Acta Universitatis Danubius. Œconomica, Vol 13, No 3 (2017)

A Probability on Heuristic sub set of

Integer Numbers that is a Metric Space

Reza Farhadian¹

Abstract: In this work we will answer to the question, is there a probability space on a set such that it is a metric space? For answering to the question, we prove that the probability that two random positive integers given from a heuristic set defined in this article are relatively prime, is a metric space. Hence, there is a probability that it is a metric. Also, we show that this probability space is a paracompact space.

Keywords: probability; metric space; paracompact; random integers

JEL Classification: C002

1. Introduction

Let _{ }, _{ } and _{ } respectively be the sample space, collection of events and probability measure. The triple _{ } is called the probability space and measure _{ } satisfies the three Kolmogorov axioms (Gut, 2013); i.e.

1. For any _{ }, there exist a number _{ }, that this is the probability of _{ };

2. _{ };

3. _{ }, for every disjoint _{ }.

Now, we propose the question, is there a probability measure such that this measure be a meteric (the metric to be defined ahead)? In number theory, two integers _{ } and _{ } that they share no common positive factors except 1, are relatively prime. In 1970, S. W, Golomb [3] studied a class of probability distributions on the integer numbers , that under the his work, in 1972, J. E. Nymann proved that if _{ } be the Riemann _{ }-function, then probability that _{ } random positive integers, are relatively prime is equal to _{ } (see (Nymann, 1972). In this paper we will prove that the probability that two random positive integers are relatively prime, is a metric space. To prove the main theorem, the following notation will be used:

1. _{ }: greater common factor between tow integers x and y.

2. _{ }: two positive integers _{ } and _{ } , are relatively prime.

3. _{ }: probability that two random positive integers _{ } and _{ }, are relatively prime.

4. _{ }: real numbers.

5. _{ } : integer numbers.

2. The Main Definitions and Lemmas

In mathematics a metric space is defined as follows [O'searcoid, 2006, Definition 1.1.1]:

Definition 1 (metric space). Let _{ }be a set and _{ } is a real function. Then _{ } is a metric space for any _{ }, if the following conditions holds

1. _{ };

2. _{   };

3. _{ } (symmetry);

4. _{ } (triangle inequality).

Hence, the function _{ } is called a metric on the set _{ }. In mathematics, there are many functions defined on their sets that with together constitute the metric spaces. For a nice study about metric space, see (Dress et al, 2001). Other related spaces are paracompact spaces. A topological space is called paracompact if it satisfies the condition that every open cover has a locally finite open refinement (Adhikari, 2016). For definitions of open cover, locally finite and open refinement, see (Adhikari, 2016). In 1968, M. E. Rudin presented a nice proof (Rudin, 1969) for the statement that every metric spaces are paracompact.

Note that, in this work, we used of the above definition for random variables _{ }, that this means that if _{ } be a set and _{ }, then _{ } is a probability function. Hence, in this article we assume that _{ }are random integers greater than 1, and function _{ } is the probability that two random positive integers, are relatively prime. Thus, for example if _{ } (_{ }), we know that _{ } and _{ }. Now, if _{ }, then we assume that we don’t know any details about common factors between _{ } and _{ }, because _{ } and _{ } are random integers, and hence by (Nymann, 1972), we have _{ }. The details mentioned are the foundation of our work and in the next definition we suggest a new heuristic subset of positive integer numbers that has the above particulars.

Definition 2. Let _{ } be a sub set of _{ }. Then_{ } have the following conditions

1. If _{ }, then _{ }.

2. For every _{ }, we have _{ }, or _{ } and _{ } are relatively prime.

Henceforth, every set follow of definition 2, called is _{ }- set.

Example 1. Let _{ } and _{ }. Then, _{ }is a _{ }-set and _{ }is not a _{ }-set.

Example 2. Let _{ } be the prime numbers set. We know that for every _{ }, we have _{ } and _{ } and _{ } are relatively prime. Therefore, _{ } is an infinite countable _{ }- set.

Note that, if _{ } be a _{ }-set, then for every _{ } we have _{ } and _{ } or _{ } and _{ }. Now, if _{ } be the probability that two random positive integers _{ } given from an ideal _{ }-set, are relatively prime, then we have

_{ } .

To prove the main theorem we need the following lemmas:

Lemma 1. Let _{ } be a ideal _{ }-set. If _{ } be tow random positive integer, Then _{ } if and only if _{ }.

Proof. We know that if _{ } then _{ }, for every positive integer _{ } and hence _{ }. On the other hand, since M is an _{ }-set and _{ }, then if _{ }, we have _{ }.

Lemma 2. Let _{ }, _{ }be tow random positive integers given from a ideal_{ }-set, Then

_{ }.

Proof. We know that _{ }, and clearly proof is complete.

Lemma 3. Let _{ }, _{ }, _{ } be three random positive integers given from a ideal_{ }-set. Then

_{ }.

Proof. We know that _{ }, _{ }, _{ } are three random positive integers given from a ideal_{ }-set. Hence, if _{ }, then _{ }, and also if _{ }then we have _{ }. So, for all cases, we have

• Case 1:

_{ }

• Case 2:

_{ }

• Case 3:

_{ }

• Case 4:

_{ }

• Case 5:

_{ }

Hence, for every random positive integers _{ } given from a _{ }-set, always we have

_{ }.

3. The Main Theorem

Theorem 1. Let _{ } be a ideal _{ }-set and_{ }. If _{ }, then _{ } is a metric space.

Proof. By the metric space definition, and using the Lemmas 1 ( for condition 2) and the Lemma 2 (for condition 3) and the Lemma 3 (for condition 4), and since always _{ } (for condition 1), the proof is complete.

Hence, we can answer to the question, is there a probability such that it is a metric?

Corollary 1. There is at least a probability such that it is a metric.

Since, every metric spaces are paracompact space (Rudin, 1969), so we have the following corollary:

Corollary 2. There is at least a probability space such that it is paracompact space.

Acknowledgements

The author wishes to thank the editor and anonymous referee for their helpful comments.

4. References

Adhikari, M.R. (2016). Basic Algebric Topology and Applications. Springer India.

Dress, A. & Huber. K.T. & Moulton, V. (2001). Metric Spaces in Pure and Applied Mathematics. Documenta Math. Quadratic Forms LSU (2001), 121-139.

Golomb, S.W. (1970). A Class of Probability Distributions on the Integer. Journal of number theory 2(1970), 189-192.

Gut, A. (2013). Probability: A Graduate course. Second Edition. London: Springer New Heidelberg Dordrecht.

Nymann, J.E. (1972). On the probability that k positive integers are Relatively prime. Journal of number theory 4(1972), 469-473.

O'searcoid, M. (2006). Metric space. Springer Science & Business Media.

Rudin, M. E. (1969). A new proof that metric spaces are paracompact. Proc. Amer. Math. Soc.

Refbacks

• There are currently no refbacks.