aarjaneiro/planetmath huggingface dataset:
Hall’s marriage theorem

Hall’s marriage theorem

Let

S

=

{

S

1

,

S

2

,

…

⁢

S

n

}

be a finite

collection

of

finite sets

. There exists a

system of distinct representatives

of

S

if and only if the following condition holds for any

T

⊆

S

:

|

∪

T

|

≥

|

T

|

As a corollary, if this condition fails to hold anywhere, then no SDR exists.

This is known as Hall’s

marriage theorem

. The name arises from a particular application of this

theorem

. Suppose we have a finite set of single men/women, and, for each man/woman, a finite collection of women/men to whom this person is attracted. An SDR for this collection would be a way each man/woman could be (theoretically) married happily. Hence, Hall’s marriage theorem can be used to determine if this is possible.

An application of this theorem to

graph theory

gives that if

G

⁢

(

V

1

,

V

2

,

E

)

is a

bipartite graph

, then

G

has a

complete matching

that

saturates

every vertex of

V

1

if and only if

|

S

|

≤

|

N

⁢

(

S

)

|

for every subset

S

⊂

V

1

.

Title

Hall’s marriage theorem

Canonical name

HallsMarriageTheorem

Date of creation

2013-03-22 12:35:14

Last modified on

2013-03-22 12:35:14

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

6

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 05D15

Synonym

marriage theorem

Related topic

Saturate

Generated on Fri Feb 9 07:58:15 2018 by

LaTeXML
Hall’s marriage theorem, proof of

Hall’s marriage theorem, proof of

We prove Hall’s

marriage theorem

by

induction

on

|

S

|

, the size of

S

.

The

theorem

is trivially true for

|

S

|

=

0

.

Assuming the theorem true for all

|

S

|

<

n

, we prove it for

|

S

|

=

n

.

First suppose that we have the stronger condition

|

∪

T

|

≥

|

T

|

+

1

for all

∅

≠

T

⊂

S

. Pick any

x

∈

S

n

as the representative of

S

n

; we must choose an SDR from

S

′

=

{

S

1

∖

{

x

}

,

…

,

S

n

-

1

∖

{

x

}

}

.

But if

{

S

j

1

∖

{

x

}

,

…

,

S

j

k

∖

{

x

}

}

=

T

′

⊆

S

′

then, by our

assumption

,

|

∪

T

′

|

≥

|

∪

i

=

1

k

S

j

i

|

-

1

≥

k

.

By the already-proven case of the theorem for

S

′

we see that we can indeed pick an SDR for

S

′

.

Otherwise, for some

∅

≠

T

⊂

S

we have the “exact” size

|

∪

T

|

=

|

T

|

.

Inside

T

itself, for any

T

′

⊆

T

⊂

S

we have

|

∪

T

′

|

≥

|

T

′

|

,

so by an already-proven case of the theorem we can pick an SDR for

T

.

It remains to pick an SDR for

S

∖

T

which avoids all elements of

∪

T

(these

elements

are in the SDR for

T

).
To use the already-proven case of the theorem (again) and do this, we must show that for any

T

′

⊆

S

∖

T

, even after discarding elements of

∪

T

there remain enough elements in

∪

T

′

: we must prove

|

∪

T

′

∖

∪

T

|

≥

|

T

′

|

.

But

|

∪

T

′

∖

∪

T

|

=

|

⋃

(

T

∪

T

′

)

|

-

|

∪

T

|

≥

(1)

≥

|

T

∪

T

′

|

-

|

T

|

=

(2)

=

|

T

|

+

|

T

′

|

-

|

T

|

=

|

T

′

|

,

(3)

using the disjointness of

T

and

T

′

. So by an already-proven case of the theorem,

S

∖

T

does indeed have an SDR which avoids all elements of

∪

T

.

This the proof.

