History

History and Mythology of Set Theory

Equation form expr-0113bbb81598d6d8

L×L\unitline \times \unitline

Read as: the Cartesian product of the unit interval with itself

Means: the Cartesian product of the unit interval with itself

1 occurrence in this chapter

Equation form expr-019a811125828efa

hm(x)h_m(x)

Read as: h sub m of x

Means: h sub m of x

1 occurrence in this chapter

Equation form expr-0416c7345677b4d9

β=0\beta = 0

Read as: beta equals zero

Means: beta equals zero

2 occurrences in this chapter

Equation form expr-082db4c82cd55c3c

hmh_m

Read as: h sub m

Means: h sub m

1 occurrence in this chapter

Equation form expr-0c988f8f79315b86

f(x)=|x|f(x) = |x|

Read as: f of x equals the absolute value of x

Means: f of x equals the absolute value of x

1 occurrence in this chapter

Equation form expr-0f1a0ea14f982693

f(c+x)f(c + x)

Read as: f of c plus x

Means: f of c plus x

1 occurrence in this chapter

Equation form expr-13d3ca4e970acf51

⟨a,b⟩\tuple{a, b}

Read as: the ordered pair a comma b

Means: the ordered pair a comma b

1 occurrence in this chapter

Equation form expr-1426674c44e8876f

a1a_1

Read as: a sub one

Means: a sub one

1 occurrence in this chapter

Equation form expr-144d219b08bf6c0e

f(12+β)−f(12)f(\nicefrac{1}{2}+\beta) - f(\nicefrac{1}{2})

Read as: f of one half plus beta, minus f of one half

Means: f of one half plus beta, minus f of one half

1 occurrence in this chapter

Equation form expr-148de9c5a7a44d19

pp

Read as: p

Means: p

5 occurrences in this chapter

Equation form expr-1b16b1df538ba12d

nn

Read as: n

Means: n

2 occurrences in this chapter

Equation form expr-1bad6b8cf97131fc

−1-1

Read as: negative one

Means: negative one

1 occurrence in this chapter

Equation form expr-1d94df5b66df527b

f(a,b)=f(c,d)f(a, b) = f(c,d)

Read as: f of a comma b equals f of c comma d

Means: f of a comma b equals f of c comma d

1 occurrence in this chapter

Equation form expr-20357fc605f38d5b

L≈S\cardeq{\unitline}{\unitsquare}

Read as: the unit interval and the unit square have the same cardinality

Means: the unit interval and the unit square have the same cardinality

3 occurrences in this chapter

Equation form expr-235b4507e6ad1587

limn→∞2(12−n)=0.\lim_{n \rightarrow \infty} 2^{(\frac{1}{2}-n)} = 0.

Read as: the limit, as n tends to infinity, of two to the power one half minus n, equals zero

Means: the limit, as n tends to infinity, of two to the power one half minus n, equals zero

1 occurrence in this chapter

Equation form expr-252f10c83610ebca

ff

Read as: f

Means: f

9 occurrences in this chapter

Equation form expr-259072e6c653cc56

f(x)f(x)

Read as: f of x

Means: f of x

1 occurrence in this chapter

Equation form expr-2d711642b726b044

xx

Read as: x

Means: x

7 occurrences in this chapter

Equation form expr-2dfab4a78393f05c

h1h_1

Read as: h sub one

Means: h sub one

6 occurrences in this chapter

Equation form expr-2e7d2c03a9507ae2

cc

Read as: c

Means: c

7 occurrences in this chapter

Equation form expr-2ed52d51edc5e0ab

h(x)h(x)

Read as: h of x

Means: h of x

2 occurrences in this chapter

Equation form expr-2f4583c8b7c73ba0

f′f'

Read as: f prime

Means: f prime

1 occurrence in this chapter

Equation form expr-317c048f143649ec

f(12+β)−f(12)β.\frac{f(\nicefrac{1}{2}+\beta) - f(\nicefrac{1}{2})}{\beta}.

