Explanation of the Christian Greek Orthodox approach and its variables using set theory, general topology, group theory, and boolean algebra.

Dr Michel Zaki Guirguis 19/09/2014Bournemouth University1

Institute of Business and LawFern BarrowPoole, BH12 5BB, UKTel:0030-210-9841550Mobile:0030-6982044429Email: guirguismichel@yahoo.co.uk

Biographical notes

I hold a PhD in Finance from Bournemouth University in the U.K. I have worked for several multinational companies including JP Morgan Chase and Interamerican Insurance and Investment Company in Greece. Through seminars, I learned how to manage and select the right mutual funds according to various clients needs. I supported and assisted the team in terms of six-sigma project and accounts reconciliation. Application of six-sigma project in JP Morgan Chase in terms of statistical analysis is important to improve the efficiency of the department. Professor Philip Hardwick and I have published a chapter in a book entitled “International Insurance and Financial Markets: Global Dynamics and Local Contingencies”, edited by Cummins and Venard at Wharton Business School (University of Pennsylvania in the US). I am working on several papers that focus on the Financial Services Sector.

1 I have left from Bournemouth University since 2006. The permanent address of the author’s is, 94, Terpsichoris road, Palaio – Faliro, Post Code: 17562, Athens – Greece.

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Abstract

In this article, we are going to use the different theorems to explain the Christian Greek Orthodox approach and its variables using set theory, general topology, group theory, and Boolean algebra. Under the Greek Orthodox approach, the set is the Greek Orthodox religion. We denote the elements as x. The elements are strong faith towards the Holy Trinity, Virgin Mary, repentance, confession, Holy Communion, prayers, fast, prostrations, vigil, almsgiving, rejection of personal will, appreciate of Byzantine music, humility and strong belief that Jesus – Christ was resurrected. By following the elements of the set theory the spiritual factors will follow an increasing function. In contrast, by following the elements that are not part of the set theory, the spiritual integration of the individual follows a decreasing function. The elements that do not constitute the set are mortal sins, pardonable sins, sins of omission, devil, Satan and demons. The point is to eliminate those elements through repentance, confession and Holy Communion and most importantly with the manipulation of thoughts under the supervision of the Greek Orthodox spiritual father. The set topology is used to illustrate the properties of the metric and normed spaces, the separation axioms, the continuous functions, compactness and connectedness sets. The Orthodox approach will be interrelated with Urysohn’s lemma and metrization theorem. In addition, we will explain how the Orthodox variables converge to the function spaces in terms of uniform and compact convergence. Under the group theory, permutations or combinations will be used in relation to a set to show how the Orthodox variables are combined with the equations of open economy macroeconomics. The point is to explain why the Orthodox variables have been chosen to achieve symmetry in group theory. The Orthodox theory will be combined with homomorphism, isomorphism and Cayley’s theorem to prove the importance of the repetition of the Orthodox variables to achieve maximum spiritual integrity. We will discuss about other groups such as groupoids. Finally, Boolean algebra will be used within the algebra set theory and binary digits to define and form abstract algebra to intersect and complement the set of the Orthodox variables in the business area. Through practical examples from the prostitute industry, the gambling industry and the war industry we will show the deviations and the spiritual falls that arise from excluding elements from the subsets and insisting in the personal will and the ancestral sin.

Keywords: Greek Orthodox church, spiritual father, manipulation of thoughts, Holy Trinity, Virgin Mary, Saints, set theory, general topology, group theory and boolean algebra.

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Introduction

Set theory is very important branch of mathematics, as it groups different objects or elements that constitute a set and consider them as one entity. The set theory is formulated and the logic that constitutes the elements and actions that are performed is union, intersection and complement. The set is denoted by a capital letter and the elements that constitute the set are included in braces, denoted by lower-case letters and separated by commas. The type of numbers that are used in the set are nonnegative integers, positive integers, set of integers, set of rational numbers, set of real numbers and set of complex numbers. In our case of the Orthodox approach, we will use nonnegative and positive integers to reveal the association of the strength in each Orthodox variable. For simplicity two subsets are used to show that each elements of C is also an element of D. Thus, C is a subset of D. . The first subset C is the Orthodox individual and the second subset D is the spiritual father. Initially, each subset will have the same elements namely strong faith towards the Holy Trinity, Virgin Mary, repentance, confession, Holy Communion, prayers, fast, prostrations, vigil, almsgiving, rejection of personal will, appreciate of Byzantine music, humility and strong belief that Jesus – Christ was resurrected. Then, through practical examples from the prostitute industry, the gambling industry and the war industry we will show the deviations and the spiritual falls that arise from excluding elements from the subsets and insisting in the personal will and the ancestral sin. We will illustrate this relationship through a Venn diagram, set operations in terms of union and intersections. Then, we will illustrate the Orthodox variables through functions by assigning the set C as the domain and the set D as the target set. In addition, we will illustrate the image and the diagram of the function. The Orthodox variables will be illustrated through a sequence and a fundamental product of a set. Moreover, we will use logical expressions in terms of conjunction, disjunction and negation, which represent the words and, or and not. These logical expressions will be used to determine if they are true or false.

