aarjaneiro/planetmath huggingface dataset:
directed segment

directed segment

Let

A

⁢

B

a

line segment

. The

directed segment

A

⁢

B

¯

is to be taken the segment

A

⁢

B

with a direction (

similar

to vectors).
The

defining property

is then

A

⁢

B

¯

=

-

B

⁢

A

¯

,

(which relates to the

property

of vectors stating that

v

and

-

v

have opposite direction and same modulus).

The

addition

of directed segments is done in a similar fashion of vectors, so the above

relation

is

equivalent

to

A

⁢

B

¯

+

B

⁢

A

¯

=

0

.

where

0

represents any segment of the form

P

⁢

P

¯

.
If it is stated that we will work with directed segments, it’s customary to omit the overlining and to just write

A

⁢

B

, convention we will follow now.

Notes.

•

The

definition

does not says

A

⁢

B

is either

positive

or negative (and it does not really matters doing so). It merely states that traveling the segment on different directions give different signs.

•

It does not make sense to compare signs of non-collinear segments. So if

A

,

B

,

C

are not on the same line (or

parallel lines

) we cannot relate the signs of

A

⁢

B

,

B

⁢

C

and

C

⁢

A

.

It can be proved considering cases that no matter the relative position of three points

A

,

B

,

C

on a line, the following

equality

holds:

A

⁢

B

+

B

⁢

C

=

A

⁢

C

.

In the above picture

A

⁢

P

+

P

⁢

B

=

A

⁢

B

. Notice that

A

⁢

B

goes to the left since

A

⁢

B

is the segment that starts at

A

and ends at

B

.
Also, taking

A

=

C

gives

A

⁢

B

+

B

⁢

A

=

A

⁢

A

=

0

which is

consistent

with the earlier remarks.

Just like undirected segments in

Euclidean geometry

(and unlike vectors), directed segments can be divided to obtain a ratio. Such ratio is the number obtained dividing the undirected segments, but taking signs int oaccount (ratio of two segments with the same direction is positive, and negative otherwise).

Given two points

A

,

B

on a line, we can locate any other point

P

on the line considering the ratio

A

⁢

P

/

P

⁢

B

. In other words,

P

=

Q

if and only if

A

⁢

P

/

P

⁢

B

=

A

⁢

Q

/

Q

⁢

B

.
Moreover, to each point

P

corresponds an extended

1

1

We use extended reals to avoid dealing with separate cases, allowing us to deal also with

points at infinity

.

real

r

=

A

⁢

P

/

P

⁢

B

and to each extended real

r

corresponds a point

P

such that

r

=

A

⁢

P

/

P

⁢

B

.

Notice that

A

⁢

P

/

P

⁢

B

is positive when

A

⁢

P

and

P

⁢

B

have the same direction, which happens if and only if

p

is between

A

and

B

. If

P

lies outside

A

⁢

P

, then

A

⁢

P

and

P

⁢

B

have negative signs and so the ratio will be negative.

Title

directed segment

Canonical

name

DirectedSegment

Date of creation

2013-03-22 14:56:59

Last modified on

2013-03-22 14:56:59

Owner

drini (3)

Last modified by

drini (3)

Numerical id

9

Author

drini (3)

Entry type

Definition

Classification

msc 51F99

Classification

msc 51M25

Related topic

CevasTheorem

Related topic

Midpoint

Generated on Fri Feb 9 22:01:46 2018 by

LaTeXML
n-section of line segment with compass and straightedge

n

-section of line segment with compass and straightedge

Task.

Let

A

⁢

B

be a given

line segment

and

n

a positive integer

>

1

. Divide

A

⁢

B

to

n

equal parts.

.

Draw a

half-line

p

beginning from

A

but not

parallel

to

A

⁢

B

. From

p

separate

n

consecutive equally long segments

A

⁢

A

1

,

A

1

⁢

A

2

,

A

2

⁢

A

3

, …,

A

n

-

1

⁢

A

n

. Draw the line

A

n

⁢

B

and denote by

B

1

,

B

2

, …,

B

n

-

1

the points of

A

⁢

B

such that

A

1

⁢

B

1

⁢

∥

A

2

⁢

B

2

∥

⁢

…

⁢

∥

A

n

-

1

⁢

B

n

-

1

∥

⁢

A

n

⁢

B

(see

compass and straightedge construction of parallel line

).
These points divide the line segment

A

⁢

B

in

n

equal segments.

Proof.