Read as: the quantity f of one half plus beta, minus f of one half, divided by beta

Means: the quantity f of one half plus beta, minus f of one half, divided by beta

1 occurrence in this chapter

Equation form expr-3c71c5ee4ef54e16

f′(c)=limx→0(f(c+x)−f(c)x) where a limit exists.{f}'(c) = \lim_{x \rightarrow 0}\left(\frac{f(c +x) - f(c)}{x}\right) \text{ where a limit exists}.

Read as: f prime of c equals the limit, as x tends to zero, of the quantity f of c plus x minus f of c, divided by x, where a limit exists

Means: f prime of c equals the limit, as x tends to zero, of the quantity f of c plus x minus f of c, divided by x, where a limit exists

1 occurrence in this chapter

Equation form expr-3e23e8160039594a

bb

Read as: b

Means: b

1 occurrence in this chapter

Equation form expr-42a18589cf97b8ca

|x|=−x|x| =-x

Read as: the absolute value of x equals negative x

Means: the absolute value of x equals negative x

1 occurrence in this chapter

Equation form expr-457e9235e6c708ba

(12n)2+(12n)2=2(12−n)\sqrt{\left(\nicefrac{1}{2^{n}}\right)^2+\left(\nicefrac{1}{2^{n}}\right)^2} = 2^{(\frac{1}{2}-n)}

Read as: the square root of the sum of two copies of one over two to the n, squared, equals two to the power one half minus n

Means: the square root of the sum of two copies of one over two to the n, squared, equals two to the power one half minus n

1 occurrence in this chapter

Equation form expr-463f2998327eb3a6

[0,1][0,1]

Read as: the closed interval from zero to one

Means: the closed interval from zero to one

1 occurrence in this chapter

Equation form expr-49273dd1c648eb11

2(12−n)2^{(\frac{1}{2}-n)}

Read as: two to the power one half minus n

Means: two to the power one half minus n

1 occurrence in this chapter

Equation form expr-4b850654a002e903

0.1˙0˙=0.1010101010…0.\dot{1}\dot{0} = 0.1010101010\ldots

Read as: binary zero point one zero recurring equals binary zero point one zero one zero one zero one zero one zero and so on

Means: binary zero point one zero recurring equals binary zero point one zero one zero one zero one zero one zero and so on

1 occurrence in this chapter

Equation form expr-4e07408562bedb8b

33

Read as: three

Means: three

1 occurrence in this chapter

Equation form expr-53619f07d192583d

f(c)=limx→cf(x)f(c) = \lim_{x \rightarrow c} f(x)

Read as: f of c equals the limit, as x tends to c, of f of x

Means: f of c equals the limit, as x tends to c, of f of x

1 occurrence in this chapter

Equation form expr-5afa20230f875253

n∈ℕn \in \Nat

Read as: n is a natural number

Means: n is a natural number

1 occurrence in this chapter

Equation form expr-5b86eb1897c44804

ℓ\ell

Read as: ell

Means: ell

2 occurrences in this chapter

Equation form expr-5d17cb9ad3fcb3a1

a=0.a1a2a3a4…b=0.b1b2b3b4…Now consider the function f:S→L given byf(a,b)=0.a1b1a2b2a3b3a4b4…a &= 0.a_1a_2a_3a_4\dots\\ b &= 0.b_1b_2b_3b_4\dots \intertext{Now consider the function $f \colon \unitsquare \to \unitline$ given by} f(a, b) & = 0.a_1b_1a_2b_2a_3b_3a_4b_4\dots

Read as: a has binary expansion zero point a sub one, a sub two, a sub three, a sub four, and so on; b has binary expansion zero point b sub one, b sub two, b sub three, b sub four, and so on; now consider the function f from the unit square to the unit interval, where f of a comma b interleaves the digits as zero point a sub one, b sub one, a sub two, b sub two, a sub three, b sub three, a sub four, b sub four, and so on

Means: a has binary expansion zero point a sub one, a sub two, a sub three, a sub four, and so on; b has binary expansion zero point b sub one, b sub two, b sub three, b sub four, and so on; now consider the function f from the unit square to the unit interval, where f of a comma b interleaves the digits as zero point a sub one, b sub one, a sub two, b sub two, a sub three, b sub three, a sub four, b sub four, and so on

