signup and view all the answers

What conclusion can be drawn from M being the least upper bound of A?

No number smaller than M can be an upper bound of A. Signup and view all the answers

Why is it impossible for M to equal M − ε for ε > 0 in the context of upper bounds?

Since M − ε would be a valid upper bound as well, contradicting the property of M. Signup and view all the answers

What does the statement that A is bounded in R imply?

A has both a minimum and maximum value in the real numbers. Signup and view all the answers

Study Notes

Algebraic Properties

For any real numbers ( x ) and ( y ), the identity ( a^2 - b^2 = (|x| - |y|)^2 - (x - y)^2 ) leads to the conclusion that ( |x| - |y| \leq |x - y| ).
When ( xy \leq |xy| ), the inequality ( a^2 \leq b^2 ) implies ( |a| \leq |b| ).

Inequalities Involving Absolute Values

Equation: ( |x^2 + 4x - 5| = x^2 + 4x - 5 ) has solutions in the intervals ( ] - \infty, -5] \cup [1, +\infty[ ).
Inequality: ( |x^2 + 4x - 5| \leq x^2 + 4x - 5 ) holds under the same conditions as above.
Strict Inequality: ( |x^2 + 4x - 5| < x^2 + 4x - 5 ) has no solutions.
Inequality: ( |x^2 + 4x - 5| \geq x^2 + 4x - 5 ) is true for all real ( x ).
Strict Inequality: ( |x^2 + 4x - 5| > x^2 + 4x - 5 ) provides solutions in the interval ( ] -5, 1[ ).

Supremum and Infimum Theorems

The supremum criterion states that ( M ) is the supremum of a set ( A ) if it is an upper bound and for any ( \epsilon > 0 ), there exists an element ( a ) in ( A ) such that ( M - \epsilon < a \leq M ).
A nonempty set ( A \subset \mathbb{R} ) is bounded by some values if it has a supremum and infimum.

Boundedness of Sets

For the set ( A = \left{ \frac{1}{m} + \frac{1}{n} : m, n \in \mathbb{Z} \backslash {0} \right} ), both supremum ( \sup A = 2 ) and infimum ( \inf A = -2 ) exist.
The sequence ( A_n = \frac{n}{n+1} ) is bounded, leading to ( \sup A = 1 ) and ( \inf A = 0 ).

Properties of Intervals

For intervals ( [a, b], [a, b[, ]a, b], ) and ( ]a, b[ ), the supremum is ( b ) and the infimum is ( a ).
The suprema and infima exist for unbounded intervals as well, based on defined properties.

Rational and Irrational Numbers

Any ( r = \frac{a}{b} ) (with ( a \in \mathbb{Z} ) and ( b \in \mathbb{N}^* )) that is a root of ( x^n = m ) leads to the conclusion that ( a ) divides ( m ) and ( b = 1 ).
This implies that rational solutions to the equation must be integers.
Any prime number ( p ) is deemed irrational when expressed as ( \sqrt{p} ).
The sum of two prime numbers ( p ) and ( q ) results in an irrational number, considering the number ( \sqrt{p} - \sqrt{q} ).

Exercises and Applications

Various exercises provide practical applications of the concepts of absolute values, inequalities, and properties of rational/irrational numbers. Specific goals include proving or demonstrating properties related to bounded sets and understanding the nature of their suprema and infima.

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Description

This quiz explores key concepts in algebra, focusing on properties of absolute values, inequalities, and theorems concerning supremum and infimum. Test your knowledge with questions covering the implications of various algebraic expressions and their solutions. Understand how these concepts apply to real numbers and their relationships.