For clarity, we prove the

theorem

only in the case

n

=

3

.

The line

A

⁢

B

intersects

the

parallel lines

A

1

⁢

B

1

,

A

2

⁢

B

2

and

A

3

⁢

B

, and thus the corresponding angles (

http://planetmath.org/CorrespondingAnglesInTransversalCutting

)

A

1

⁢

B

1

⁢

A

,

A

2

⁢

B

2

⁢

A

and

A

3

⁢

B

⁢

A

are equal. Similarly the angles

A

⁢

A

1

⁢

B

1

,

A

⁢

A

2

⁢

B

2

and

A

⁢

A

3

⁢

B

are equal. Because of the equal angles, the

triangle

A

⁢

A

2

⁢

B

2

is

similar

to the triangle

A

⁢

A

3

⁢

B

with the ratio of similarity

2

:

3

. Therefore

A

⁢

B

2

=

2

3

⁢

A

⁢

B

;

B

2

⁢

B

=

1

3

⁢

A

⁢

B

.

Also the triangle

A

⁢

A

1

⁢

B

1

is similar to the triangle

A

⁢

A

3

⁢

B

with the line ratio

1

:

3

, whence

A

⁢

B

1

=

1

3

⁢

A

⁢

B

;

B

1

⁢

B

2

=

1

3

⁢

A

⁢

B

.

The equations show that the points

B

1

and

B

2

divide the line segment

A

⁢

B

in 3 equal segments.

Title

n

-

section

of line segment with

compass

and straightedge

Canonical name

NsectionOfLineSegmentWithCompassAndStraightedge

Date of creation

2013-03-22 17:24:41

Last modified on

2013-03-22 17:24:41

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

12

Author

pahio (2872)

Entry type

Algorithm

Classification

msc 51F99

Classification

msc 51M05

Classification

msc 51-00

Related topic

CompassAndStraightedgeConstructionOfParallelLine

Generated on Fri Feb 9 22:01:48 2018 by

LaTeXML
pentagram

pentagram

A

pentagram

is the figure formed by the five

diagonals

of a

regular

(

http://planetmath.org/RegularPolygon

)

pentagon

.  The name comes from the Greek

π

⁢

ε

⁢

ν

⁢

τ

⁢

α

⁢

γ

⁢

ϱ

⁢

α

⁢

μ

⁢

μ

⁢

o

⁢

ν

. Its are

π

⁢

ε

⁢

ν

⁢

τ

⁢

ε

‘five’ and

γ

⁢

ϱ

⁢

α

⁢

μ

⁢

μ

⁢

o

⁢

ς

‘stoke, ’.  These Greek are transliterated as

pentagrammon

,

pente

, and

grammos

, respectively.

In the picture below, a

regular pentagon

is drawn dashed in black, the pentagram is drawn in blue, and the regular pentagon in the middle of the pentagram is shaded in cyan.

All pentagrams are

similar

.  If the length of the pentagon diagonal is

d

, then each side of the small pentagon in the middle of the pentagram is of length

(

5

-

2

)

⁢

d

.

Title

pentagram

Canonical

name

Pentagram

Date of creation

2013-03-22 17:08:52

Last modified on

2013-03-22 17:08:52

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

11

Author

pahio (2872)

Entry type

Definition

Classification

msc 51F99

Generated on Fri Feb 9 22:01:50 2018 by

LaTeXML
similarity in geometry

similarity in geometry

Two figures

K

and

K

′

in a

Euclidean plane

or space (

http://planetmath.org/EuclideanVectorSpace

) are

similar

iff there exists a

bijection

f

from the set of points of

K

onto the set of points of

K

′

such that, for any

P

,

Q

∈

K

, the ratio

P

′

⁢

Q

′

P

⁢

Q

of the lengths of the

line segments

P

′

⁢

Q

′

and

P

⁢

Q

is always the same number

k

, where

P

′

=

f

⁢

(

P

)

and

Q

′

=

f

⁢

(

Q

)

.

The number

k

is called the

ratio of similarity

or the

line ratio

of the figure

K

′

with respect to the figure

K

(N.B. the in which the figures are mentioned!).  The

similarity

of

K

and

K

′

is often denoted by

K

′

∼

K

(

or

K

∼

K

′

)

.

Examples

•

All squares are similar.

•

All cubes are similar.

•

All circles are similar.

•

All

parabolas

are similar.

•

All

sectors of circle

with equal

central angle

are similar.

•

All spheres are similar.

•

All

equilateral triangles

are similar.

