aarjaneiro/planetmath huggingface dataset:
Pascal’s triangle is symmetrical along its central column

Pascal’s triangle is symmetrical along its central column

As a

consequence

of Pascal’s rule, we see that Pascal’s

triangle

is symmetrical along its

central

column (the column containing the

central binomial coefficients

). Expressing

individual

values in Pascal’s triangle

T

as

T

⁢

(

n

,

k

)

, with

n

and

k

being

integers

obeying the

relation

-

1

<

k

≤

n

, this means that each

T

⁢

(

n

,

k

)

=

T

⁢

(

n

,

n

-

k

)

.

Since Pascal’s triangle is essentially a table in which to look up

binomial coefficients

,

T

⁢

(

n

,

k

)

=

(

n

k

)

.

From Pascal’s rule it follows that

T

⁢

(

n

,

k

)

=

T

⁢

(

n

-

1

,

k

-

1

)

+

T

⁢

(

n

-

1

,

k

)

.

Obviously

T

⁢

(

0

,

k

)

=

1

because there is only one way to choose no items from a

collection

of

k

items; likewise,

T

⁢

(

k

,

k

)

=

1

because there is only one way to choose

k

items from a collection of

k

items. Therefore, the leftmost and rightmost column of Pascal’s triangle are filled with 1’s. Almost as obvious is the fact that

T

⁢

(

1

,

k

)

=

k

because there are

k

ways to choose just one item from a collection of

k

items; likewise,

T

⁢

(

k

-

1

,

k

)

=

k

because there are

k

ways to choose all but one item from a collection of

k

items since leaving out one item in turn can only be done

k

times in such a collection.

From the foregoing, row 1 of Pascal’s triangle is 1, 1, row 2 is 1, 2, 1 and row 3 is 1, 3, 3, 1. From Pascal’s rule it follows that even-numbered rows (with an

odd number

of columns, and their highest, central value at

T

⁢

(

k

2

,

k

)

) will be symmetrical along the central value if the previous row was also symmetrical, while odd-numbered rows (with an

even number

of columns, and the highest, central value at both

T

⁢

(

k

-

1

2

,

k

)

and

T

⁢

(

k

+

1

2

,

k

)

will be symmetrical about the central values if the previous row was symmetrical. Since the first three rows are symmetrical, all the following rows are also symmetrical.

Title

Pascal’s triangle is symmetrical along its central column

Canonical name

PascalsTriangleIsSymmetricalAlongItsCentralColumn

Date of creation

2013-03-22 19:00:14

Last modified on

2013-03-22 19:00:14

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Corollary

Classification

msc 05A19

Generated on Thu Feb 8 20:38:29 2018 by

LaTeXML
proof of Pascal’s rule

proof of Pascal’s rule

The

definition

of

(

n

k

)

is the number of

k

-subsets out from an

n

-set. Using this combinatorial meaning the proof is straightforward.

Let

x

a distinct element from the

n

-set. As previously stated,

(

n

k

)

counts the number of subsets with

k

elements, chosen from the set with

n

elements. Now, some of these subsets will contain

x

and some others don’t.

The number of

k

-subsets not containing

x

is

(

n

-

1

k

)

, since we need to choose

k

elements from the

n

-

1

elements different from

x

.

The number of

k

-subsets containing

x

is

(

n

-

1

k

-

1

)

, because if it is given that

x

is in the subset, we only need to choose the remaining

k

-

1

elements from the

n

-

1

elements that are different from

x

.

Thus

(

n

k

)

=

(

n

-

1

k

)

+

(

n

-

1

k

-

1

)

.

Title

proof of Pascal’s rule

Canonical

name

ProofOfPascalsRule

Date of creation

2013-03-22 15:03:11

Last modified on

2013-03-22 15:03:11

Owner

drini (3)

Last modified by

drini (3)

Numerical id

5

Author

drini (3)

Entry type

Proof

Classification

msc 05A19

Generated on Thu Feb 8 20:38:31 2018 by

LaTeXML
Vandermonde identity

Vandermonde identity

Theorem 1

(

[

1

]

24.1.1 formula A. II.).

For any

p

and

q

and any

k

with

0

≤

k

≤

p

+

q

,

(

p

+

q

k

)

=

∑

i

=

0

k

(

p

i

)

⁢

(

q

k

-

i

)

.

(*)

Proof.

