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| title | TARGET DECK | FILE TAGS | tags | |
|---|---|---|---|---|
| Bounds | Obsidian::STEM | calculus::bounds |
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Overview
Suppose S is a nonempty set of real numbers and suppose there are numbers L and U such that L \leq x \leq U for all x \in S. Then S is said to be bounded below by L and bounded above by U. The number L is said to be a lower bound for S; the number U is said to be an upper bound for S.
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Basic
Let \varnothing \subset S \subseteq \mathbb{R}. What does it mean for S to be bounded below by B?
Back: For all x \in S, B \leq x.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
What does it mean for \varnothing \subseteq \mathbb{R} to be bounded above by B?
Back: N/A.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Suppose S \subseteq \mathbb{R} is bounded below by B. What kind of mathematical object is S?
Back: A nonempty set.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Suppose S \subseteq \mathbb{R} is unbounded above. What kind of mathematical object is S?
Back: A set (possibly empty).
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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%%ANKI Basic Is the set of positive real numbers bounded below? Back: Yes. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Let \varnothing \subset S \subseteq \mathbb{R}. What does it mean for S to be bounded above by B?
Back: For all x \in S, x \leq B.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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%%ANKI Basic Is the set of positive real numbers bounded above? Back: No. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
What are the upper bounds of interval [0, 1] \subseteq \mathbb{R}?
Back: All real numbers x \geq 1.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
What are the upper bounds of interval (0, 1) \subseteq \mathbb{R}?
Back: All real numbers x \geq 1.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
What are the lower bounds of interval [0, 1] \subseteq \mathbb{R}?
Back: All real numbers x \leq 0.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
What are the lower bounds of interval (0, 1) \subseteq \mathbb{R}?
Back: All real numbers x \leq 0.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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If L \in S, then L is the minimum element of S (denoted L = \mathop{\text{min}} S). Likewise, if U \in S, then U is the maximum element of S (denoted U = \mathop{\text{max}}S). A set with no lower bound is said to be unbounded below. A set with no upper bound is said to be unbounded above.
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Basic
What is a maximum element of set \varnothing \subset S \subseteq \mathbb{R}?
Back: A member of S that is also an upper bound.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
What is the maximum element of interval [0, 1] \subseteq \mathbb{R}?
Back: 1.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
What is the maximum element of interval (0, 1) \subseteq \mathbb{R}?
Back: N/A.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
What is a minimum element of set \varnothing \subset S \subseteq \mathbb{R}?
Back: A member of S that is also a lower bound.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
END%%
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Basic
What is the minimum element of interval (0, 1) \subseteq \mathbb{R}?
Back: N/A.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
END%%
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Basic
What is the minimum element of interval [0, 1] \subseteq \mathbb{R}?
Back: 0.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
How is a maximum element of \varnothing \subset S \subseteq \mathbb{R} denoted?
Back: As \mathop{\text{max}} S.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
How is a minimum element of \varnothing \subset S \subseteq \mathbb{R} denoted?
Back: As \mathop{\text{min}} S.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Least Upper Bounds
A number B is called a least upper bound (or supremum) of a nonempty set S if B is an upper bound for S and no number less than B is an upper bound for S. This is denoted as B = \mathop{\text{lub}}S or B = \mathop{\text{sup}} S.
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Basic
Let \varnothing \subset S \subseteq \mathbb{R}. What is a least upper bound of S?
Back: An upper bound B for S such that no number less than B is also an upper bound for S.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
What is an alternative term for a least upper bound of S?
Back: A supremum of S.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
How is a least upper bound for S denoted?
Back: As \mathop{\text{lub}} S.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
END%%
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Basic
How is a supremum for S denoted?
Back: As \mathop{\text{sup}} S.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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%%ANKI Basic What distinguishes a supremum from a least upper bound? Back: They are synonyms of one another. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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%%ANKI Basic What distinguishes a supremum from a maximum? Back: A supremum is not necessarily a member of the reference set. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
What is the least upper bound of interval [0, 1] \subseteq \mathbb{R}?
Back: 1.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
END%%
%%ANKI
Basic
What is the supremum of interval (0, 1) \subseteq \mathbb{R}?
Back: 1.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
END%%
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Basic
How many upper bounds can a nonempty subset of \mathbb{R} have?
Back: 0 or more.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
END%%
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Basic
How many supremums can a nonempty subset of \mathbb{R} have?
Back: 0 or 1.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
END%%
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Basic
Is a maximum of a nonempty subset S of \mathbb{R} a supremum of S?
Back: Yes.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Is a least upper bound of a nonempty subset S of \mathbb{R} a maximum of S?
Back: Not necessarily.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Let S \subseteq \mathbb{R} have a supremum. If h > 0, why does there exist an x \in S such that x > \mathop{\text{sup}} S - h?
Back: Otherwise \mathop{\text{sup}}S - h is an upper bound less than \mathop{\text{sup}}S.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Let S \subseteq \mathbb{R} have a supremum. If h > 0, why does there exist an x \in S such that x < \mathop{\text{sup}} S - h?