Title

Hall’s marriage theorem, proof of

Canonical name

HallsMarriageTheoremProofOf

Date of creation

2013-03-22 12:45:12

Last modified on

2013-03-22 12:45:12

Owner

mps (409)

Last modified by

mps (409)

Numerical id

10

Author

mps (409)

Entry type

Proof

Classification

msc 05D15

Generated on Fri Feb 9 07:58:17 2018 by

LaTeXML
saturate

saturate

Let

G

⁢

(

V

,

E

)

be a graph and

M

a

matching

in

G

. A vertex

v

∈

V

⁢

(

G

)

is said to be

saturated

by

M

if there is an edge in

M

incident

to

v

. A vertex

v

∈

V

⁢

(

G

)

with no such edge is said to be

unsaturated by

M

. We also say that

M

saturates

v

.

Title

saturate

Canonical

name

Saturate

Date of creation

2013-03-22 13:57:57

Last modified on

2013-03-22 13:57:57

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

4

Author

mathcam (2727)

Entry type

Definition

Classification

msc 05D15

Synonym

saturates

Synonym

saturated

Related topic

HallsMarriageTheorem

Related topic

BipartiteMatching

Related topic

Matching

Generated on Fri Feb 9 07:58:19 2018 by

LaTeXML
system of distinct representatives

system of distinct representatives

Let

S

=

{

S

1

,

S

2

,

…

⁢

S

n

}

be a finite

collection

of

finite sets

. A

system of distinct representatives

, or

SDR

, of

S

is a set

x

1

∈

S

1

,

x

2

∈

S

2

,

…

⁢

x

n

∈

S

n

such that

x

i

≠

x

j

⁢

whenever

⁢

i

≠

j

(i.e., each choice must be unique).

Title

system of distinct representatives

Canonical name

SystemOfDistinctRepresentatives

Date of creation

2013-03-22 12:35:11

Last modified on

2013-03-22 12:35:11

Owner

vampyr (22)

Last modified by

vampyr (22)

Numerical id

4

Author

vampyr (22)

Entry type

Definition

Classification

msc 05D15

Synonym

SDR

Generated on Fri Feb 9 07:58:21 2018 by

LaTeXML
tight

tight

A bound is

tight

if it can be realized. A bound that is not
tight is sometimes said to be

slack

.

For example, let

𝒮

be the collection of all finite subsets
of

ℝ

2

in

general position

. Define a function

g

:

ℕ

→

ℕ

as follows. First, let the weight of
an element

S

of

𝒮

be the size of its largest convex subset,
that is,

w

⁢

(

S

)

=

max

⁡

{

|

T

|

:

T

⊂

S

⁢

and

T

is convex

}

.

The function

g

is defined by

g

⁢

(

n

)

=

min

⁡

{

|

S

|

:

w

⁢

(

S

)

≥

n

}

,

that is,

g

⁢

(

n

)

is the smallest number such that any collection of

g

⁢

(

n

)

points in general position contains a convex

n

-gon. (By the
Erdős-Szekeres

theorem

(

http://planetmath.org/HappyEndingProblem

),

g

⁢

(

n

)

is
always finite, so

g

is a

well-defined

function.) The bounds for

g

due to Erdős and Szekeres are

2

n

-

2

+

1

≤

g

⁢

(

n

)

≤

(

2

⁢

n

-

4

n

-

2

)

+

1

.

The lower bound is tight because for each

n

, there is a set of

2

n

-

2

points in general position which contains no convex

n

-gon.
On the other hand, the upper bound is believed to be slack. In fact,
according to the Erdős-Szekeres

conjecture

, the formula for

g

⁢

(

n

)

is exactly the lower bound:

g

⁢

(

n

)

=

2

n

-

2

+

1

.

Title

tight

Canonical

name

Tight

Date of creation

2013-03-22 16:56:23

Last modified on

2013-03-22 16:56:23

Owner

mps (409)

Last modified by

mps (409)

Numerical id

6

Author

mps (409)

Entry type

Definition

Classification

msc 05D99

Defines

slack

Generated on Fri Feb 9 07:58:26 2018 by

LaTeXML
algebraic independence of elementary symmetric polynomials

algebraic independence of elementary symmetric polynomials

Theorem

.

