^{1}

^{2}

^{*}

^{3}

In the present paper, geometry of the Boolean space
B^{n} in terms of Hausdorff distances between subsets and subset sums is investigated. The main results are the algebraic and analytical expressions for representing of classical figures in
B^{n} and the functions of distances between them. In particular, equations in sets are considered and their interpretations in combinatory terms are given.

Let

It is clear that

The Hausdorff distance has essential role in many problems of discrete analysis [

First, we present the following simple properties of the Hausdorff distance:

1)

2)

3) Если

4)

Let us note that, generally speaking, the Hausdorff distance does not satisfy the triangle inequality:

which is demonstrated in the following picture:

But inequality (1) holds true if

^{n}

Let

Thus, we have the following two equvalent interpretations for

1)

2)

Examples.

1)

2)

3) If

Theorem 1.

Proof. We consider two cases.

a)

and, consequently,

b)

From here we have:

As

Then, taking into account that:

we have:

The theorem is proved.

Let:

Taking into account that the sphere of the radius p with the center at

we get the following corollary.

Corrollary. If

The value of the function

Theorem 2. If

The general form of the standard generating function for the distance between the subsets

The summation in (2) is over all pairs of the subsets

Let us consider a few examples.

1)

In this case, we have:

Thus,

2)

In this case:

As:

for arbitrary points

Then:

Consequently, the distance between the zero point and an arbitrary subset Y equals the minimal weight of the points which are in Y.

Hence,

where

where

From (5), we get the following statement:

Lemma1. If

For

Theorem 3. The following formula holds true:

Proof. By definition:

From this and Lemma 1, taking into account (3) and (4), we get:

The theorem is proved.

If

Corollary 1. The following formula holds true:

Corollary 2. The following holds true:

Corollary 3. The formula for

Proof. From (6) we get for p = 1:

Corollary 4. For q = 2, the following formula holds true:

Proof. By definition and from Corollary 2, we derive:

Transforming the terms in (7), we get:

Then, using the following formulas:

Let us “compress” the sum:

By definition, we have:

Furthermore, if:

then:

Further:

And:

From here it follows that

Then:

Taking this and (8) into account, we get:

And the generating function:

can be expressed by the following parameters:

2) if

Hence, the set

The cardinality of this family is:

2) The number of all m-element subsets having the distance r from M is:

Summarizing all the previous, we get the following statement.

Theorem 4. The following expression is true:

Let

The operation “+” is defined in the family

Besides, the following inequality holds true:

Here both limits are reachable.

The properties of “+” are as follows:

1)

2)

3)

4)

5)

6)

7)

Examples.

If X is a subspace in

1)

2)

Let the following holds true:

Then

Thus, there is certain analogy between the norm of the sum of points and the distance between those points, as well as between the norm of the sum of the sets and the distance between those sets.

In the general form, the following statement connecting the operations “∪” и+, is true:

A facet or interval in

Every interval J can be written in the form of a word of the length n, in the alphabet

Examples.

If

If the operation “*” is introduced on the alphabet A:

then the sum

Examples.

1) If

2) If

The distance between the intervals

Let

Statement 1.

Thus, the distance between the intervals

Examples.

1) Let

Then

Let

Let us consider the direct product

Theorem 5. The following formula is true:

where

Let us consider the matrix

Lemma 2. The following expression is true:

where

Proof. According to the definition:

where

and

It follows from (9):

The internal sum in (10) equals the number of such pairs

Example.

1) Let

The total number of the edges in

all letters of the alphabet

From here, we get:

2) In the general form, if

Thus, the sum of the pairwise distances between the intervals in

In the general form, the following statement holds true.

Lemma 3. The following formula is true:

Proof. By definition:

Thus, the above introduced parameter

Lemma4.

Proof. If

Then, from

From here we get:

or:

Hence,

And if

Hence,

Theorem 6. The following expression is true:

Proof. We have from Lemma 4:

Then, we have from Lemma 3:

And the proof is over.

Formula (12) defines the rule of “addition” for arbitrary spheres in the space

Let

According to definition,

Then, the following statement is true.

Lemma 5. The following expression holds true:

Proof. First, let us note the following.

If

In standard terms, the symmetric group

If

Taking this into account, to describe the set

We discuss the following outline:

Here

In the general case, the situation is absolutely analogous, and the weights are arranged as follows:

for

Here the condition for z holds true:

Examples.

1)

2)

3)

Theorem 7. The following expression is true:

where

Proof. From Lemmas 3 and 5, we have:

The proof is over.

As usual, let

Statement 2.

and:

Statement 3. Let

And if:

then the following equality is true:

Proof. We assume the contrary, that is,

Then, we have:

Hence,

This contradicts the initial condition and the proof is over.

The following example shows that condition (12) is not necessary.

Example.

Let

The “simplest” equation by sets is the following:

where

Equation (14) always has the trivial solution:

The significance of Equation (14) is explained by the following circumstances.

1) The standard problems of covering and partitioning in the Boolean space B^{n} [

2) For certain additional conditions, the solution of Equation (14) forms a perfect pair (perfect code) in the additive channel of communication [

3) The set of all solutions of Equation (14) coincides with the class of equivalence of the additive channel of communication [

Examples.

1) If

2) If A is a subspace of

The following statements are true:

Statement 4. If the equations

Proof. Let the pairs

Statement 5. For

has the solution

Statement 6. For

where

Statement 7. For

Statement 8. The sets of solutions

Below, when discussing Equation (14), without violating generality, we may assume

Statement 9. The equation

Proof. If

From here it follows that the equation

Statement 10. The equation

Let

holds iff:

then the following statement is true.

Statement 11. If

Statement 12. If A is a subspace from

tion

In an additive channel of communication [

Let

We introduce the following parameters:

Such definition of

One can easily prove that:

Statement 13. For the subspace

Proof. It follows from (15) that it is sufficient to prove that the following equality is true:

Let

On the other hand, it follows from Statement 11 that

Theorem 8. The following estimations are true:

1)

2)

Proof. We have:

From this and definition of addition of sets we get:

Consequently:

To prove the 2nd estimation, we consider such subspaces

Let:

where

Let us prove that

We have:

As (Statement 12)

Then, using:

we get:

Hence, taking this and Statement 13 into account we get:

The statement is proved.

Examples.

1)

2)

but actually:

3)

We consider the set:

We have:

4)

We have:

But actually

Suggestion. For each

Vladimir Leontiev,Garib Movsisyan,Zhirayr Margaryan, (2016) Algebra and Geometry of Sets in Boolean Space. Open Journal of Discrete Mathematics,06,25-40. doi: 10.4236/ojdm.2016.62004