The set topology is used as continuation of the set theory to show the interrelation of the Orthodox approach in relation to the different theorems. Initially, we will use the Heine – Borel theorem of closed and bounded interval to show the validity of the model in relation to the Orthodox variables. Then, we will use topologies induced by continuous functions to show the spiritual falls and the deviations from the initial elements that constitute the set, which is the Greek Christian Orthodox religion. Then, an introduction to metric space will be given in order to better understand the separation axiom. Metric space is a set that is composed from elements that the distance between them is well defined. All the distances together are named a metric of the set. On the other hand, separation axioms are restrictions made to metric spaces. For example, in Venn diagram we could assign open or closed disjoint spaces. We will mention Hausdorff spaces, the distribution of open sets and whether a space is separable or second countable. Then, our effort is to address the metrization problem in Topology and interrelate it with the Greek Orthodox approach. The Urysohn’s lemma theorem was an effort to solve this puzzle by trying to find a metric d on X, which induces the Topology T space and has a countable base. The problem will be addressed within open and closed disjoint spaces. In addition, we will explain how the Orthodox variables converge to the function spaces in terms of uniform and compact convergence. Compactness is based upon the Heine – Borel theorem in terms

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that every open cover of a closed and bounded interval is reducible to a finite cover. In addition, connectedness between the Orthodox variables exists if the two subsets of a topological space X are not disjoint non-empty open subsets whose union is the original subset.

Under the group theory, permutations or combinations will be used in relation to a set to show how the Orthodox variables are combined with the equations of open economy macroeconomics. The point is to explain why the Orthodox variables have been chosen to achieve symmetry in group theory. Mapping with the same domain will be used to show the direction of the Orthodox variables. The Orthodox theory will be combined with homomorphism, isomorphism and Cayley’s theorem to prove the importance of the repetition of the Orthodox variables to achieve maximum spiritual integrity. We are interested on the bijective function of Arthur Cayley theorem between the elements of two sets, namely the Orthodox individual and the spiritual father. We want to spot the deviation and the spiritual falls based on the paired and unpaired elements between the two sets by rearranging the elements of a set into particular sequence. We will discuss about other groups such groupoids. The purpose of groupoids is to combine a set of integers Orthodox variables with positive and nonnegative signs of a pair, (G,μ) and a binary operation, μ in G. The purpose of binary operation is to assign only two values to the Orthodox variables according to their spiritual direction. The pair of (G,μ) that consists of a non-empty set G is known as the carrier of the groupoid. The purpose of using this repetitive task is to test the spiritual integrity of the Christian Orthodox individual. Then, we want to check if two groupoids have the property of isomorphism or homomorphism in terms of similarity.

Finally, Boolean algebra will be used within the algebra set theory and binary digits of groupoids to define and form abstract algebra to intersect and complement the set of the Orthodox variables in the business area. The point is to find if there are similarities in terms of isomorphism between the subsets of the Orthodox prayer and the spiritual father. In addition, we are looking for duality in terms of symmetry by interchanging the identity of the elements of the Orthodox variables. Thus, we are trying to find true conditions based on logic to eliminate the wicked thoughts and improve the manipulation of thoughts.

The rest of the paper is organized as follows. Section 1 describes the building blocks of the Greek Orthodox Church. Section 2 describes the successful manipulation of thoughts. Section 3 describes the set theory. Section 4 describes the set topology. Section 5 describes the group theory. Section 6 describes the principles of the Orthodox approach in relation to Boolean algebra. Section 7 summarizes and concludes.

1. Building blocks of the Greek Orthodox Church

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The building blocks that constitute and sustain the faith of the Greek Orthodox Christian is the belief in the Holy Trinity, Holy communion, the powerful presence of Theotokos or Virgin Mary, the fleet of Saints in terms of hermits, ascetics, spiritual sacrifice, the David’s Psalms, Holy Bible, Byzantine music, fasting, the cross and the resurrection of Jesus - Christ.

The Holy Trinity is composed from the father, the son and the Holy Spirit. The unity in one person is not static but dynamic interrelation of the Holy Spirit under the roof of the grace of the son and the love of the father. The unity of the Holy Trinity supports you with harmony, love and peace of yourself, primarily, and then with the fellow - citizens and the environment.

The Holy Communion is an essential part of the Liturgy taken inside the church every Sunday and during celebration of Saints days. Repentance and confession, which constitute a second baptism, should take place before the Holy Communion. It helps the individual’s to strengthen his/her faith to the Holy Trinity and to protect him/her from the devil. If the individual commune the Divine Mysteries without the appropriate preparation that will be mentioned from the spiritual father then the sins become larger and worse.

Theotokos or Virgin Mary, the mother of Jesus-Christ, occupies an essential place in the heart of the Greek Orthodox Church. It symbolizes the second Eve and selected from God to continue the salvation of the mankind through her son, who represents the new Adam. She protects the Christian Orthodox from the various types of demons and encourages him/her during difficult times with patience and faith. For a more detailed analysis of the various types of demons, please refer to the book of Saint John Sinaet (2006), named the Ladder. Unfortunately, the ancestral sin of Adam has destroyed the divine relationship and created disequilibrium in balancing the evil from the good thoughts. This condition or tendency of the heart towards sin remains even after the baptism has removed the ancestral sin. The mystery of Chrism, which follows the mystery of baptism, seals the newborn with the grace of the Holy Spirit. Thus, spiritual sacrifice with the grace of the Holy Spirit is the next step to stabilize the spiritual effort. It helps the person to reconcile and sustain a healthy mind through the examples of hermits, ascetics and Ηoly Fathers of the Greek Orthodox Church.