1 occurrence in this chapter

Equation form expr-5d2fdba2ed4efda7

h3h_3

Read as: h sub three

Means: h sub three

4 occurrences in this chapter

Equation form expr-5feceb66ffc86f38

00

Read as: zero

Means: zero

8 occurrences in this chapter

Equation form expr-686d29dd27b7437a

(a,b)(a,b)

Read as: the open interval from a to b

Means: the open interval from a to b

1 occurrence in this chapter

Equation form expr-6b86b273ff34fce1

11

Read as: one

Means: one

4 occurrences in this chapter

Equation form expr-6d49baa31f16b806

β>0\beta > 0

Read as: beta is greater than zero

Means: beta is greater than zero

1 occurrence in this chapter

Equation form expr-6f9812f4fc9586c2

2×22 \times 2

Read as: two by two

Means: two by two

1 occurrence in this chapter

Equation form expr-6faa4156d145f5df

f(a,b)=0.1˙0˙f(a, b) = 0.\dot{1}\dot{0}

Read as: f of a comma b equals binary zero point one zero recurring

Means: f of a comma b equals binary zero point one zero recurring

1 occurrence in this chapter

Equation form expr-6ff8333bbb047008

x<0x < 0

Read as: x is less than zero

Means: x is less than zero

1 occurrence in this chapter

Equation form expr-70a45b4b9a6f64c6

f′(0)f'(0)

Read as: f prime of zero

Means: f prime of zero

1 occurrence in this chapter

Equation form expr-70dfa6bc0a72c1ca

2n×2n2^n\times 2^n

Read as: two to the n by two to the n

Means: two to the n by two to the n

1 occurrence in this chapter

Equation form expr-714b615136bd2142

|x−c|<δ|x - c| < \delta

Read as: the absolute value of x minus c is less than delta

Means: the absolute value of x minus c is less than delta

1 occurrence in this chapter

Equation form expr-7433fa59f4ea9ae9

(∀ε>0)(∃δ>0)∀x(|x−c|<δ→|g(x)−ℓ|<ε).(\forall\epsilon > 0)(\exists \delta > 0)\forall x \left(|x - c| < \delta \lif |g(x) - \ell| < \epsilon \right).

Read as: for every positive epsilon, there exists a positive delta such that, for every x, if the absolute value of x minus c is less than delta, then the absolute value of g of x minus ell is less than epsilon

Means: for every positive epsilon, there exists a positive delta such that, for every x, if the absolute value of x minus c is less than delta, then the absolute value of g of x minus ell is less than epsilon

1 occurrence in this chapter

Equation form expr-74bf6daef0c8b4f8

b1b_1

Read as: b sub one

Means: b sub one

1 occurrence in this chapter

Equation form expr-765109b1ce4b0d52

L\unitline

Read as: the unit interval

Means: the unit interval

6 occurrences in this chapter

Equation form expr-77942b679fbbf550

f(a,b)=0.1˙0˙f(a,b) = 0.\dot{1}\dot{0}

Read as: f of a comma b equals binary zero point one zero recurring

Means: f of a comma b equals binary zero point one zero recurring

1 occurrence in this chapter

Equation form expr-785ae493560363c1

hnh_n

Read as: h sub n

Means: h sub n

7 occurrences in this chapter

Equation form expr-7e7942106b5a439c

|x|=x|x| = x

Read as: the absolute value of x equals x

Means: the absolute value of x equals x

1 occurrence in this chapter

Equation form expr-7ee112153a906f00

1,2,3,4,5\ytext

Read as: the vertical-axis tick values one through five

Means: the vertical-axis tick values one through five

2 occurrences in this chapter

Equation form expr-84295e79ef191208

limx→cg(x)=ℓ.\lim_{x \rightarrow c}g(x) = \ell.