Let

P

and

Q

be

disjoint sets

with

|

P

|

=

p

and

|

Q

|

=

q

. Then the
left-hand side
of Equation (*) is equal to the number of subsets of

P

∪

Q

of size

k

.
To build a subset of

P

∪

Q

of size

k

, we first decide how many

elements

, say

i

with

0

≤

i

≤

k

,
we will select from

P

. We can then select those elements in

(

p

i

)

ways. Once we have done so, we must select the
remaining

k

-

i

elements from

Q

, which we can do in

(

q

k

-

i

)

ways. Thus there are

(

p

i

)

⁢

(

q

k

-

i

)

ways to select a subset of

P

∪

Q

of size

k

subject to the

restriction

that exactly

i

elements come from

P

. Summing over all possible

i

completes

the proof.
∎

References

1

Abramowitz, M., and I. A. Stegun, eds.

Handbook of Mathematical

Functions

. National Bureau of Standards, Dover, New York, 1974.

Title

Vandermonde identity

Canonical name

VandermondeIdentity

Date of creation

2013-03-22 14:08:49

Last modified on

2013-03-22 14:08:49

Owner

mps (409)

Last modified by

mps (409)

Numerical id

8

Author

mps (409)

Entry type

Theorem

Classification

msc 05A19

Related topic

PascalsRule

Generated on Thu Feb 8 20:38:34 2018 by

LaTeXML
Gaussian polynomials

Gaussian polynomials

For an

indeterminate

q

and

integers

n

≥

m

≥

0

we define the following:

(a)

(

m

)

q

=

q

m

-

1

+

q

m

-

2

+

⋯

+

1

for

m

>

0

,

(b)

(

m

!

)

q

=

(

m

)

q

⁢

(

m

-

1

)

q

⁢

⋯

⁢

(

1

)

q

for

m

>

0

, and

(

0

!

)

q

=

1

,

(c)

(

n

m

)

q

=

(

n

!

)

q

(

m

!

)

q

⁢

(

(

n

-

m

)

!

)

q

.
If

m

>

n

then we define

(

n

m

)

q

=

0

.

The

expressions

(

n

m

)

q

are called

q

-

binomial coefficients

or

Gaussian polynomials

.

Note: if we replace

q

with 1, then we obtain the familiar integers,

factorials

, and binomial coefficients. Specifically,

(a)

(

m

)

1

=

m

,

(b)

(

m

!

)

1

=

m

!

,

(c)

(

n

m

)

1

=

(

n

m

)

.

(d)

(

m

m

)

q

=

1

.

Title

Gaussian polynomials

Canonical

name

GaussianPolynomials

Date of creation

2013-03-22 11:49:49

Last modified on

2013-03-22 11:49:49

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

10

Author

mathcam (2727)

Entry type

Definition

Classification

msc 05A30

Classification

msc 05A10

Classification

msc 16S36

Classification

msc 26A09

Classification

msc 26A18

Classification

msc 15A04

Synonym

q-binomial coefficients

Related topic

ContentOfPolynomial

Generated on Thu Feb 8 20:38:36 2018 by

LaTeXML
principle of inclusion-exclusion

principle of inclusion-exclusion

The

principle of inclusion-exclusion

provides a way of methodically counting the union of possibly non-disjoint sets.

Let

C

=

{

A

1

,

A

2

,

…

⁢

A

N

}

be a finite

collection

of

finite sets

. Let

I

k

represent

the set of

k

-fold

intersections

of members of

C

(e.g.,

I

2

contains all possible intersections of two sets chosen from

C

).

Then

|

⋃

i

=

1

N

A

i

|

=

∑

j

=

1

N

(

(

-

1

)

(

j

+

1

)

⁢

∑

S

∈

I

j

|

S

|

)

For example:

|

A

∪

B

|

=

|

A

|

+

|

B

|

-

|

A

∩

B

|

|

A

∪

B

∪

C

|

=

|

A

|

+

|

B

|

+

|

C

|

-

(

|

A

∩

B

|

+

|

A

∩

C

|

+

|

B

∩

C

|

)

+

|

A

∩

B

∩

C

|

The principle of inclusion-exclusion, combined with de Morgan’s laws, can be used to count the

intersection of sets

as well. Let

A

be some

universal set

such that

A

k

⊆

A

for each

k

, and let

A

k

¯

represent the

complement

of

A

k

with respect to

A

. Then we have

|

⋂

i

=

1

N

A

i

|

=

|

⋃

i

=

1

N

A

i

¯

¯

|

thereby turning the problem of finding an intersection into the problem of finding a union.