Nonexamples

•

Not all

rectangles

are similar.

•

Not all rhombi are similar.

•

Not all rectangular prisms are similar.

•

Not all

ellipses

are similar.

•

Not all

ellipsoids

are similar.

•

Not all

triangles

are similar.

Properties

•

The corresponding angles (consisting of corresponding points) of two similar figures are equal.

•

The lengths of any corresponding arcs of two similar figures are proportional in the ratio

k

.

•

The areas of two similar regions are proportional in the ratio

k

2

when

k

is the line ratio of the regions.

•

The volumes of two similar solids are proportional in the ratio

k

3

when

k

is the line ratio of the solids.

Remarks

•

In any Euclidean space

E

, the

relation

(

http://planetmath.org/Relation

) of similarity (denoted

∼

) on the set of figures in

E

is an

equivalence relation

.

•

If one pair of corresponding line segments in the similar figures

K

and

K

′

are equal, then all pairs of corresponding line segments are equal, i.e. the figures have also equal : They are

congruent

(

http://planetmath.org/

Congruence

) (

K

′

≅

K

).

Title

similarity in geometry

Canonical

name

SimilarityInGeometry

Date of creation

2013-03-22 17:08:48

Last modified on

2013-03-22 17:08:48

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

19

Author

pahio (2872)

Entry type

Definition

Classification

msc 51F99

Classification

msc 51M05

Classification

msc 51-00

Synonym

similarity

Synonym

similitude

Related topic

Homothety

Related topic

ProportionEquation

Related topic

HarmonicMeanInTrapezoid

Defines

similar

Defines

ratio of similarity

Defines

similitude ratio

Defines

line ratio

Generated on Fri Feb 9 22:01:53 2018 by

LaTeXML
similarity of triangles

similarity of triangles

The following

theorems

are valid in

Euclidean geometry

:

Theorem AA.

If one

triangle

has a pair of angles that are

congruent

to a pair of angles in another triangle, then the two triangles are

similar

.

Theorem .

If a pair of sides of a triangle are proportional to a pair of sides in another triangle and if the angles included by the side-pairs are congruent, then the triangles are similar.

Theorem .

If the sides of a triangle are proportional to the sides of another triangle, then the triangles are similar.

The

AA theorem

may be regarded as the

definition

of the similarity of triangles. In some texts, the AA theorem is assumed as a postulate. The other two theorems may be proved by using the

law of cosines

for determining the the ratios other sides (for ) and the angles.

In

hyperbolic geometry

and

spherical geometry

,

similar triangles

are congruent. (See the

AAA theorem

for more details.) Thus, the theorem and theorem are invalid in these .

Title

similarity of triangles

Canonical

name

SimilarityOfTriangles

Date of creation

2013-03-22 17:49:41

Last modified on

2013-03-22 17:49:41

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

12

Author

pahio (2872)

Entry type

Theorem

Classification

msc 51F99

Classification

msc 51M05

Classification

msc 51-00

Synonym

similar triangles

Related topic

HarmonicMeanInTrapezoid

Related topic

AreaOfSphericalCalotteByMeansOfChord

Related topic

InterceptTheorem

Defines

AA

Defines

AA postulate

Defines

AA theorem

Generated on Fri Feb 9 22:01:55 2018 by

LaTeXML
Weizenbock’s inequality

Weizenbock’s inequality

In a

triangle

with sides

a

,

b

,

c

, and with area

A

, the following

inequality

holds:

a

2

+

b

2

+

c

2

≥

4

⁢

A

⁢

3

The proof goes like this: if

s

=

a

+

b

+

c

2

is the

semiperimeter

of the
triangle, then from Heron’s

formula

we have:

A

=

s

⁢

(

s

-

a

)

⁢

(

s

-

b

)

⁢

(

s

-

c

)

But by

squaring

the latter and expanding the parentheses we obtain:

16

⁢

A

2

=

2

⁢

(

a

2

⁢

b

2

+

a

2

⁢

c

2

+

b

2

⁢

c

2

)

-

(

a

4

+

b

4

+

c

4

)

Thus, we only have to prove that:

(

a

2

+

b

2

+

c

2

)

2

≥

3

⁢

[

2

⁢

(

a

2

⁢

b

2

+

a

2

⁢

c

2

+

b

2

⁢

c

2

)

-

(

a

4

+

b

4

+

c

4

)

]

or equivalently:

4

⁢

(

a

4

+

b

4

+

c

4

)

≥

4

⁢

(

a

2

⁢

b

2

+

a

2

⁢

c

2

+

b

2

⁢

c

2

)

which is trivially

equivalent

to:

(

a

2

-

b

2

)

2

+

(

a

2

-

c

2

)

2

+

(

b

2

-

c

2

)

2

≥

0

Equality is achieved if and only if

a

=

b

=

c

(i.e. when the triangle is

equilateral

) .