Let

s

1

,

s

2

,

…

,

s

n

be the

elementary symmetric polynomials

in

n

variables

t

1

,

t

2

,

…

,

t

n

over a commutative ring

R

.
Then

s

1

,

s

2

,

…

,

s

n

are

algebraically independent

elements of

R

⁢

[

t

1

,

t

2

,

…

,

t

n

]

over

R

.

Title

algebraic independence of elementary symmetric polynomials

Canonical name

AlgebraicIndependenceOfElementarySymmetricPolynomials

Date of creation

2013-03-22 14:49:11

Last modified on

2013-03-22 14:49:11

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

5

Author

mclase (549)

Entry type

Theorem

Classification

msc 05E05

Generated on Fri Feb 9 07:58:28 2018 by

LaTeXML
elementary symmetric polynomial

elementary symmetric polynomial

The

coefficient

of

x

n

-

k

in the

polynomial

(

x

+

t

1

)

⁢

(

x

+

t

2

)

⁢

⋯

⁢

(

x

+

t

n

)

is called the

k

th

elementary symmetric polynomial

in the

n

variables

t

1

,

…

,

t

n

. The elementary symmetric polynomials can also be constructed by taking the sum of all possible degree

k

monomials

in

t

1

,

…

,

t

n

having distinct factors.

The first few examples are:

n

=

1

:

t

1

n

=

2

:

t

1

+

t

2

t

1

⁢

t

2

n

=

3

:

t

1

+

t

2

+

t

3

t

1

⁢

t

2

+

t

2

⁢

t

3

+

t

1

⁢

t

3

t

1

⁢

t

2

⁢

t

3

Title

elementary symmetric polynomial

Canonical

name

ElementarySymmetricPolynomial

Date of creation

2013-03-22 12:09:01

Last modified on

2013-03-22 12:09:01

Owner

djao (24)

Last modified by

djao (24)

Numerical id

9

Author

djao (24)

Entry type

Definition

Classification

msc 05E05

Generated on Fri Feb 9 07:58:30 2018 by

LaTeXML
proof of algebraic independence of elementary symmetric polynomials

proof of algebraic independence of elementary symmetric polynomials

Geometric proof, works when R is a division ring.

Consider the quotient field Q of R and then the

algebraic closure

K

of

Q

.

Consider the substitution map that

associates

to values

t

1

,

…

,

t

n

∈

K

n

the symmetric functions in these variables

s

1

,

…

,

s

n

.

ϕ

:

K

n

→

K

n

(

t

i

)

↦

(

s

i

)

Because

K

is

algebraic

closed this map is

surjective

. Indeed, fix values

v

i

, then on an algebraic closed field there are roots

t

i

such that

X

n

+

∑

i

v

i

⁢

X

i

=

Π

i

⁢

(

X

+

t

i

)

And by developing the right-hand side we get

v

i

=

s

i

.

Then we consider the

transposition

morphism

of

algebras

ϕ

*

:

ϕ

*

:

R

⁢

[

S

1

,

…

,

S

n

]

→

R

⁢

[

T

1

,

…

,

T

n

]

f

↦

f

∘

ϕ

The capital letters are there to emphasize the

S

i

and

T

i

are variables and

R

⁢

[

S

1

,

…

,

S

n

]

and

R

⁢

[

T

1

,

…

,

T

n

]

are regarded as

function

algebras over

K

n

.

The

theorem

stating that the symmetric functions are

algebraically independent

is no more than saying that this morphism is

injective

.
As a matter of fact,

ϕ

*

⁢

(

S

i

)

is the

i

t

⁢

h

symmetric function in the

T

i

, and

ϕ

*

is clearly a morphism of algebras.