Title

principle of inclusion-exclusion

Canonical name

PrincipleOfInclusionexclusion

Date of creation

2013-03-22 12:33:25

Last modified on

2013-03-22 12:33:25

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

9

Author

Mathprof (13753)

Entry type

Theorem

Classification

msc 05A99

Synonym

inclusion-exclusion principle

Related topic

SieveOfEratosthenes2

Related topic

BrunsPureSieve

Generated on Thu Feb 8 20:38:38 2018 by

LaTeXML
difference set

difference set

Definition

. Let

A

be a finite abelian group of order

n

. A subset

D

of

A

is said to be a

difference set

(in

A

) if there is a

positive integer

m

such that every non-zero element of

A

can be expressed as the

difference

of elements of

D

in exactly

m

ways.

If

D

has

d

elements, then we have the

equation

m

⁢

(

n

-

1

)

=

d

⁢

(

d

-

1

)

.

In the equation, we are counting the number of pairs of distinct elements of

D

. On the left hand side, we are counting it by noting that there are

m

⁢

(

n

-

1

)

pairs of elements of

D

such that their difference is non-zero. On the right hand side, we first count the number of elements in

D

2

, which is

d

2

, then subtracted by

d

, since there are

d

pairs of

(

x

,

y

)

∈

D

2

such that

x

=

y

.

A difference set with parameters

n

,

m

,

d

defined above is also called a

(

n

,

d

,

m

)

-difference set. A difference set is said to be

non-trivial

if

1

<

d

<

n

-

1

. A difference set is said to be

planar

if

m

=

1

.

Difference sets versus

square designs

. Recall that a square design is a

τ

-

(

ν

,

κ

,

λ

)

-design (

http://planetmath.org/Design

) where

τ

=

2

and the number

ν

of points is the same as the number

b

of blocks. In a general design,

b

is related to the other numbers by the equation

b

⁢

(

κ

τ

)

=

λ

⁢

(

ν

τ

)

.

So in a square design, the equation reduces to

b

⁢

κ

⁢

(

κ

-

1

)

=

λ

⁢

ν

⁢

(

ν

-

1

)

, or

λ

⁢

(

ν

-

1

)

=

κ

⁢

(

κ

-

1

)

,

which is identical to the equation above for the difference set. A square design with parameters

λ

,

ν

,

κ

is called a square

(

ν

,

κ

,

λ

)

-design.

One can show that a subset

D

of an

abelian group

A

is an

(

n

,

d

,

m

)

-difference set iff it is a square

(

n

,

d

,

m

)

-design where

A

is the set of points and

{

D

+

a

∣

a

∈

A

}

is the set of blocks.

Title

difference set

Canonical name

DifferenceSet

Date of creation

2013-03-22 16:50:04

Last modified on

2013-03-22 16:50:04

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 05B10

Defines

non-trivial difference set

Defines

planar difference set

Generated on Thu Feb 8 20:38:55 2018 by

LaTeXML
construction of magic square of odd length

construction of magic square of odd length

The construction of an

n

×

n

magic square

with

n

being odd is simple. (The letter

x

will be used as a placeholder).

First, place the number 1 in the middle square of the top row.

[

x

x

1

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

]

The 2 goes in the bottommost row, in the column to the right of the one where we put the 1. Each subsequent number goes in the row above, in the column to the right, wrapping around as needed.

[

x

x

1

x

x

x

5

x

x

x

4

x

x

x

x

x

x

x

x

3

x

x

x

2

x

]

But if that spot already has a number, the next number goes in the square below instead.

[

17

x

1

8

15

x

5

7

14

16

4

6

13

x

x

10

12

x

x

3

11

x

x

2

9

]

And so on and so forth. Of course this procedure can be rotated.

Title

construction of magic square of odd length

Canonical name

ConstructionOfMagicSquareOfOddLength

Date of creation

2013-03-22 16:25:19

Last modified on

2013-03-22 16:25:19

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Algorithm

Classification

msc 05B15

Generated on Thu Feb 8 20:38:57 2018 by

LaTeXML
example of Latin squares

example of Latin squares

It is easily shown that the multiplication table (

Cayley-table

) of a group has exactly these properties and thus are

Latin squares

. However the converse is not true, ie. not all Latin squares are multiplication tables for a group (the smallest counter example is a Latin square of order 5).