See also the

Hadwiger-Finsler inequality

, from which this result follows as a corollary.

Title

Weizenbock’s inequality

Canonical name

WeizenbocksInequality

Date of creation

2013-03-22 13:19:33

Last modified on

2013-03-22 13:19:33

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 51F99

Related topic

HadwigerFinslerInequality

Generated on Fri Feb 9 22:01:57 2018 by

LaTeXML
a line segment has at most one midpoint

a line segment has at most one midpoint

(this proof is not correct yet)

Theorem 1

.

In an

ordered geometry

a

line segment has at most one midpoint

.

Proof.

Let

[

p

,

q

]

be a

closed line segment

and suppose

m

and

m

′

are

midpoints

.
If

m

:

p

:

q

then

[

m

,

p

]

<

[

m

,

q

]

so

m

is not a midpoint. Similarly we cannot have

p

:

q

:

m

, so we have

p

:

m

:

q

. And also,

p

:

m

′

:

q

. Suppose

m

≠

m

′

. Without loss of
generality we can assume

p

:

m

:

m

′

and

m

:

m

′

:

q

. But then

[

p

,

m

′

]

>

[

p

,

m

]

≅

[

m

,

q

]

>

[

m

′

,

q

]

so that

[

p

,

m

′

]

≇

[

m

′

,

q

]

, a

contradiction

. Hence

m

=

m

′

.
∎

Title

a line segment has at most one midpoint

Canonical name

ALineSegmentHasAtMostOneMidpoint

Date of creation

2013-03-22 17:17:37

Last modified on

2013-03-22 17:17:37

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

9

Author

Mathprof (13753)

Entry type

Theorem

Classification

msc 51G05

Generated on Fri Feb 9 22:02:00 2018 by

LaTeXML
betweenness relation

betweenness relation

1

Definition

Let

A

be a set. A

ternary relation

B

on

A

is said to be a

betweenness relation

if it has the following properties:

O1

if

(

a

,

b

,

c

)

∈

B

, then

(

c

,

b

,

a

)

∈

B

; in other words, the set

B

⁢

(

b

)

=

{

(

a

,

c

)

∣

(

a

,

b

,

c

)

∈

B

}

is a

symmetric relation

(

http://planetmath.org/

Symmetric

) for

each

b

; thus, from now on, we may say, without any ambiguity, that

b

is

between

a

and

c

if

(

a

,

b

,

c

)

∈

B

;

O2

if

(

a

,

b

,

a

)

∈

B

, then

a

=

b

;

O3

for each

a

,

b

∈

A

, there is a

c

∈

A

such that

(

a

,

b

,

c

)

∈

B

;

O4

for each

a

,

b

∈

A

, there is a

c

∈

A

such that

(

a

,

c

,

b

)

∈

B

;

O5

if

(

a

,

b

,

c

)

∈

B

and

(

b

,

a

,

c

)

∈

B

, then

a

=

b

;

O6

if

(

a

,

b

,

c

)

∈

B

and

(

b

,

c

,

d

)

∈

B

, then

(

a

,

b

,

d

)

∈

B

;

O7

if

(

a

,

b

,

d

)

∈

B

and

(

b

,

c

,

d

)

∈

B

, then

(

a

,

b

,

c

)

∈

B

.

Title

betweenness relation

Canonical name

BetweennessRelation

Date of creation

2013-03-22 17:18:44

Last modified on

2013-03-22 17:18:44

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

6

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 51G05

Synonym

axioms of order

Related topic

SomeTheoremsOnTheAxiomsOfOrder

Generated on Fri Feb 9 22:02:02 2018 by

LaTeXML
complementary angles

complementary angles

Two angles are called

complementary angles

of each other, if their sum is the

right angle

π

2

, i.e.

90

∘

.

For example, the

acute angles

of a

right triangle

are complement angles of each other, since the

angle sum

in any triangle is

180

∘

.

The sine of an angle is equal to the cosine of the complement angle, and vice versa.

The

tangent

of an angle equals to the cotangent of the complement angle, and vice versa (provided that no one of the angles is a multiple of the

straight angle

).