Read as: the limit, as x tends to c, of g of x equals ell

Means: the limit, as x tends to c, of g of x equals ell

1 occurrence in this chapter

Equation form expr-88eb9c29d4c1ef70

hn+1h_{n+1}

Read as: h sub n plus one

Means: h sub n plus one

1 occurrence in this chapter

Equation form expr-89f611ae025c19cc

x∈Sx \in \unitsquare

Read as: x is a point of the unit square

Means: x is a point of the unit square

1 occurrence in this chapter

Equation form expr-8bc4bfc2bdebfaac

ε\epsilon

Read as: epsilon

Means: epsilon

8 occurrences in this chapter

Equation form expr-90377b7753361eda

S≈L\cardeq{\unitsquare}{\unitline}

Read as: the unit square and the unit interval have the same cardinality

Means: the unit square and the unit interval have the same cardinality

1 occurrence in this chapter

Equation form expr-97adf0ad98b6ff13

h2h_2

Read as: h sub two

Means: h sub two

7 occurrences in this chapter

Equation form expr-9c0d85e5f93ed930

bn=dnb_n = d_n

Read as: b sub n equals d sub n

Means: b sub n equals d sub n

1 occurrence in this chapter

Equation form expr-a2277e0b98ac28a5

g(x)g(x)

Read as: g of x

Means: g of x

2 occurrences in this chapter

Equation form expr-a6fa461b11cbf1dd

1.000…1.000\ldots

Read as: binary one point zero zero zero and so on

Means: binary one point zero zero zero and so on

1 occurrence in this chapter

Equation form expr-a83dd0ccbffe39d0

II

Read as: the open interval I

Means: the open interval I

1 occurrence in this chapter

Equation form expr-a924fddb17bec52b

1,2,3,4\xtext

Read as: the horizontal-axis tick values one through four

Means: the horizontal-axis tick values one through four

2 occurrences in this chapter

Equation form expr-aa95e76b466c6462

L≼S\cardle{\unitline}{\unitsquare}

Read as: the cardinality of the unit interval is at most the cardinality of the unit square

Means: the cardinality of the unit interval is at most the cardinality of the unit square

1 occurrence in this chapter

Equation form expr-aaa9402664f1a41f

hh

Read as: h

Means: h

11 occurrences in this chapter

Equation form expr-ac4be96771c0e36c

S≼L\cardle{\unitsquare}{\unitline}

Read as: the cardinality of the unit square is at most the cardinality of the unit interval

Means: the cardinality of the unit square is at most the cardinality of the unit interval

3 occurrences in this chapter

Equation form expr-acabacb741471ece

c=12c = \nicefrac{1}{2}

Read as: c equals one half

Means: c equals one half

2 occurrences in this chapter

Equation form expr-aeb41586998433be

|x||x|

Read as: the absolute value of x

Means: the absolute value of x

2 occurrences in this chapter

Equation form expr-b1c2f0420aca8443

h(x)=limn→∞hn(x)h(x) &= \lim_{n \rightarrow \infty} h_n(x)

Read as: h of x equals the limit, as n tends to infinity, of h sub n of x

Means: h of x equals the limit, as n tends to infinity, of h sub n of x

1 occurrence in this chapter

Equation form expr-b880bf8b5c28c347

ℝ2\Real^2

Read as: the real plane

Means: the real plane

3 occurrences in this chapter

Equation form expr-bbb9f8aa8fec5523

12\nicefrac{1}{2}

Read as: one half

Means: one half

1 occurrence in this chapter

Equation form expr-bc5f0d8034c42b69

S\unitsquare

Read as: the unit square

Means: the unit square

12 occurrences in this chapter

Equation form expr-bfdaa023faec0e4e

b2b_2

Read as: b sub two

Means: b sub two

1 occurrence in this chapter

Equation form expr-c2047e180552bf51

f(c)f(c)