Spiritual sacrifice is defined in terms of rejecting personal will and following the will of God through the Ten Commandments. Fear, defined in a positive manner, not to break the commandments and keep an active ascetic profile is a substantial factor that keeps alive the faith towards God. At this point, it is worth to distinguish between obedience and pure discipline. Obedience created and based on love to approach God has complete different interpretation and result from the pure discipline that is cultivated and developed in other institutions. Obedience inside the Greek Orthodox Church has a positive and healthy characteristic. It is based on the freedom to approach the Holy Trinity through the applications of the Ten Commandments and continuous prays from the individual’s and the spiritual father. Pure discipline without religious characteristic has negative effects, as most of the time, is based on unbalancing power and imposition of force in a negative way to dominate on the situation. A good example, to mention is the Holy monk Porfirios that teaches us to pray for our children and fellow - colleagues when we face a problem with them. The results will be much better by approaching the Theotokos rather than by believing or

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depending solely on our capabilities. Obedience will help the individual to achieve various stages of spiritual successes. In addition, the individual’s need to learn how to recognize, manipulate the thoughts by rejecting the bad and accepting the good ones. This can only be achieved through prays to the Holy Trinity, Virgin Mary and the help that you get from the spiritual father.

The David’s Psalms contain different types of hymn to God. For example, Psalm 50 is a prayer for forgiveness and Psalm 136 is a hymn of thanksgiving. The Holy Bible comprised from the Old and New Testament is an essential core of the base of the Greek Orthodox Church. They display the main event of prophets, Jesus-Christ, his mother and the apostles.

Byzantine music constitutes a fundamental part of the Greek Orthodox Church. The Divine Liturgy, Vespers, the hymns of David’s Psalms, The lamentations, the small and great canons of the Theotokos and the service of the supplicatory canons to Archangels and Saints are based on Byzantine music. The requirements to protect one of our sense gates, such as hear, favour this type of music. It is essential to protect us from the evil thoughts, as we will see in the later section of the manipulation of thoughts.

Fasting is an integrated and essential part of the life of the Greek Orthodox Christian. During the ecclesiastical year, there are various periods of fasts. The main fasts are Wednesday’s and Friday’s fast, the fast before the birth of Jesus-Christ, the fast of Great Lent, the fast during the month of August, which is concerned with Virgin Mary, the elevation of the cross. All fasts and especially the fast of Great Lent, which starts with the Publican and the Pharisee Gospel passage, help the individuals to strengthen their faith and spiritual effort through continuous repentances and prostrations. Fast should be avoided if the person faces bodily illness. To fast or not has to be discussed and solved with the help of the spiritual father.

The cross and resurrection constitute the base of the win of Jesus-Christ from death. They are based on the belief that your spiritual effort in this life is a continuous condition that will count after your death, and will have an eternal characteristic. Thus, after your death, you can not change anything in terms of your actions, good or bad, that were reflected in your attitude, behaviour and saying that were seen or unseen. Death is just a temporary change of status until the judgment day. Thus, if the spiritual effort and less the physical is based on the commandments, sincere and fair progress throughout the life of person, under the supervision of the spiritual father, then, there are increased chances to safeguard the purity of the soul. The fact that, the author’s use the expression increased chances is because the life of the individual’s until his/her death is a continuous effort to safeguard the will of God with ultimate objective to save his/her soul. Thus, there are many spiritual falls and illnesses of the soul that could be recovered through the mystery of repentance followed by confession.

On the other hand, if his/her effort was not based and blessed under the Holy Trinity and no confession took place in the whole phase of the life of the person, then, the reward will be eternal hell. By the term eternal hell, I mean the outer darkness, the unquenchable fire, the unsleeping worm and the gnashing of teeth followed by innumerable torments. Τhe remembrance of the term eternal hell helps tremendously

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every individual to obey better to the Ten Commandments and avoid a sinful lifestyle. The fact that the individual’s has compromised to stay in a sinful condition leads him/her to eternal hell. Thus, as God is love, it will be unfair to change the status or habits of the person and adapt him/her to new reality, namely, in the upper Jerusalem with the Saints. An excellent source of reference is the Sacred Augustine known as ‘the son of tears’. In the book written by Mperati (2005), there is a clear illustration of how poor decisions away from God can destroy the personality of the individual’s and how repentance and confession could save his/her soul.

2. The manipulation of thoughts

Thoughts constitute the invisible part of the mind of the person that is liable to the continuous flow of thousands of decisions that affect his/her life, the life of others and the results of his/her actions in the society. The organs constitute the human body and work in perfect harmony to sustain life. In a similar way, thoughts have to be regulated, structured and manipulated with the functioning of the heart to achieve equilibrium in terms of faith to the Holy Trinity. The problem is that after the ancestral sin, the mind has lost the initial pure functionality, which was to praise God and get divided into good, vain and evil thoughts. The biggest problem that we are facing, nowadays, is how to distinguish between them, balance them and isolate the evil thoughts that create substantial damage in our life in terms of the decisions that we take in a daily basis.

The golden rule, to firstly start this discussion, is that without the presence of God in terms of prays and guidance of the spiritual father under the Greek Orthodox Church, we can not trust and be totally sure that what we are thinking or saying is correct or good. The devil and the various demons could affect the will and ambition of the individual’s and captures him/her under a spirit of conceit or haughtiness. Lack of humility and the mixture of thoughts according to daily problems and circumstances create anxiety and despair, which should be avoided by continuous prays to Virgin Mary. The ultimate objective is our soul salvation. By setting our actions, as an example, we contribute to the salvation of the persons that are surrounding us irrelevant from their age, sex, religious background, nationality and literacy level.

Vain thoughts are created from the relaxation of the mind that God is not an active observer in our lives, hearts and thoughts. The lack of faith and deviation from the basic necessities of life reinforce vanity thoughts at the expense of the peace of the soul. The successors of vain thoughts are evil thoughts that start to erode the building blocks of the soul until the final destruction of the healthy functionality of the mind and the corruption of heart with evil thoughts.