The

conclusion

is then straightforward from the surjectivity of

ϕ

because if

f

∘

ϕ

=

0

for some

f

, then by surjectivity of

ϕ

it means that

f

was zero in the first place. In other words the kernel of

ϕ

*

is

reduced

to 0.

Title

proof of algebraic independence of elementary symmetric polynomials

Canonical name

ProofOfAlgebraicIndependenceOfElementarySymmetricPolynomials

Date of creation

2013-03-22 17:38:28

Last modified on

2013-03-22 17:38:28

Owner

lalberti (18937)

Last modified by

lalberti (18937)

Numerical id

4

Author

lalberti (18937)

Entry type

Proof

Classification

msc 05E05

Generated on Fri Feb 9 07:58:38 2018 by

LaTeXML
reduction algorithm for symmetric polynomials

reduction algorithm for symmetric polynomials

We give here an algorithm for reducing a

symmetric polynomial

into a

polynomial

in the

elementary symmetric polynomials

.

We define the

height

of a

monomial

x

1

e

1

⁢

⋯

⁢

x

n

e

n

in

R

⁢

[

x

1

,

…

,

x

n

]

to be

e

1

+

2

⁢

e

2

+

⋯

+

n

⁢

e

n

. The height of a polynomial is defined to be the maximum height of any of its monomial terms, or 0 if it is the

zero polynomial

.

Let

f

be a symmetric polynomial. We reduce

f

into elementary symmetric polynomials by

induction

on the height of

f

. Let

c

⁢

x

1

e

1

⁢

⋯

⁢

x

n

e

n

be the monomial term of maximal height in

f

. Consider the polynomial

g

:=

f

-

c

⁢

s

1

e

n

-

e

n

-

1

⁢

s

2

e

n

-

1

-

e

n

-

2

⁢

⋯

⁢

s

n

-

1

e

2

-

e

1

⁢

s

n

e

1

where

s

k

is the

k

–th elementary symmetric polynomial in the

n

variables

x

1

,

…

,

x

n

. Then

g

is a symmetric polynomial of lower height than

f

, so by the

induction hypothesis

,

g

is a polynomial in

s

1

,

…

,

s

n

, and it follows immediately that

f

is also a polynomial in

s

1

,

…

,

s

n

.

Title

reduction algorithm for symmetric polynomials

Canonical name

ReductionAlgorithmForSymmetricPolynomials

Date of creation

2013-03-22 12:11:17

Last modified on

2013-03-22 12:11:17

Owner

djao (24)

Last modified by

djao (24)

Numerical id

6

Author

djao (24)

Entry type

Proof

Classification

msc 05E05

Classification

msc 12F10

Defines

height

Generated on Fri Feb 9 07:58:40 2018 by

LaTeXML
star product

star product

The

star product

of two

graded posets

(

P

,

≤

P

)

and

(

Q

,

≤

Q

)

, where

P

has a unique

maximal element

1

^

and

Q

has a unique minimal element

0

^

, is the poset

P

*

Q

on the set

(

P

∖

1

^

)

∪

(

Q

∖

0

^

)

. We define the

partial order

≤

P

*

Q

by

x

≤

y

if and only if:

1.

{

x

,

y

}

⊂

P

, and

x

≤

P

y

;

2.

{

x

,

y

}

⊂

Q

, and

x

≤

Q

y

; or

3.

x

∈

P

and

y

∈

Q

.

In other words, we pluck out the top of

P

and the bottom of

Q

, and require that everything in

P

be smaller than everything in

Q

.
For example, suppose

P

=

Q

=

B

2

.