Read as: f of c

Means: f of c

1 occurrence in this chapter

Equation form expr-c8f73c0f05f3a5cc

a,b∈La, b \in \unitline

Read as: a and b belong to the unit interval

Means: a and b belong to the unit interval

1 occurrence in this chapter

Equation form expr-ca978112ca1bbdca

aa

Read as: a

Means: a

2 occurrences in this chapter

Equation form expr-ce94ebae6d217018

c+βc + \beta

Read as: c plus beta

Means: c plus beta

1 occurrence in this chapter

Equation form expr-d203ba01eef4198c

δ\delta

Read as: delta

Means: delta

2 occurrences in this chapter

Equation form expr-d4735e3a265e16ee

22

Read as: two

Means: two

1 occurrence in this chapter

Equation form expr-d4d0833bd88a4973

limx→cg(x)=ℓ\lim_{x \rightarrow c} g(x) = \ell

Read as: the limit, as x tends to c, of g of x equals ell

Means: the limit, as x tends to c, of g of x equals ell

1 occurrence in this chapter

Equation form expr-d54c7c4c96d78a01

an=cna_n = c_n

Read as: a sub n equals c sub n

Means: a sub n equals c sub n

1 occurrence in this chapter

Equation form expr-d6a6c3181acd3c81

m>nm > n

Read as: m is greater than n

Means: m is greater than n

2 occurrences in this chapter

Equation form expr-d99e76cb3ff6c737

a=ca = c

Read as: a equals c

Means: a equals c

1 occurrence in this chapter

Equation form expr-dd9a69efe2620fa9

a=0.1˙1˙=0.111111…b=0a&= 0.\dot{1}\dot{1} = 0.111111\ldots\\ b&= 0

Read as: a equals binary zero point one recurring, which is binary zero point one one one one one one and so on; b equals zero

Means: a equals binary zero point one recurring, which is binary zero point one one one one one one and so on; b equals zero

1 occurrence in this chapter

Equation form expr-de33cf012125a5fa

f′(c)=f(c+β)−f(c)β where β is infinitesimal.{f'}(c) = \frac{f(c+\beta) - f(c)}{\beta} \text{ where $\beta$ is infinitesimal.}

Read as: f prime of c equals the quantity f of c plus beta minus f of c, divided by beta, where beta is infinitesimal

Means: f prime of c equals the quantity f of c plus beta minus f of c, divided by beta, where beta is infinitesimal

1 occurrence in this chapter

Equation form expr-e5cda8318149bdb8

a2a_2

Read as: a sub two

Means: a sub two

1 occurrence in this chapter

Equation form expr-e5d8d2df689e3009

f(x)=x24+12f(x) = \nicefrac{x^2}{4} + \nicefrac{1}{2}

Read as: f of x equals x squared over four plus one half

Means: f of x equals x squared over four plus one half

1 occurrence in this chapter

Equation form expr-e6e34eb705b0bf07

b=db = d

Read as: b equals d

Means: b equals d

1 occurrence in this chapter

Equation form expr-e9a6c6d19d6658fc

x∈Ix \in I

Read as: x belongs to the open interval I

Means: x belongs to the open interval I

1 occurrence in this chapter

Equation form expr-eb66c745743b0e60

f′(c)f'(c)

Read as: f prime of c

Means: f prime of c

2 occurrences in this chapter

Equation form expr-ec1e44838a8410a7

2n×2n2^n \times 2^n

Read as: two to the n by two to the n

Means: two to the n by two to the n

3 occurrences in this chapter

Equation form expr-eff1cbd137d2d155

|g(x)−ℓ|<ε|g(x) - \ell| < \epsilon

Read as: the absolute value of g of x minus ell is less than epsilon

Means: the absolute value of g of x minus ell is less than epsilon

1 occurrence in this chapter

Equation form expr-f1a8d30b38f0ae9a

34\nicefrac{3}{4}

Read as: three quarters

Means: three quarters

1 occurrence in this chapter

Equation form expr-f266ff6292423043

a=0.1˙1˙=1a = 0.\dot{1}\dot{1} = 1

Read as: a equals binary zero point one recurring, which equals one

Means: a equals binary zero point one recurring, which equals one

1 occurrence in this chapter

Equation form expr-f3f3804480e8551a

β\beta

Read as: beta

Means: beta

5 occurrences in this chapter

Equation form expr-fadb3925eb4b6364

x≥0x \geq 0

Read as: x is greater than or equal to zero

Means: x is greater than or equal to zero

1 occurrence in this chapter

Parabola with three secant triangles

Coordinate diagram for the function f of x equals x squared over four plus one half. The horizontal axis is x, and the vertical axis is f of x. The horizontal tick labels use the horizontal-axis tick values one through four; the vertical tick labels use the vertical-axis tick values one through five. A black upward-opening parabola is shown. Three right secant triangles share the point where x is one half. The red triangle has base three and slope one; the smaller blue triangle has base two and slope three quarters; the smallest green triangle has base one and slope one half. End diagram.

