17 Questions
What is the name given to the open interval (a - ε, a + ε) around a point a in the real line?
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
What are the open intervals (a - ε, a) and (a, a + ε) called?
Left and right neighborhoods of a
What is the set R̄ = R ∪ {+∞, -∞} called?
The extended real line
What are the open intervals (α, +∞) and (-∞, -α) called?
α-neighborhoods of +∞ and -∞ in the extended real line
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
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.
Which of the following statements about the set of natural numbers $\mathbb{N}$ is true?
$+\infty$ is not a cluster point of $\mathbb{N}$.
What is the definition of the subtraction operation on $\mathbb{R$?
For $a, b \in \mathbb{R$ $a - b = a + (-b)$.
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 $a \cdot \frac{1}{b$.
Which of the following statements about the set of natural numbers $\mathbb{N$ is true?
$+\infty$ is the infimum of $\mathbb{N$.
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.
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.
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.
Which of the following is true about the well-order property of integer numbers?
Both (a) and (b) are true.
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 $\epsilon > 0$.
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.
Learn about neighborhoods in real numbers, denoted by ε and defined as open intervals around a point. Understand how to determine if a point is within the ε-neighborhood of another point based on the distance between them.
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