Title

complementary angles

Canonical

name

ComplementaryAngles

Date of creation

2013-03-22 17:18:07

Last modified on

2013-03-22 17:18:07

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Definition

Classification

msc 51G05

Classification

msc 51F20

Synonym

complement angles

Synonym

complementary

Related topic

ExplementaryAngle

Related topic

SupplementaryAngles

Related topic

GoniometricFormulae

Related topic

ConvexAngle

Related topic

Explementary

Generated on Fri Feb 9 22:02:04 2018 by

LaTeXML
linear ordered geometry

linear ordered geometry

An

incidence geometry

A

=

(

P

,

n

,

I

)

is a

linear ordered geometry

if there is a

strict betweenness
relation

B

defined on the points

P

0

of

A

, such that

Col1

(

p

,

q

,

r

)

∈

B

only if

p

,

q

, and

r

are

collinear

(all

incident

with a common
line

ℓ

∈

P

1

);

Col2

for any pairwise distinct

collinear points

p

,

q

,

r

, at least one of

(

p

,

q

,

r

)

,

(

q

,

r

,

p

)

, or

(

r

,

p

,

q

)

∈

B

,

We denote the linear ordered geometry by

(

A

,

B

)

.

Title

linear ordered geometry

Canonical name

LinearOrderedGeometry

Date of creation

2013-03-22 17:19:35

Last modified on

2013-03-22 17:19:35

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

4

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 51G05

Generated on Fri Feb 9 22:02:05 2018 by

LaTeXML
Pasch’s theorem

Pasch’s theorem

Theorem

.

(Pasch) Let

△

⁢

a

⁢

b

⁢

c

be a

triangle

with
non-collinear vertices

a

,

b

,

c

in a

linear ordered geometry

.
Suppose a line

ℓ

intersects

one side, say

open line segment

a

⁢

b

¯

, at a
point

strictly between

a

and

b

, then

ℓ

also intersects exactly one of the following:

b

⁢

c

¯

⁢

,

a

⁢

c

¯

⁢

,

c

.

Proof.

First, note that vertices

a

and

b

are on opposite sides of line

ℓ

. Then either

c

lies on

ℓ

, or

c

does not. if

c

does not, then it must lie on either side (half plane) of

ℓ

. In other words,

c

and

a

must be on the opposite sides of

ℓ

, or

c

and

b

must be on the opposite sides of

ℓ

.
If

c

and

a

are on the opposite sides,

ℓ

has a non-empty

intersection

with

a

⁢

c

¯

. But if

c

and

a

are on the opposite sides, then

c

and

b

are on the same side, which means that

b

⁢

c

¯

does not intersect

ℓ

.
∎

Remark

A companion property states that if line

ℓ

passes through

one vertex

a

of a triangle

△

⁢

a

⁢

b

⁢

c

and at least one other point on

△

⁢

a

⁢

b

⁢

c

, then it must intersect exactly one of the following:

b

⁢

,

c

⁢

,

b

⁢

c

¯

.

Of course, if

ℓ

passes through

b

,

a

⁢

b

¯

must lie on

ℓ

. Similarly,

a

⁢

c

¯

lies on

ℓ

if

ℓ

passes through

c

.

Title

Pasch’s

theorem

Canonical name

PaschsTheorem

Date of creation

2013-03-22 15:32:09

Last modified on

2013-03-22 15:32:09

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

13

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 51G05

Related topic

Angle

Related topic

OrderedGeometry

Generated on Fri Feb 9 22:02:10 2018 by

LaTeXML
some theorems on strict betweenness relations

some theorems on strict betweenness relations

Let

B

be a

strict betweenness relation

. In the following the sets

B

*

p

⁢

q

,

B

p

*

q

,

B

p

⁢

q

⁣

*

,

B

p

⁢

q

,

B

⁢

(

p

,

q

)

are defined in the entry about some

theorems

on the

axioms of order

.

Theorem 1

.

Three

elements

are
in a strict betweenness relation only if they are pairwise distinct.

Theorem 2

.

If

B

is strict, then

B

*

p

⁢

q

,

B

p

*

q

and

B

p

⁢

q

⁣

*

are

pairwise disjoint

.
Furthermore, if

p

=

q

then all three sets are empty.

Theorem 3

.

If

B

is strict, then

B

p

⁢

q

∩

B

q

⁢

p

=

B

p

*

q

and

B

p

⁢

q

∪

B

q

⁢

p

=

B

⁢

(

p

,

q

)

.