Evil thoughts are derived from external and internal causes. The external causes are related to five senses such as sight, hear, smell, taste, and touch. Each sense has to be carefully monitored and protected through holy gates that are installed to each one according to his/her spiritual effort to keep and implement precisely the will of God. Thus, according to the degree of your effort you get blessed and better protected from indecent sights, obscene words and irritated perfumes. In addition, the person manages to control excess and beyond necessity food and drink and improper touches of his/her body with soft materials. They result to relaxations of the body and creations of improper fantasies and sinful thoughts.

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The internal causes of evil thoughts are related to imagination, passions, demons, and the already corrupted nature created after that the ancestral sin took place. Let’s analyze each cause. Imagination, which is fed mainly by sight and hears, affects the way by which we perceive, understand and structure the images around us. It also affects our future prospect, consumption habits, and ambitions. The stimulus that affect our senses and create an internal picture of the external world has to be based on religious background in terms of fast, prays, and consultation with the spiritual father in a continuous basis. If the stimulus is wrong, then they are going to create a malfunction of the mind and erosion of the soul with many negative effects. As a result of the acceptance of bad thoughts, we can mention minimum efficiency, spiritual pollution and therefore physical pollution of the environment.

Passions created outside the Greek Orthodox Church creates negative externalities with high costs as there is no balance between the various emotions such as love, hate, pleasure and pain. The faith inside the Orthodox Church is to love God with all your heart, soul, power and intellect part of your mind. This is the first commandment. The second commandment is to love your neighbour as you love yourself. The face of the fellow-citizen reflects the Holy Trinity. The first two commandments are very important to understand that the person is image or picture of God with ultimate objective to achieve resemblance. Thus, the various emotions are eliminated or better controlled as they are managed in line with the Commandments. Once the individual manage to get blessing from the application of the first two Commandments, then, he/she will progress towards the application of the remaining eight Commandments. Finally, the individual’s will get help to balance the various emotions with the help of the spiritual father and the examples that the Holy Fathers of the Greek Orthodox Church.

Demons analyzed in an excellent way in the book of Saint John Sinaet (2006). This book is a masterpiece that teaches you the art to defend yourself from evil thoughts. He teaches you how to stop the various attacks and get out from the various snares that the demons are trying to set up around the individual’s heart. They are trying to steal the treasure from the inner part of the roots of the person heart with ultimate objective to lead and sustain him/her in a sinful situation until his/her death. Τhey try to take advantage from the fact that passions are recrudescent. Thus, they are trying to guarantee that the individual’s will get eternal hell rather than salvation of his/her soul.

Finally, it is the already corrupted human nature after the fall from paradise and recorded as the ancestral sin. The purity of the soul, intellect and innocent understanding of life has been eroded and shacked substantially. The evil thoughts started to occupy a predetermined place in the mind, heart and soul of various persons. The lack of proper faith, in terms of a Greek Orthodox Approach, that is trying to find an equilibrium point to reconcile the damage, has resulted to even worse results. People are loosing their identity in a daily basis. Globalization of different cultures, ideas and nationalities without an integrated roof of clear guidelines of a viable religious approach has created further complexity. The success of decisions making in terms of economic, social, and religious outcomes will have to be based upon our interaction with the Orthodox Church primarily. For a detailed analysis of

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the manipulation of thoughts and senses please refer to Saint Nikodemos the Hagiorite (1819).

Section 3 describes the set theory

Under the Greek Orthodox approach, the set is the Greek Orthodox religion. We denote the elements as x. The elements are strong faith towards the Holy Trinity, Virgin Mary, repentance, confession, Holy Communion, prayers, fast, prostrations, vigil, almsgiving, rejection of personal will, appreciate of Byzantine music, humility, and strong believe that Jesus - Christ was resurrected. We will use nonnegative and positive integers to reveal the association of the strength in each Orthodox variable. For simplicity two subsets are used to show that each elements of C is also an element of D. Thus, C is a subset of D and D is a subset of C. . The first subset C is the Orthodox individual and the second subset D is the spiritual father. The set is the Greek Orthodox religion. Initially, each subset will have the same elements, namely, strong faith towards the Holy Trinity, Virgin Mary, repentance, confession, Holy Communion, prayers, fast, prostrations, vigil, almsgiving, rejection of personal will, appreciate of Byzantine music, humility, and strong believe that Jesus - Christ was resurrected. Then, through practical examples from the prostitute industry, the gambling industry and the war industry we will show the deviations and the spiritual falls that arise from excluding elements from the subsets and insisting in the personal will and the ancestral sin.

After listing the elements, we could write in a mathematical format the set.

S = { Strong faith towards the Holy Trinity, Virgin Mary, repentance, confession, Holy Communion, prayers, fast, prostrations, vigil, almsgiving, rejection of personal will, appreciate Byzantine music, humility, and strong believe that Jesus - Christ was resurrected}

Where x is strong faith towards the Holy Trinity, Virgin Mary, repentance, confession, Holy Communion, prayers, fast, prostrations, vigil, almsgiving, rejection of personal will, appreciate of Byzantine music, humility, and strong believe that Jesus - Christ was resurrected.

Thus,

x Є S which indicates that x is an element of S.

x Є S which indicates that x is not an element of S. Thus, the elements that do not constitute the set are mortal sins, pardonable sins, sins of omission, devil, Satan and demons. The point is to eliminate those elements through repentance, confession and Holy Communion and most importantly with the manipulation of thoughts under the supervision of the Greek Orthodox spiritual father.