Title

example of Latin squares

Canonical name

ExampleOfLatinSquares

Date of creation

2013-03-22 13:06:41

Last modified on

2013-03-22 13:06:41

Owner

jgade (861)

Last modified by

jgade (861)

Numerical id

6

Author

jgade (861)

Entry type

Example

Classification

msc 05B15

Generated on Thu Feb 8 20:38:59 2018 by

LaTeXML
Franklin magic square

Franklin magic square

One day in 1771,

Benjamin Franklin

, tired of hearing political debates, amused himself by creating the following

magic square

, now called a

Franklin magic square

:

[

52

61

4

13

20

29

36

45

14

3

62

51

46

35

30

19

53

60

5

12

21

28

37

44

11

6

59

54

43

38

27

22

55

58

7

10

23

26

39

42

9

8

57

56

41

40

25

24

50

63

2

15

18

31

34

47

16

1

64

49

48

33

32

17

]

The

magic constant

is 260. Furthermore, half any row or column (positions 1 to 4 or 5 to 8) equals half the magic constant. Two centuries later, Joseph Madachy realized that some half diagonals from the corner to the center also give 260.

Some other 8 by 8 magic squares with these properties are also called Franklin magic squares.

Title

Franklin magic square

Canonical name

FranklinMagicSquare

Date of creation

2013-03-22 16:24:52

Last modified on

2013-03-22 16:24:52

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 05B15

Classification

msc 01A50

Synonym

Franklin square

Generated on Thu Feb 8 20:39:01 2018 by

LaTeXML
Graeco-Latin squares

Graeco-Latin squares

Let

A

=

(

a

i

⁢

j

)

and

B

=

(

b

i

⁢

j

)

be two

n

×

n

matrices. We define their

as the matrix whose

(

i

,

j

)

th entry is the pair

(

a

i

⁢

j

,

b

i

⁢

j

)

.

A

Graeco-Latin square

is then the join of two

Latin squares

.

The name comes from Euler’s use of Greek and Latin letters to differentiate the entries on each array.

An example of Graeco-Latin square:

(

a

⁢

α

b

⁢

β

c

⁢

γ

d

⁢

δ

d

⁢

γ

c

⁢

δ

b

⁢

α

a

⁢

β

b

⁢

δ

a

⁢

γ

d

⁢

β

c

⁢

α

c

⁢

β

d

⁢

α

a

⁢

δ

b

⁢

γ

)

Title

Graeco-Latin squares

Canonical

name

GraecoLatinSquares

Date of creation

2013-03-22 12:14:36

Last modified on

2013-03-22 12:14:36

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

9

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 05B15

Defines

join

Generated on Thu Feb 8 20:39:03 2018 by

LaTeXML
Latin square

Latin square

A

Latin square

of

n

is an

n

×

n

array such that each column and each row are made with the same

n

symbols, using every one exactly once .

Examples.

(

a

b

c

d

c

d

a

b

d

c

b

a

b

a

d

c

)

(

1

2

3

4

4

3

2

1

2

1

4

3

3

4

1

2

)

Title

Latin square

Canonical

name

LatinSquare

Date of creation

2013-03-22 12:14:34

Last modified on

2013-03-22 12:14:34

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

7

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 05B15

Generated on Thu Feb 8 20:39:04 2018 by

LaTeXML
magic constant

magic constant

Given a

magic square

, magic cube, etc., the sum of any row, column or diagonal is called the

magic constant

of that magic square, cube, etc.

In the case of a standard

n

×

n

magic square that uses the integers from 1 to

n

2

, the magic constant is

1

n

⁢

∑

i

=

1

n

2

i

,

while that for a magic cube is

1

n

2

⁢

∑

i

=

1

n

3

i

.

We can then generalize to higher

dimensions

d

thus:

1

n

d

-

1

⁢

∑

i

=

1

n

d

i

.

So, for dimension

d

the magic constant is

n

⁢

(

n

d

+

1

)

2

. For instance, a

Franklin magic square

(

n

=

8

,

d

=

2

) has magic constant

8

⁢

(

8

2

+

1

)

2

=

260

.

In a trivial sense, an

n

×

n

sudoku puzzle has a magic constant of

n

2

.

Title

magic constant

Canonical

name

MagicConstant

Date of creation

2013-03-22 16:24:57

Last modified on

2013-03-22 16:24:57

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 05B15

Generated on Thu Feb 8 20:39:06 2018 by

LaTeXML
magic square

magic square

A

magic square

of order

n

is an

n

×

n

array using each one of the numbers

1

,

2

,

3

,

…

,

n

2

once and such that the sum of the numbers in each row, column or main diagonal is the same.

