20 KiB
title | TARGET DECK | FILE TAGS | tags | |
---|---|---|---|---|
Bounds | Obsidian::STEM | calculus::bounds |
|
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
.
%%ANKI
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).
END%%
%%ANKI
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).
END%%
%%ANKI
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).
END%%
%%ANKI
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).
END%%
%%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).
END%%
%%ANKI
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).
END%%
%%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).
END%%
%%ANKI
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).
END%%
%%ANKI
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).
END%%
%%ANKI
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).
END%%
%%ANKI
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).
END%%
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.
%%ANKI
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).
END%%
%%ANKI
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).
END%%
%%ANKI
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).
END%%
%%ANKI
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%%
%%ANKI
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%%
%%ANKI
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).
END%%
%%ANKI
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).
END%%
%%ANKI
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).
END%%
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
.
%%ANKI
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).
END%%
%%ANKI
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).
END%%
%%ANKI
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%%
%%ANKI
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).
END%%
%%ANKI Basic What distinguishes a supremum from a least upper bound? Back: N/A. 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).
END%%
%%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).
END%%
%%ANKI
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%%
%%ANKI
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%%
%%ANKI
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%%
%%ANKI
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).
END%%
%%ANKI
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).
END%%
%%ANKI
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).
END%%
%%ANKI
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).
END%%
%%ANKI
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).
END%%
%%ANKI
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).
END%%
%%ANKI
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).
END%%
%%ANKI
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 supremums.
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
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 infimums.
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
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).
END%%
%%ANKI
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).
END%%
%%ANKI
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).
END%%
%%ANKI
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).
END%%
%%ANKI
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).
END%%
%%ANKI
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).
END%%
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).
END%%
%%ANKI
Basic
Consider \varnothing \subseteq \mathbb{R}
. Why doesn't the completeness axiom of real numbers apply?
Back: \varnothing
is not 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).
END%%
%%ANKI
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).
END%%
%%ANKI
Basic
Consider \mathbb{R}^+
. Why doesn't the completeness axiom apply?
Back: \mathbb{R}^+
is not bounded above.
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
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).
END%%
%%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).
END%%
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
.
%%ANKI
Basic
Let \varnothing \subset S \subseteq \mathbb{R}
. What is a greatest lower 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%%
%%ANKI
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%%
%%ANKI
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%%
%%ANKI
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).
END%%
%%ANKI Basic What distinguishes a greatest lower bound from an infimum? Back: N/A. 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).
END%%
%%ANKI Basic What distinguishes an infimum from a minimum? Back: An infimum 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%%
%%ANKI
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%%
%%ANKI
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%%
%%ANKI
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).
END%%
Bibliography
- Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).