By following the elements of the set theory the spiritual factors will follow an increasing function. In contrast, by following the elements that are not part of set theory, the spiritual integration of the individual follows a decreasing function.

Thus, the positive integers that describe each variable have been listed as follows:

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Strong faith towards the Holy Trinityvery weak =0; weak=1; medium=2.5; strong=4; verystrong=5;Strong faith towards the holy trinity=verystrong – veryweak

Strong faith towards Virgin Maryvery weak =0; weak=1; medium=2.5; strong=4; verystrong=5;Strong faith towards Virgin Mary=verystrong – veryweak

Repentancevery weak =0; weak=1; medium=2.5; strong=4; verystrong=5;Repentance= verystrong - veryweak

Confessionvery weak =0; weak=1; medium=2.5; strong=4; verystrong=5;Confession= verystrong - veryweak

HolyCommunionvery weak =0; weak=1; medium=2.5; strong=4; verystrong=5;Holy communion= verystrong - veryweak

Prayervery weak =0; weak=1; medium=2.5; strong=4; verystrong=5;Prayer= verystrong - veryweak

Fastveryweak =0; weak=1; medium=2.5; strong=4; verystrong=5;Fast= verystrong - veryweak

Prostrationsveryweak =0; weak=1; medium=2.5; strong=4; verystrong=5;Prostrations= verystrong - veryweak

Vigilveryweak =0; weak=1; medium=2.5; strong=4; verystrong=5;Vigil= verystrong - veryweak

Almsgivingveryweak =0; weak=1; medium=2.5; strong=4; verystrong=5;Almsgiving= verystrong - veryweak

Rejection of personal willveryweak =0; weak=1; medium=2.5; strong=4; verystrong=5;Rejection of personal will= verystrong - veryweak

Appreciate Byzantine musicveryweak =0; weak=1; medium=2.5; strong=4; verystrong=5;Appreciate byzantine music= verystrong – veryweak

Humility

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veryweak =0; weak=1; medium=2.5; strong=4; verystrong=5;Humility= verystrong – veryweak

Strong believe that Jesus - Christ was resurrectedveryweak =0; weak=1; medium=2.5; strong=4; verystrong=5;Strong believe that Jesus Christ was resurrected= verystrong - veryweak

For simplicity two subsets are used to show that each elements of C is also an element of D. Thus, C is a subset of D and D is a subset of C. The nonnegative and positive integers of the subset C of the Orthodox individual are C = {0,1,2.5,4,5} and the values of the spiritual father are D = {0,1,2.5,4,5}. The mathematical notation is

. A very simple example is that by excluding in the subset the repentance, confession, fast, almsgiving and Holy Communion, then, the individual get disconnected from the spiritual father and starts to cultivate the bad habits and adopt a sinful lifestyle plenty of self-interest and greed.

The prostitute industry, the gambling industry and the war industry face the same spiritual problems from excluding fundamental elements from the subset and insisting in the personal will and the ancestral sin. For example, the soldier depends on dirty strategic plans based on weapons and killing. The female prostitute depends on her body and her capability to provide professional sex. The gambler put all his strength to increase fun by trying to win the black jack and the roulette in the Casino of Las Vegas or Macao. The gambler ignores the law of large number in probability theory that his/her chances to win are very limited. In other words, the gambler loose huge amounts of money and win very few and most of the times he/she records only losses and psychological disorders. In addition, he / she ignores that the money could be used to help the poors, the beggars and the prostitutes. They ignore the great mercy and the inner peace that is provided from the Holy Trinity and from the help of the Virgin Mary by following in a continuous and disciplined way the variables that constitute the subsets. Repentance should be followed by a prepared confession in an Orthodox Church or monastery and with the continuous prayers of the Orthodox individual and the spiritual father the spiritual improvement will start to appear.

Only the Christian servants inside the Orthodox have more chances and opportunity to achieve and maximize spiritual integrity. For example, by excluding the strong belief that Jesus Christ was resurrected, they ignore the commandments and the fundamental concepts of the Gospel, which is based on, divine love. The faith inside the Orthodox Church is to love God with all your heart, soul, and power and intellect part of your mind. This is the first commandment. The second commandment is to love your neighbour as you love yourself. The face of the fellow-citizen reflects the Holy Trinity. The first two commandments are very important to understand that the person is image or picture of God with ultimate objective to achieve resemblance. If the gambler, the prostitute and the soldier were honest with themselves and follow strictly the commandments of the Gospel, then, they have more chances to protect their body and achieve spiritual integrity through the help of the Holy Trinity, Virgin Mary, the spiritual father, the Saints and the archangel Michael and Gabriel. In addition to their effort, the government has to make sure that it provides the necessary financial support for their education and employment. We are not discussing about solo or individual efforts, but first not second or third class communication between the government, the Orthodox Church and the households.

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We will illustrate this relationship through a Venn diagram, set operations in terms of union and intersections. Two events C and D are mutually exclusive if the occurrence of C precludes or prevents the occurrence of D. When one event takes place, the other will not. Two events C and D are not mutually exclusive if the occurrence of C does not preclude or prevent the occurrence of D.

Source: author’s illustration. Source: author’s illustration

Then, we will illustrate the Orthodox variables through functions by assigning the set C as the domain and the set D as the target set. In addition, we will illustrate the image and the diagram of the function. The Orthodox variables will be illustrated through a sequence and a fundamental product of a set through the following tables.

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C D

C D

P(C or D) = P(C) + P(D)

P(C or D) = P(C) + P(D) – P(C and D)

Table 1 shows the variables that are included under the subset C of the Orthodox individual and the subset of the spiritual father D.