Source

Graph of the absolute-value function

Coordinate diagram of the absolute-value function. The horizontal axis is x, and the vertical axis is the absolute value of x. The horizontal tick command uses the printed horizontal-axis tick values, including the duplicated positive-one position; the vertical tick command uses the vertical-axis tick values one and two. The thick graph is a V shape, descending with slope negative one to the origin and ascending with slope one from the origin. The source tick list prints positive one twice; that duplication is preserved and disclosed. End diagram.

Source

First six Hilbert-curve stages

Six square panels show the first six Hilbert-curve approximations in source order: stages one, two, and three across the top row, then stages four, five, and six across the bottom row. Each red curve lies in a gray square. The step sizes halve from stage to stage, while the recursive order rises from one through six, making the curve increasingly dense. No arrow direction, coordinate label, or filled region is printed. End diagram.

Source

Cantor's line-and-square theorem

The theorem states that the unit interval and the unit square have the same cardinality. The following source proof first attempts a binary-digit interleaving injection, then records its surjectivity failure and a completion using the Schroeder-Bernstein theorem.

Source

Binary expansions and the interleaving map

a has binary expansion zero point a sub one, a sub two, a sub three, a sub four, and so on; b has binary expansion zero point b sub one, b sub two, b sub three, b sub four, and so on; now consider the function f from the unit square to the unit interval, where f of a comma b interleaves the digits as zero point a sub one, b sub one, a sub two, b sub two, a sub three, b sub three, a sub four, b sub four, and so on

Source

Recurring binary counterexample inputs

The display sets a to binary zero point one recurring, which equals binary zero point one one one one one one and so on, and sets b to zero. It exhibits why choosing a single binary expansion prevents the proposed interleaving map from reaching every recurring target.

Source

First Hilbert-curve approximation

First Hilbert-curve approximation in a square divided by a two by two grid. A black first-order Hilbert path connects the four quarter-cells in the order bottom left, top left, top right, bottom right. Short red segments connect the two bottom endpoints to the square boundary. End diagram.

Source

Second Hilbert-curve approximation

Second Hilbert-curve approximation in a square divided by a four by four grid. Four scaled black copies of the first-order core occupy the quarter-squares. The lower-left copy is rotated two hundred seventy degrees, the two upper copies are unrotated, and the lower-right copy is rotated ninety degrees. Three green segments join consecutive copies, and two red segments connect the outer endpoints to the left and right boundary. End diagram.

Source

Third Hilbert-curve approximation

Third Hilbert-curve approximation in a square divided by an eight by eight grid. Four scaled black copies of the second-order core occupy the quarter-squares. The lower-left copy is rotated two hundred seventy degrees, the two upper copies are unrotated, and the lower-right copy is rotated ninety degrees. The source draws three distinct green connector positions, but repeats the right connector command once at the same coordinates; two red segments connect the outer endpoints to the bottom boundary. End diagram.

Source

Pointwise limit defining the Hilbert curve

For each point x, h of x is defined as the limit, as n tends to infinity, of h sub n of x.

Source

Definition of a curve

A curve is a continuous map from the unit interval to the real plane. The surrounding source then gives an informal epsilon and delta continuity argument for the Hilbert curve.

