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?

<p>The extended real line (A)</p> Signup and view all the answers

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

<p>α-neighborhoods of +∞ and -∞ in the extended real line (B)</p> 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)?

<p>The well-order property (B)</p> Signup and view all the answers

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

<p>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)</p> Signup and view all the answers

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

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

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

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

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

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

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

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

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

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

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

<p>The infimum of $M$ satisfies the two conditions given in Proposition 1.3.2. (C)</p> 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?

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

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

<p>Both (a) and (b) are true. (A)</p> Signup and view all the answers

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

<p>The set of all points $x$ in $\mathbb{R}$ such that $|x - a| &lt; \epsilon$, where $\epsilon &gt; 0$. (D)</p> 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?

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

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