Orthodox individual ( C) Spiritual father (D)

Source: author’s illustration.

According to the above table, the function f is defined from C ={ Strong faith towards the Holy Trinity, Virgin Mary, Holy Communion, prayer, fast, vigil, appreciate of Byzantine music, humility , and strong believe that Jesus - Christ was resurrected} into

D ={ Strong faith towards the Holy Trinity, Virgin Mary, repentance, confession, Holy Communion, prayer, fast, prostrations, vigil, almsgiving, rejection of personal will appreciate of Byzantine music, humility, and strong believe that Jesus - Christ was resurrected}

The image of f is the set { Strong faith towards the Holy Trinity, Virgin Mary, Holy Communion, prayer, fast, vigil, appreciate of Byzantine music, humility , and strong believe that Jesus - Christ was resurrected} The missing variables from C and the fact that they do not belong to the image of f are repentance, confession, prostrations, almsgiving and rejection of personal will. These elements or variables are not the image of any element in C under the f function.

Finally, we will use logical expressions in terms of conjuction, disjunction and negation, which represent the words and, or and not. These logical expressions will be used to determine if they are true or false. We will start with the conjunction function

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Faith towards the Holy TrinityVirgin MaryHoly CommunionPrayerFastVigilAppreciate of Byzantine musicHumilityStrong believe that Jesus Christ was resurrected

Faith towards the Holy TrinityVirgin MaryRepentanceConfessionHoly CommunionPrayerFastProstrationsVigilAlmsgivingRejection of personal will Appreciate of Byzantine musicHumilityStrong believe that Jesus - Christ was resurrected

based on the above function example. Table 2 shows the conjunction function “and” between the Orthodox individual and the spiritual father.

Table 2 shows the conjunction function “and” between the Orthodox individual and the spiritual father.

Orthodox individual (C) Spiritual father (D) C^DFaith towards the Holy Trinity

Faith towards the Holy Trinity

T

Virgin Mary Virgin Mary THoly Communion Holy Communion TPrayer Prayer TFast Fast TVigil Vigil TAppreciate of Byzantine music

Appreciate of Byzantine music

T

Humility Humility TStrong believe that Jesus Christ was resurrected

Strong believe that Jesus - Christ was resurrected

T

No almsgiving Almsgiving FAcceptance of personal will

Rejection of personal will F

No prostrations Prostrations FNo confession Confession FNo repentance Repentance FSource: author’s illustration.T represents true statement that both variables are represented in both subsets.F represents false statement that both variables are not represented in both subsets.

In the above example, the conjunction, the disjunction and negation function is the interrelation of the false and true statement of the variables of the subsets C and D.

Section 4 describes the set topology

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The set topology is used as continuation of the set theory to show the interrelation of the Orthodox approach in relation to the different theorems. Initially, we will use the Heine – Borel theorem of closed and bounded interval to show the validity of the model in relation to the Orthodox variables.

Under the Greek Orthodox approach, the set H is the Greek Orthodox religion. We denote the elements as x. The set is divided into two subsets, namely, the Orthodox individual, C and the spiritual father, D. For simplicity two subsets are used to show that each elements of C is also an element of D. Thus, C is a subset of D and D is a subset of C. . The set H has a closed and bounded interval of nonnegative and positive integers of subsets in terms of H = [c,d]. The reason of being bounded and closed is that because they have two open infinite complements interval. The complements are ∞,c) ∞). The closed and bounded condition should be met by the set H in order that the theorem of Heine - Borel is valid and true. The set H is consistent of the elements x. Thus, H = {x i}.The elements are strong faith towards the Holy Trinity, Virgin Mary, repentance, confession, Holy Communion, prayers, fast, prostrations, vigil, almsgiving, rejection of personal will, appreciate Byzantine music, humility, and strong believe that Jesus - Christ was resurrected. The two subsets and the union of elements are defined as follows:

The maximum and minimum association of strength is used to show the degree that the Orthodox individual has strong faith towards the Holy Trinity, the Saints and towards the mother of Jesus – Christ.

Then, we will use topologies induced by continuous functions to show the spiritual falls and the deviations from the initial elements that constitute the set, which is the Greek Christian Orthodox religion. Let’s assume the topology on the set H = {x1,x2,…….xn} Where T = {H,Ø, {x1}, {x2,x4,x6}, {xn}}. The subsets C and D are joint, as they have common elements. Therefore, an empty set does not exist. Let’s use the nonnegative and positive continuous integers of the subsets C and D and denote them by X = {0,1,2.5,4,5}. Then, the function f: X (H,T)

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f

Thus, the defining subbase is equal to = {f-1 [H]: H ЄT} Or the defining subbase is = {X, Ø, {x1, x2, x3}, {x4},{xn}}

The variables should be combined with the positive continuous integers. It is an example to show the important relation to keep close connection with the Orthodox variables under the supervision of the spiritual father. Please remember that the strength of keeping and developing the degree of these variables remain to the grace of the Holy Trinity. The Orthodox individual is trying with the help of the spiritual father to achieve spiritual integrity. The degree of this association lies on what the Holy Trinity will offer as spiritual gifts to the Orthodox Christian servant. For example, Saint Nectarios did not study mathematics, but he was offered spiritual gifts from heaven due to the Orthodox and respectful way that he lived in earth. He was following the Gospel in a strict and disciplined way.