Source

Cross-reference reference-001734

George Berkeley 1734, §XIII

Source occurrence

Cross-reference reference-001735

(George Berkeley, 1734, §IX)

Source occurrence

Cross-reference reference-001736

the Weierstrass function (external resource; internet required)

Source occurrence

Cross-reference reference-001737

Fernando Q. Gouvêa 2011

Source occurrence

Cross-reference reference-001738

Fernando Q. Gouvêa (2011)

Source occurrence

Cross-reference reference-001739

the section More Myth than History

Source occurrence

Cross-reference reference-001740

the section Cantor on the Line and the Plane

Source occurrence

Cross-reference reference-001741

1890

Source occurrence

Cross-reference reference-001742

1891

Source occurrence

Cross-reference reference-001743

the appendix on Hilbert's space-filling curves

Source occurrence

Cross-reference reference-001744

Fernando Q. Gouvêa (2011)

Source occurrence

Cross-reference reference-001745

Fernando Q. Gouvêa

Source occurrence

Cross-reference reference-001746

Fernando Q. Gouvêa

Source occurrence

Cross-reference reference-001747

the section Pathologies

Source occurrence

Cross-reference reference-001748

Marcus Giaquinto (2007)

Source occurrence

Cross-reference reference-001749

the section Pathologies

Source occurrence

Cross-reference reference-001750

the Schroeder-Bernstein theorem

Source occurrence

Cross-reference reference-001751

the theorem that the unit interval and unit square have equal cardinality

Source occurrence

Cross-reference reference-001752

the section Pathologies

Source occurrence

Cross-reference reference-001753

the theorem that the unit interval and unit square have equal cardinality

Source occurrence

Cross-reference reference-001754

Nicholas J Rose (2010)

Source occurrence

Cross-reference reference-001755

the section Rigorous Definition of Limits

Source occurrence

Source disclosures

Ordered structures

Parabola with three secant triangles

Structure: diagram tikz.

Coordinate diagram for the function f of x equals x squared over four plus one half. The horizontal axis is x, and the vertical axis is f of x. The horizontal tick labels use the horizontal-axis tick values one through four; the vertical tick labels use the vertical-axis tick values one through five. A black upward-opening parabola is shown. Three right secant triangles share the point where x is one half. The red triangle has base three and slope one; the smaller blue triangle has base two and slope three quarters; the smallest green triangle has base one and slope one half. End diagram.

Read the source-bound structure in context

Graph of the absolute-value function

Structure: diagram tikz.

Coordinate diagram of the absolute-value function. The horizontal axis is x, and the vertical axis is the absolute value of x. The horizontal tick command uses the printed horizontal-axis tick values, including the duplicated positive-one position; the vertical tick command uses the vertical-axis tick values one and two. The thick graph is a V shape, descending with slope negative one to the origin and ascending with slope one from the origin. The source tick list prints positive one twice; that duplication is preserved and disclosed. End diagram.

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First six Hilbert-curve stages

Structure: diagram tikz.

Six square panels show the first six Hilbert-curve approximations in source order: stages one, two, and three across the top row, then stages four, five, and six across the bottom row. Each red curve lies in a gray square. The step sizes halve from stage to stage, while the recursive order rises from one through six, making the curve increasingly dense. No arrow direction, coordinate label, or filled region is printed. End diagram.

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First Hilbert-curve approximation

Structure: diagram tikz.

First Hilbert-curve approximation in a square divided by a two by two grid. A black first-order Hilbert path connects the four quarter-cells in the order bottom left, top left, top right, bottom right. Short red segments connect the two bottom endpoints to the square boundary. End diagram.

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Second Hilbert-curve approximation

Structure: diagram tikz.

Second Hilbert-curve approximation in a square divided by a four by four grid. Four scaled black copies of the first-order core occupy the quarter-squares. The lower-left copy is rotated two hundred seventy degrees, the two upper copies are unrotated, and the lower-right copy is rotated ninety degrees. Three green segments join consecutive copies, and two red segments connect the outer endpoints to the left and right boundary. End diagram.

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Third Hilbert-curve approximation

Structure: diagram tikz.

Third Hilbert-curve approximation in a square divided by an eight by eight grid. Four scaled black copies of the second-order core occupy the quarter-squares. The lower-left copy is rotated two hundred seventy degrees, the two upper copies are unrotated, and the lower-right copy is rotated ninety degrees. The source draws three distinct green connector positions, but repeats the right connector command once at the same coordinates; two red segments connect the outer endpoints to the bottom boundary. End diagram.

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