Back: N/A. This is not necessarily the case.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Let S \subseteq \mathbb{R} have an infimum. If h > 0, why does there exist an x \in S such that x > \mathop{\text{inf}} S + h?
Back: N/A. This is not necessarily the case.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Let S \subseteq \mathbb{R} have an infimum. If h > 0, why does there exist an x \in S such that x < \mathop{\text{inf}} S + h?
Back: Otherwise \mathop{\text{inf}}S + h is a lower bound greater than \mathop{\text{inf}}S.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Let A, B \subseteq \mathbb{R} have supremums. What set C satisfies \mathop{\text{sup}}C = \mathop{\text{sup}}A + \mathop{\text{sup}}B?
Back: C = \{a + b \mid a \in A, b \in B\}
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Let A, B \subseteq \mathbb{R}. When is \mathop{\text{sup}} \,\{a + b \mid a \in A, b \in B\} defined?
Back: When A and B both have a supremum.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Let A, B \subseteq \mathbb{R}. When is \mathop{\text{inf}} \,\{a + b \mid a \in A, b \in B\} defined?
Back: When A and B both have an infimum.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Let A, B \subseteq \mathbb{R} have infimums. What set C satisfies \mathop{\text{inf}}C = \mathop{\text{inf}}A + \mathop{\text{inf}}B?
Back: C = \{a + b \mid a \in A, b \in B\}
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Let S, T \subseteq \mathbb{R} be nonempty sets such that \forall s \in S, \forall t \in T, s \leq t. Does S have a supremum?
Back: Yes.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Let S, T \subseteq \mathbb{R} be nonempty sets such that \forall s \in S, \forall t \in T, s \leq t. Does T have a supremum?
Back: Indeterminate.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Let S, T \subseteq \mathbb{R} be nonempty sets such that \forall s \in S, \forall t \in T, s \leq t. Does S have an infimum?
Back: Indeterminate.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Let S, T \subseteq \mathbb{R} be nonempty sets such that \forall s \in S, \forall t \in T, s \leq t. Does T have an infimum?
Back: Yes.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Let S, T \subseteq \mathbb{R} be nonempty sets such that \forall s \in S, \forall t \in T, s \leq t. How does \mathop{\text{sup}} S compare to \mathop{\text{inf}} T?
Back: \mathop{\text{sup}}S \leq \mathop{\text{inf}}T
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Completeness Axiom
Every nonempty set S of real numbers which is bounded above has a supremum; that is, there is a real number B such that B = \mathop{\text{sup}} S.
%%ANKI Basic What does the completeness axiom of real numbers state? Back: Every nonempty set of real numbers bounded above has a supremum. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Consider \varnothing \subseteq \mathbb{R}. Why doesn't the completeness axiom of real numbers apply?
Back: It only applies to nonempty sets.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Consider (0, 1) \subseteq \mathbb{R}. Why doesn't the completeness axiom of real numbers apply?
Back: N/A. It does.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Consider \mathbb{R}^+. Why doesn't the completeness axiom apply?
Back: It only applies to nonempty sets that are bounded above.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Basic
Consider (-\infty, 0). Why doesn't the completeness axiom apply?
Back: N/A. It does.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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%%ANKI Basic What arbitrary decision was made when formulating the completeness axiom of real numbers? Back: Whether to use supremums or infimums. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Greatest Lower Bounds
A number B is called a greatest lower bound (or infimum) of a nonempty set S if B is a lower bound for S and no number greater than B is a lower bound for S. This is denoted as B = \mathop{\text{glb}} S or B = \mathop{\text{inf}} S.
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Basic
Let \varnothing \subset S \subseteq \mathbb{R}. What is a greatest upper bound of S?
Back: A lower bound B for S such that no number greater than B is also a lower bound for S.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
END%%
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Basic
What is an alternative term for a greatest lower bound of S?
Back: An infimum of S.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
END%%
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Basic
How is a greatest lower bound for S denoted?
Back: As \mathop{\text{glb}} S.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
END%%
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Basic
How is an infimum for S denoted?
Back: As \mathop{\text{inf}} S.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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%%ANKI Basic What distinguishes a greatest lower bound from an infimum? Back: They are synonyms of one another. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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%%ANKI Basic What distinguishes an infimum from a minimum? Back: A supremum is not necessarily a member of the reference set. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
END%%
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Basic
What is the infimum of interval [0, 1] \subseteq \mathbb{R}?
Back: 0.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
END%%
%%ANKI
Basic
What is the greatest lower bound of interval (0, 1) \subseteq \mathbb{R}?
Back: 0.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
END%%
%%ANKI
Basic
How many lower bounds can a nonempty subset of \mathbb{R} have?
Back: 0 or more.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
END%%
%%ANKI
Basic
How many greatest lower bounds can a nonempty subset of \mathbb{R} have?
Back: 0 or 1.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
END%%
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Basic
Is an infimum of a nonempty subset S of \mathbb{R} a minimum of S?
Back: Not necessarily.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
END%%
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Basic
Is a minimum of a nonempty subset S of \mathbb{R} a greatest lower bound of S?
Back: Yes.
Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
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Bibliography
- Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).