Then, an introduction to metric space will be given in order to better understand the separation axiom. Metric space is a set that is composed from elements that the distance between them is well defined. All the distances together are named a metric of the set. We will include topological properties on the metric space to include closed sets of the Orthodox variables. We will incorporate these Orthodox variables in propositional logic to form axioms of propositional variables that are associated with conjunction, disjunction or negation function. For example, the variable the strong faith towards the Holy Trinity (Repentance Confession Strong faith towards the Holy Trinity) and lack of fasting (reduced humility negationReduced faith towards the Holy Trinity and increase of the ancestral sin). On the other hand, separation axioms are restrictions made to metric spaces. For example, in Venn diagram we could assign open or closed disjoint spaces. Let’s assume that the subsets C, which is the Orthodox individual and the subset D, which is the spiritual father are closed joint or disjoint subsets of a metric space X. If the two subsets are disjoint,

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0

1

2.5

4

5

x1

x2

x3

Keep adding variables

then, it will exist disjoint open sets F and Y such that . The Venn diagram illustrates this relationship as follows:

This is not desirable, as we are looking for a joint closed relationship between subset C and D in a metric space X with the same topology on X. In other words, we would like that circle C and D interacts to achieve spiritual integrity through the help and intercede of the Virgin Mary and the Holy Trinity. It will be ideal if the subset C is entirely in the subset D. In other words, . In reality and in modern society, this could not happen. Therefore, in practical world, the Venn diagram takes the second format. The Venn diagram will take the following shapes:

Source: author’s illustration.

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CDF

Y

DC

U

Ideal situation

Source: author’s illustration.

We will mention Hausdorff spaces, the distribution of open sets and whether a space is separable or second countable. But what is a Hausdorff space? A separated or T2

space is a topological space in which distinct points have disjoint neighbourhoods. A neighbourhood is a set containing the point where one can move that point in some degree without leaving the set. Each pair of distinct points belongs to disjoint open sets. Therefore, if X is a topological space and q is a point in X, a neighbourhood of q is a subset, C, of X that includes an open set U containing q. Thus, the mathematical expression is as follows:

Є U

Let T1 be a topology on the real line R. Thus, T = {X, Ø, {x1}, {x2, x3}, {xn}} is based on the set Orthodox religion H that is defined as H = {x1, x2, x3,…xn} The closed subsets of C, Orthodox individual and D, spiritual father are also of H in a way that C,D,H satisfy [R].Similarly, H is a Hausdorff space of T2 and T3 and T4 spaces of T1 normal space.

The Orthodox variables are forming a neighbourhood system of positive nonnegative integers and their associated variables.

Є U

Є U

C D

Virgin Mary + Holy Trinity

U

Example of practical modern society. Some elements are in both subsets with the same intensity and others are missing from both of the subsets.

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Then, our effort is to address the metrization problem in Topology and interrelate it with the Greek Orthodox approach. This problem in topology is based on finding necessary and sufficient topological conditions for a topological space to be metrizable. The Urysohn’s lemma theorem was an effort to solve this puzzle by trying to find a metric d on X which induces the Topology T space and has a countable base. The problem will be addressed within open and closed disjoint spaces. A normal space exists if any disjoint closed subsets can be separated by a function. The Orthodox variables avoid this problem as the topological space T of the set H does not consist of more than one point in each single variable. We are discussing about homeomorphic variables that are practiced from both the Orthodox individual and the spiritual father with aim to achieve maximum spiritual integrity and salvation of the soul. The Urysohn’s lemma theorem assumed two disjoint closed subsets F1 and F2 of a normal space X. Then, there exists a continuous function f: X [0,1] that f[F 1] ={0} and f[F2] = {1}. According to this theorem every second countable normal T1

space is metrizable and homeomorphic. The Orthodox variables are homeomorphic and have a continuous function based on the effort and the help that the Orthodox individual is getting. The difference is how the Orthodox variables are used from the Orthodox individual and the spiritual father. The correct way of using them facilitate the successful prayer to demand things that are for his/her spiritual benefit and for his/her salvation.

For example,

Appreciate of Byzantine musicveryweak =0; weak=1; medium=2.5; strong=4; verystrong=5;Appreciate byzantine music= verystrong – veryweak

The continuous function could be f: X [0,2.5] that f[F1] ={0} and f[F2] = {2.5}

The continuous function should be used to combine the results of the subsets of the Orthodox individual and the spiritual father. Deviations and spiritual falls should be discussed and adjusted with the spiritual father.

In addition, we will explain how the Orthodox variables converge to the function spaces in terms of uniform and compact convergence. Compactness is based upon the Heine – Borel theorem in terms that every open cover of a closed and bounded interval is reducible to a finite cover. Again, we are discussing about closed and bounded interval on the real line R. The subsets C and D of a topological space X are compacts if every open cover of C and D is reducible to a finite cover. If C is compact and and , where the Gi are open sets, then, the individual could select a finite number of the open sets such as Gi,1,………..,Gi,m in order that

and The Orthodox variables are continuous images of a compact set H of a closed unit interval. In addition, connectedness between the Orthodox variables exists if the two subsets of a topological space X are not disjoint non-empty open subsets whose union is the original subset. We are discussing about a set and subsets of Orthodox variables that are connected and not separated by empty sets.

Section 5 describes the group theory

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Under the group theory, permutations will be used in relation to a set to show how the Orthodox variables are combined with the equations of open economy macroeconomics. Permutation is used in mathematics to arrange the members or elements of a set into a particular order. For example, for the set, H = {1,4,5}, there are six permutations. These are as follows:(1,4,5), (1,5,4), (4,1,5), (4,5,1), (5,1,4), (5,4,1). They could be the Orthodox variables fasting, prayers, almsgiving and appreciate of Byzantine music. The Orthodox variables have been chosen to achieve symmetry in the group theory. They are all equally important and arranging them in a particular sequence is not important. In this case, it is better to use combinations as the order of selecting the variables does not matter. The variables are used in harmony and in symmetry towards the soul salvation of the humble Christian servant. The nonnegative and positive integers are used as an example to show the spiritual association between them.