Example:

(

8

1

6

3

5

7

4

9

2

)

It’s easy to prove that the sum is always

1

2

⁢

n

⁢

(

n

2

+

1

)

. So in the example with

n

=

3

the sum is always

1

2

⁢

(

3

×

10

)

=

15

.

One way to generalize this concept is to allow any numbers in the entries, instead of

1

,

2

,

…

,

n

.

Title

magic square

Canonical

name

MagicSquare

Date of creation

2013-03-22 12:14:39

Last modified on

2013-03-22 12:14:39

Owner

drini (3)

Last modified by

drini (3)

Numerical id

6

Author

drini (3)

Entry type

Definition

Classification

msc 05B15

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proof that Hadamard matrix has order 1 or 2 or 4n

proof that Hadamard matrix has order 1 or 2 or 4n

Let

m

be the order of a

Hadamard matrix

. The matrix

[

1

]

shows that order 1
is possible, and the entry has a

2

×

2

Hadamard matrix
, so assume

m

>

2

.

We can assume that the first row of the matrix is all 1’s by multiplying
selected columns by

-

1

. Then permute columns as needed to arrive at a
matrix whose first three rows have the following form, where

P

denotes a

submatrix

of one row
and all 1’s and

N

denotes a submatrix of one row and all

-

1

’s.

x

y

z

w

[

P

⏞

P

⏞

P

⏞

P

⏞

P

P

N

N

P

N

P

N

]

Since the rows are

orthogonal

and there are

m

columns we have

{

x

+

y

+

z

+

w

=

m

x

+

y

-

z

-

w

=

0

x

-

y

+

z

-

w

=

0

x

-

y

-

z

+

w

=

0

.

Adding the 4 equations together we get

4

⁢

x

=

m

.

so that

m

must be divisible by 4.

Title

proof that Hadamard matrix has order 1 or 2 or 4n

Canonical name

ProofThatHadamardMatrixHasOrder1Or2Or4n

Date of creation

2013-03-22 16:50:56

Last modified on

2013-03-22 16:50:56

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

10

Author

Mathprof (13753)

Entry type

Proof

Classification

msc 05B20

Classification

msc 15-00

Generated on Thu Feb 8 20:39:10 2018 by

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criterion for a near-linear space being a linear space

criterion for a near-linear space being a linear space

Theorem

Suppose

𝒮

is

near-linear space

with

v

points and

b

lines, and

s

i

is the number of points in the

i

th line, for

i

=

1

,

…

,

b

. Then

∑

i

=

1

b

s

i

⁢

(

s

i

-

1

)

≤

v

⁢

(

v

-

1

)

,

and equality holds if and only if

𝒮

is a

linear space

.

Proof

Let

N

be the number of

ordered pairs

of points that are joined by a line. Clearly

N

can be no more than

v

⁢

(

v

-

1

)

, and

N

=

v

⁢

(

v

-

1

)

if and only if every pair of points are joined by a line. Since two points in a near-linear space are on at most one line, we can label each pair by the line to which the two points belong to. We thus have a

partition

of the

N

pairs into

b

groups, and each group is associated with a distinct line. The group corresponding to the line consisting of

s

i

points contributes

s

i

⁢

(

s

i

-

1

)

to the total sum. Therefore

∑

i

=

1

b

s

i

⁢

(

s

i

-

1

)

=

N

≤

v

⁢

(

v

-

1

)

.

Title

criterion for a near-linear space being a linear space

Canonical name

CriterionForANearlinearSpaceBeingALinearSpace

Date of creation

2013-03-22 14:32:47

Last modified on

2013-03-22 14:32:47

Owner

kshum (5987)

Last modified by

kshum (5987)

Numerical id

9

Author

kshum (5987)

Entry type

Theorem

Classification

msc 05B25

Generated on Thu Feb 8 20:39:13 2018 by

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tactical decomposition

tactical decomposition

Let

ℐ

be an

incidence structure

with point set

𝒫

and block set

ℬ

.
Let

X

𝒫

be a

partition

of

𝒫

into classes

𝒫

i

, and

X

ℬ

a partition
of

ℬ

into classes

ℬ

j

. Let

#

⁢

(

p

,

ℬ

j

)

denote for a moment the number
of blocks in class

ℬ

j

incident

with point

p

, and

#

⁢

(

b

,

𝒫

i

)

the
number of points in class

𝒫

i

incident with block

b

. Now the pair

(

X

𝒫

,

X

ℬ

)

is said to be

•

point-tactical

iff

#

⁢

(

p

,

ℬ

j

)

is for any

p

the
same for all

ℬ

j

, and is the same for all

p

within a
class

𝒫

i

,

•

block-tactical

iff

#

⁢

(

b

,

𝒫

i

)

is for any

b

the
same for all

𝒫

i

, and is the same for all

b

within a
class

ℬ

j

,

•

a

tactical decomposition

if both hold.