Theorem 4

.

If

B

is strict, then for any

p

,

q

∈

A

,

p

≠

q

,

B

*

p

⁢

q

,

B

p

*

q

and

B

p

⁢

q

⁣

*

are

infinite

.

Title

some theorems on strict betweenness relations

Canonical name

SomeTheoremsOnStrictBetweennessRelations

Date of creation

2013-03-22 17:18:59

Last modified on

2013-03-22 17:18:59

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

6

Author

Mathprof (13753)

Entry type

Theorem

Classification

msc 51G05

Related topic

StrictBetweennessRelation

Generated on Fri Feb 9 22:02:15 2018 by

LaTeXML
some theorems on the axioms of order

some theorems on the axioms of order

Let

B

be a

betweenness relation

on a set

A

.

Theorem 1

.

If

(

a

,

b

,

c

)

∈

B

and

(

a

,

c

,

d

)

∈

B

, then

(

a

,

b

,

d

)

∈

B

.

Theorem 2

.

For each pair of elements

p

,

q

∈

A

,
we can define five sets:

1.

B

*

p

⁢

q

:=

{

r

∈

A

∣

(

r

,

p

,

q

)

∈

B

}

,

2.

B

p

*

q

:=

{

r

∈

A

∣

(

p

,

r

,

q

)

∈

B

}

,

3.

B

p

⁢

q

⁣

*

:=

{

r

∈

A

∣

(

p

,

q

,

r

)

∈

B

}

,

4.

B

p

⁢

q

:=

B

p

*

q

∪

{

q

}

∪

B

p

⁢

q

⁣

*

, and

5.

B

⁢

(

p

,

q

)

:=

B

*

p

⁢

q

∪

{

p

}

∪

B

p

⁢

q

.

Then

(1)

B

*

p

⁢

q

=

B

q

⁢

p

⁣

*

.

(2)

B

p

*

q

=

B

q

*

p

.

(3)

The intersection of any pair of the first three sets contains at most one element, either

p

or

q

.

(4)

Each of the sets can be partially ordered.

(5)

The

partial order

on

B

p

⁢

q

and

B

⁢

(

p

,

q

)

extends that of the subsets.

Title

some theorems on the axioms of order

Canonical name

SomeTheoremsOnTheAxiomsOfOrder

Date of creation

2013-03-22 17:18:47

Last modified on

2013-03-22 17:18:47

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

6

Author

Mathprof (13753)

Entry type

Theorem

Classification

msc 51G05

Related topic

BetweennessRelation

Generated on Fri Feb 9 22:02:17 2018 by

LaTeXML
strict betweenness relation

strict betweenness relation

1

Definition

A

strict betweenness relation

is a

betweenness relation

that

satisfies

the following axioms:

O

⁢

2

′

(

p

,

q

,

p

)

∉

B

)

for

each

pair of points

p

and

q

.

O

⁢

3

′

for each

p

,

q

∈

A

such that

p

≠

q

, there is an

r

∈

A

such that

(

p

,

q

,

r

)

∈

B

.

O

⁢

4

′

for each

p

,

q

∈

A

such that

p

≠

q

, there is an

r

∈

A

such that

(

p

,

r

,

q

)

∈

B

.

O

⁢

5

′

if

(

p

,

q

,

r

)

∈

B

, then

(

q

,

p

,

r

)

∉

B

.

2

Remarks

•

A very simple example of a strict betweenness relation is the

empty set

.
In

∅

, all the conditions are

vacuously

satisfied.
The empty set, in this

context

, is called the

trivial

strict betweenness relation.

•

Any strict betweenness relation can be enlarged to a betweenness
relation by including all triples of the forms

(

p

,

p

,

q

)

,

(

p

,

q

,

p

)

,

or

(

p

,

q

,

q

)

.

•

Conversely

, any betweenness relation can be

reduced

to a strict betweenness relation by removing all triples of the
forms just listed. However, it is possible that the “derived”
strict betweenness relation is trivial.

•

From axiom

O

⁢

2

′

we have

(

p

,

p

,

p

)

∉

B

.

Title

strict betweenness relation

Canonical

name

StrictBetweennessRelation

Date of creation

2013-03-22 17:18:56

Last modified on

2013-03-22 17:18:56

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

7

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 51G05

Related topic

SomeTheoremsOnStrictBetweennessRelations

Generated on Fri Feb 9 22:02:19 2018 by

LaTeXML