\xymatrix

⁢

&

⁢

1

^

P

⁢

\ar

⁢

@

-

[

d

⁢

l

]

⁢

\ar

⁢

@

-

[

d

⁢

r

]

⁢

&

⁢

&

⁢

&

⁢

&

⁢

1

^

Q

⁢

\ar

⁢

@

-

[

d

⁢

l

]

⁢

\ar

⁢

@

-

[

d

⁢

r

]

⁢

&

⁢

a

P

⁢

\ar

⁢

@

-

[

d

⁢

r

]

⁢

&

⁢

&

⁢

b

P

⁢

\ar

⁢

@

-

[

d

⁢

l

]

⁢

&

⁢

&

⁢

a

Q

⁢

\ar

⁢

@

-

[

d

⁢

r

]

⁢

&

⁢

&

⁢

b

Q

⁢

\ar

⁢

@

-

[

d

⁢

l

]

⁢

&

⁢

0

^

P

⁢

&

⁢

&

⁢

&

⁢

&

⁢

0

^

Q

⁢

&

Then

P

*

Q

is the poset with the

Hasse diagram

below.

\xymatrix

⁢

&

⁢

1

^

Q

⁢

\ar

⁢

@

-

[

d

⁢

l

]

⁢

\ar

⁢

@

-

[

d

⁢

r

]

⁢

&

⁢

a

Q

⁢

\ar

⁢

@

-

[

d

]

⁢

\ar

⁢

@

-

[

d

⁢

r

⁢

r

]

⁢

&

⁢

&

⁢

b

Q

⁢

\ar

⁢

@

-

[

d

⁢

l

⁢

l

]

⁢

\ar

⁢

@

-

[

d

]

⁢

a

P

⁢

\ar

⁢

@

-

[

d

⁢

r

]

⁢

&

⁢

&

⁢

b

P

⁢

\ar

⁢

@

-

[

d

⁢

l

]

⁢

&

⁢

0

^

P

⁢

&

The star product of

Eulerian posets

is Eulerian.

References

1

Stanley, R.,

Flag

f

-vectors and the

cd

-index

, Math. Z. 216 (1994), 483-499.

Title

star product

Canonical name

StarProduct

Date of creation

2013-03-22 14:09:17

Last modified on

2013-03-22 14:09:17

Owner

mps (409)

Last modified by

mps (409)

Numerical id

4

Author

mps (409)

Entry type

Definition

Classification

msc 05E99

Classification

msc 06A11

Related topic

Poset

Related topic

GradedPoset

Generated on Fri Feb 9 07:58:54 2018 by

LaTeXML
equality

equality

In any set

S

, the

equality

, denoted by “

=

”, is a

binary relation

which is

reflexive

,

symmetric

,

transitive

and antisymmetric, i.e. it is an antisymmetric

equivalence relation

on

S

, or which is the same thing, the equality is a symmetric

partial order

on

S

.

In fact, for any set

S

, the smallest equivalence relation on

S

is the equality (by smallest we that it is

contained

in every equivalence relation on

S

). This offers a

definition

of “equality”. From this, it is clear that there is only one equality relation on

S

.  Its

equivalence classes

are all

singletons

{

x

}

where

x

∈

S

.

The

concept

of equality is essential in almost all branches of mathematics. A few examples will suffice:

1

+

1

=

2

e

i

⁢

π

=

-

1

ℝ

⁢

[

i

]

=

ℂ

(The second example is Euler’s

identity

.)

Remark 1.

Although the four characterising ,

reflexivity

,

symmetry

(

http://planetmath.org/Symmetric

),

transitivity

and antisymmetry (

http://planetmath.org/Antisymmetric

), determine the equality on

S

uniquely, they cannot be thought to form the definition of the equality, since the concept of antisymmetry already the equality.

Remark 2.

An equality (

equation

) in a set

S

may be true regardless to the values of the

variables

involved in

the equality; then one speaks of an

identity

or

identic equation

in this set.  E.g.

(

x

+

y

)

2

=

x

2

+

y

2

is an identity in a field with

characteristic

(

http://planetmath.org/Characteristic

)

2

.