An incidence structure admitting a tactical decomposition with a single point class

𝒫

0

=

𝒫

is called

resolvable

and

X

ℬ

its

resolution

. Note

#

⁢

(

p

,

ℬ

j

)

is now a constant throughout. If the constant is 1 the
resolution is called a

parallelism

.

Example of point- and block-tactical: let

ℐ

be simple (it’s a

hypergraph

) and let

(

X

𝒫

,

X

ℬ

)

partition

𝒫

and

ℬ

into a single class
each. This is point-tactical for a regular hypergraph, and block-tactical for
a uniform hypergraph.

Example of parallelism: an

affine plane

(lines are the blocks, with

parallel

ones in the same class).

A natural example of a tactical decomposition is provided by the

automorphism
group

G

of

ℐ

. It induces a tactical decomposition with as point classes
the orbits of

G

acting on

𝒫

and as block classes the orbits of

G

acting
on

ℬ

.

Trivial example of a tactical decomposition: a partition into singleton
point and block classes.

The term tactical decomposition (

taktische Zerlegung

in German) was
introduced by Peter Dembowski.

Title

tactical decomposition

Canonical name

TacticalDecomposition

Date of creation

2013-03-22 15:11:02

Last modified on

2013-03-22 15:11:02

Owner

marijke (8873)

Last modified by

marijke (8873)

Numerical id

5

Author

marijke (8873)

Entry type

Definition

Classification

msc 05B25

Related topic

IncidenceStructures

Defines

point-tactical

Defines

block-tactical

Generated on Thu Feb 8 20:39:18 2018 by

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matroid independence axioms

matroid independence axioms

Hassler Whitney’s

definition

of a

matroid

was based upon the idea of

independent sets

and was given in terms of the following three axioms:

A

finite set

E

along with a

collection

ℐ

of subsets of

E

is a matroid if

M

=

(

E

,

ℐ

)

meets the following criteria:

(I1)

∅

∈

ℐ

;

(I2) If

I

1

∈

ℐ

and

I

2

⊆

I

1

, then

I

2

∈

ℐ

;

(I3) If

I

1

,

I

2

∈

ℐ

with

|

I

1

|

<

|

I

2

|

, then there is some

e

∈

I

2

such that

I

1

∪

{

e

}

∈

ℐ

.

The third axiom, (I3), is

equivalent

to the following alternative axiom:

(I3*) If

S

,

T

∈

ℐ

and

S

,

T

⊂

U

⊂

E

and

S

and

T

are both maximal subsets of

U

with the

property

that they are in

ℐ

, then

|

S

|

=

|

T

|

.

Proof.

Suppose (I3) holds, and that

S

and

T

are maximal

independent

subsets of

A

⊆

E

. Also assume, without loss of generality, that

|

S

|

<

|

T

|

. Then there is some

e

∈

T

such that

S

∪

{

e

}

∈

ℐ

, but

S

⊂

S

∪

{

e

}

⊆

A

, contradicting maximality of

S

.

Now suppose that (I3*) holds, and assume that

S

,

T

∈

ℐ

with

|

S

|

<

|

T

|

. Let

A

=

S

∪

T

. Then

S

cannot be maximal in

A

by (I3*), so there must be

elements

e

i

∈

A

such that

S

∪

{

e

i

}

∈

ℐ

is maximal, and by construction these

e

i

∈

T

. So (I3) holds.
∎

Title

matroid independence axioms

Canonical

name

MatroidIndependenceAxioms

Date of creation

2013-03-22 14:44:05

Last modified on

2013-03-22 14:44:05

Owner

sgraves (6614)

Last modified by

sgraves (6614)

Numerical id

6

Author

sgraves (6614)

Entry type

Proof

Classification

msc 05B35

Synonym

matroid independent sets

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