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Neighborhoods in Real Numbers

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Questions and Answers

What is the name given to the open interval (a - ε, a + ε) around a point a in the real line?

Extended real neighborhood of a
Neighborhood of a (correct)
One-sided neighborhood of a
Left and right neighborhood of a

Which of the following statements about the neighborhood U_a(ε) is true?

x ∈ U_a(ε) if and only if |x - a| ≤ ε (correct)
U_a(ε) is a closed interval
U_a(ε) is always a finite set
x ∈ U_a(ε) if and only if x is a real number

What are the open intervals (a - ε, a) and (a, a + ε) called?

Extended real neighborhoods of a
One-sided neighborhoods of a
Infimum and supremum neighborhoods of a
Left and right neighborhoods of a (correct)

What is the set R̄ = R ∪ {+∞, -∞} called?

The extended real line (A) Signup and view all the answers

What are the open intervals (α, +∞) and (-∞, -α) called?

α-neighborhoods of +∞ and -∞ in the extended real line (B) Signup and view all the answers

Which property of the real numbers ensures that every non-empty subset of the real numbers has a smallest element (infimum)?

The well-order property (B) Signup and view all the answers

What is the definition of a cluster point of a set $M \mathbb{R$?

A point $a \in \mathbb{R$ is a cluster point of $M$ if every neighborhood of $a$ contains at least one point of $M$ other than $a$ itself. (C) Signup and view all the answers

Which of the following statements about the set of natural numbers $\mathbb{N}$ is true?

$+\infty$ is not a cluster point of $\mathbb{N}$. (A) Signup and view all the answers

What is the definition of the subtraction operation on $\mathbb{R$?

For $a, b \in \mathbb{R$ $a - b = a + (-b)$. (B) Signup and view all the answers

Which of the following statements about the division operation on $\mathbb{R$ is true?

For $a, b \in \mathbb{R$ with $b \neq 0$, the division $a \div b$ is defined as $a \cdot \frac{1}{b$. (A) Signup and view all the answers

Which of the following statements about the set of natural numbers $\mathbb{N$ is true?

$+\infty$ is the infimum of $\mathbb{N$. (B) Signup and view all the answers

Which of the following is true about the supremum of a set $M$ bounded from above?

The supremum of $M$ satisfies the two conditions given in Proposition 1.3.1. (C) Signup and view all the answers

Which of the following is true about the infimum of a set $M$ bounded from below?

The infimum of $M$ satisfies the two conditions given in Proposition 1.3.2. (C) Signup and view all the answers

According to the Well-order property of a finite set (Proposition 1.3.3), which of the following is true?

Every finite set $M \subset \mathbb{R}$ has a maximum and a minimum. (A) Signup and view all the answers

Which of the following is true about the well-order property of integer numbers?

Both (a) and (b) are true. (A) Signup and view all the answers

What is the definition of a neighborhood of a point $a$ in $\mathbb{R}$?

The set of all points $x$ in $\mathbb{R}$ such that $|x - a| < \epsilon$, where $\epsilon > 0$. (D) Signup and view all the answers

Consider the set $M = {x \in \mathbb{Z} | x^3 < 2453, x \text{ is not divisible by } 2, 3, 17}$. Which of the following is true about this set?

The set $M$ has a maximum element. (A) Signup and view all the answers

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