Title

equality

Canonical

name

Equality

Date of creation

2013-03-22 18:01:26

Last modified on

2013-03-22 18:01:26

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

14

Author

pahio (2872)

Entry type

Topic

Classification

msc 06-00

Related topic

Equation

Defines

equality relation

Defines

identity

Defines

identic equation

Generated on Fri Feb 9 09:49:59 2018 by

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equivalence relation

equivalence relation

An

equivalence relation

∼

on a set

S

is a

relation

that is:

Reflexive

.

a

∼

a

for all

a

∈

S

.

Symmetric

.

Whenever

a

∼

b

, then

b

∼

a

.

Transitive

.

If

a

∼

b

and

b

∼

c

then

a

∼

c

.

If

a

and

b

are related this way we say that they are

equivalent

under

∼

.
If

a

∈

S

, then the set of all elements of

S

that are equivalent to

a

is called the

equivalence class

of

a

. The set of all equivalence classes under

∼

is written

S

/

∼

.

An equivalence relation on a set induces a

partition

on it.

Conversely

, any partition induces an equivalence relation. Equivalence relations are important, because often the set

S

can be ’transformed’ into another set (

quotient space

) by considering each equivalence class as a single unit.

Two examples of equivalence relations:

1. Consider the set of integers

ℤ

and take a positive integer

m

. Then

m

induces an equivalence relation by

a

∼

b

when

m

divides

b

-

a

(that is,

a

and

b

leave the same

remainder

when divided by

m

).

2. Take a group

(

G

,

⋅

)

and a

subgroup

H

. Define

a

∼

b

whenever

a

⁢

b

-

1

∈

H

. That defines an equivalence relation. Here equivalence classes are called cosets.

Title

equivalence relation

Canonical

name

EquivalenceRelation

Date of creation

2013-03-22 11:48:27

Last modified on

2013-03-22 11:48:27

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

15

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06-00

Classification

msc 03D20

Related topic

QuotientGroup

Related topic

EquivalenceClass

Related topic

Equivalent

Related topic

EquivalenceRelation

Related topic

Partition

Related topic

MathbbZ_n

Defines

equivalent

Defines

equivalence class

Generated on Fri Feb 9 09:50:01 2018 by

LaTeXML
topic entry on order theory

topic entry on order theory

This is an entry-in-progress for a new

topic entry on order theory

.

1.

index of inequalities

2.

partial ordering

,

inductive order

,

linear ordering

,

pre-ordering

,

well-ordering

3.

semilattices

4.

general

lattice theory

(a)

modular

lattices

(b)

complemented

lattices

(c)

distributive

lattices

5.

lattices with

operators

6.

Boolean algebras

7.

ordered

algebraic structures

8.

applications to

topology

,

analysis

9.

applications to mathematical logic, algebraic logic

10.

applications to computer science, formal concept analysis

Title

topic entry on order theory

Canonical name

TopicEntryOnOrderTheory

Date of creation

2013-03-22 18:00:54

Last modified on

2013-03-22 18:00:54

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Topic

Classification

msc 06-00

Generated on Fri Feb 9 09:50:10 2018 by

LaTeXML
dense total order

dense total order

A

total order

(

S

,

<

)

is

dense

if whenever

x

<

z

in

S

, there exists at least one element

y

of

S

such that

x

<

y

<

z

. That is, each nontrivial

closed interval

has nonempty

interior

.

A subset

T

of a total order

S

is

dense in

S

if for every

x

,

z

∈

S

such that

x

<

z

, there exists some

y

∈

T

such that

x

<

y

<

z

. Because of this, a dense total order

S

is sometimes said to be

dense in itself

.

For example, the integers with the usual order are not dense, since there is no integer

strictly between

0

and

1

. On the other hand, the rationals

ℚ

are dense, since whenever

r

and

s

are rational numbers, it follows that

(

r

+

s

)

/

2

is a rational number strictly between

r

and

s

.
Also, both

ℚ

and the irrationals

ℝ

∖

ℚ

are dense in

ℝ

.

It is usually convenient to assume that a dense order has at least two elements. This allows one to avoid the trivial cases of the one-point order and the empty order.

Title

dense total order

Canonical name

DenseTotalOrder

Date of creation

2013-03-22 16:40:48

Last modified on

2013-03-22 16:40:48

Owner

mps (409)

Last modified by

mps (409)

Numerical id

8

Author

mps (409)

Entry type

Definition

Classification

msc 06A05

Synonym

dense linear order

Related topic

LinearContinuum

Defines

dense

Defines

dense in

Defines

dense in itself

Generated on Fri Feb 9 09:50:15 2018 by

LaTeXML
example of law of trichotomy on ¡

example of law of trichotomy on ¡

From the axiomatic

definition

of the real numbers, “

<

” is a relation on

ℝ

which

satisfies

the

law of trichotomy

. That is, for all

a

,

b

∈

ℝ

,
exactly one of the following is true:

•

a

<

b

•

b

<

a

•

a

=

b

As an axiom, this is sometimes expressed with

b

=

0

. That is, for all

a

∈

ℝ

,

a

is either

positive

, negative, or zero.

Title

example of law of trichotomy on ¡

Canonical

name

ExampleOfLawOfTrichotomyOn

Date of creation

2013-03-22 17:15:07

Last modified on

2013-03-22 17:15:07

Owner

me_and (17092)

Last modified by

me_and (17092)

Numerical id

5

Author

me_and (17092)

Entry type

Example

Classification

msc 06A05

Classification

msc 03E20

Generated on Fri Feb 9 09:50:16 2018 by

LaTeXML
infimum and supremum of sum and product

infimum and supremum of sum and product

Suppose that the real functions

f

and

g

are defined on an

interval

Δ

.  Then on this interval

•

inf

⁡

(

f

+

g

)

≧

inf

⁡

f

+

inf

⁡

g

•

sup

⁡

(

f

+

g

)

≦

sup

⁡

f

+

sup

⁡

g

If

f

and

g

are also nonnegative on

Δ

, we can write

•

inf

⁡

(

f

⁢

g

)

≧

inf

⁡

f

⋅

inf

⁡

g

•

sup

⁡

(

f

⁢

g

)

≦

sup

⁡

f

⋅

sup

⁡

g

Title

infimum and supremum of sum and product

Canonical name

InfimumAndSupremumOfSumAndProduct

Date of creation

2013-03-22 19:01:25

Last modified on

2013-03-22 19:01:25

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

4

Author

pahio (2872)

Entry type

Theorem

Classification

msc 06A05

Classification

msc 26D15

Related topic

ProductAndQuotientOfFunctionsSum

Generated on Fri Feb 9 09:50:18 2018 by

LaTeXML
law of trichotomy

law of trichotomy

The

law of trichotomy

for a

binary relation

R

on a set

S

is the property that

•

for all

x

,

y

∈

S

, exactly one of the following holds:

x

⁢

R

⁢

y

or

y

⁢

R

⁢

x

or

x

=

y

.

A binary relation satisfying the law of trichotomy is sometimes said to be

trichotomous

.
Trichotomous binary relations are

equivalent

to

tournaments

,
although the study of tournaments is usually restricted to the finite case.

A

transitive

trichotomous binary relation is called a

total order

, and is typically written

<

.

The law of trichotomy for

cardinal numbers

is equivalent (in ZF) to the

axiom of choice

(

http://planetmath.org/AxiomOfChoice

).

Title

law of trichotomy

Canonical name

LawOfTrichotomy

Date of creation

2013-03-22 14:13:46

Last modified on

2013-03-22 14:13:46

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

9

Author

yark (2760)

Entry type

Definition

Classification

msc 06A05

Classification

msc 03E20

Defines

trichotomy

Defines

trichotomous

Generated on Fri Feb 9 09:50:20 2018 by

LaTeXML