As an example to show the equilibrium between the IS – LM curve, we use the following systems of simultaneous equations:

IS equilibrium : Y = C + I Y = 40 +0.7Y + 80 -70i Y -0.7Y = 40 + 80 – 70i 0.3Y + 70i -120 = 0 (1)

LM equilibrium: Ms = 0.7Y+40 -100i 100 = 0.7Y +40 -100i 0.7Y -60 – 100i = 0 (2)

Where: Y is the national income. i is the interest rate. I is the investment level. Ms is the money supply. C is expenditure on consumption.

We solve the simultaneous equations (1) and (2)

0.3Y + 70i -120 = 00.7Y – 100i -60 = 0

We multiply equation (1) by 7 and equation (2) by 3 and we subtract to eliminate Y. Thus, we have the following system of equations:

2.1Y +490i – 840 = 02.1Y- 300i - 180 = 0 0 790i – 660 = 0

790i = 660 or i =0.84

We then substitute i to equation (1) to find Y. Thus, we have:

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0.3Y + 70(0.84) – 120 =00.3Y + 58.8 -120 =00.3Y -61.2 = 00.3Y = 61.2Y = 61.2 /0.3 = 204 million Euros

Thus the good and the money market will be in equilibrium when Y = 204 million Euros and i = 0.84.

Graph 1 illustrates the relationship of the IS, LM curve the balance of payments trading account and the Orthodox variables.

Interest rate (r) Balance trade line

Equilibrium LM1

Orthodox variablesrE LM0

Humility

IS0 IS1

0 YE Income ( Y) Source: author’s illustration

The Greek government should pay particular attention to this graph to incorporate the Greek Orthodox variables with the macro factors, combinations and the trade balance to achieve equilibrium with full employment at a given interest and national income rate. All the Orthodox variables are equally important to achieve equilibrium and integrity in the life of the Orthodox prayer.

Mapping with the same domain will be used to show the direction of the Orthodox variables. The Orthodox theory will be combined with homomorphism, isomorphism and Cayley’s theorem to prove the importance of the repetition of the Orthodox variables to achieve maximum spiritual integrity. If the subsets C and D are non empty sets, then, a mapping from C into D has unique elements of D to each element of C. We denote the elements of the variables as x1, x2,……xn For example, the prayers from the subset D of the spiritual father strengthen the spiritual effort of the Orthodox individual in subset C to do more almsgiving. Thus, if the direction of the variable is a mapping from D to C, then this expression is denoted as:

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Prayers: D C. Thus, D is the domain and C is the codomain of the variable prayers. Similarly, we will think about the rest of the Orthodox variables. For example, sincere confession for the different sins of the Orthodox individual facilitate the prayers of the spiritual father in subset D to attract the great mercy from the Holy Trinity with always the help of the Virgin Mary.

We are interested on the bijective function of Arthur Cayley theorem between the elements of two sets, namely the Orthodox individual and the spiritual father. We want to spot the deviation and the spiritual falls based on the paired and unpaired elements between the two sets by rearranging the elements of a set into particular sequence according to the guidance of the spiritual father. For example, a bijective function f: C D between the elements of the subsets of the Orthodox individual and the spiritual father. Every element is paired exactly with the element of the other subset. There are no unpaired elements. The purpose of the bijective function is to match and strengthen the effort of both the individual Orthodox and the spiritual father to save their soul and achieve spiritual integrity with the grace of the Holy Spirit. A diagram that could illustrate this function is as follows:

Source: author’s illustration

Finally, we will discuss about other groups such groupoids. The purpose of groupoids is to combine a set of integers Orthodox variables with positive and nonnegative signs and a binary operation expressed as an addition or multiplication. A groupoid is a pair (G,μ), that consists of a non-empty set G, called the carrier and a binary operation μ in G. The purpose of binary operation is to assign only two values to the Orthodox variables according to their spiritual direction. The interrelation of the binary operation and the positive and nonnegative integers or the carrier forms a groupoid. The purpose of using this repetitive task is to test the spiritual integrity of the Christian Orthodox individual. For example, we assume that the subset C ={1,2,3} and the binary operation μ is defined by the following table:

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1

2

3

x1

x2

x3

4x4

1 4 5

1 1 5 4

4 4 1 1

5 5 4 5

Source: author’s illustration

Thus, the multiplication pair (G,h)μ is a groupoid. The numbers could help the Orthodox individual to adjust his/her spiritual efforts to achieve spiritual integrity. Then, we want to check if two groupoids have the property of isomorphism or homomorphism in terms of similarity. Let’s consider positive and nonnegative integers of two binary operations zero and one. Thus, the Orthodox variables will be combined with a value of zero if the individual Orthodox is not practicing them and a value of 1 if there is spiritual effort taking place with the help of the spiritual father. Thus, a groupoid is a pair of Orthodox variables in subset C associated with a binary operation in terms of zero or one. The objective of the Orthodox religion is to form groupoid of Orthodox individual’s with a positive pair one that shows interest towards keeping the commandments of the Gospel and safeguarding their soul. It will be optimal that the subsets C and D have the same carriers and same binary operation to achieve maximum spiritual integrity. In other words, they preserve the property of isomorphic and homomorphic. I will show an example through a graph that the value of the variable is zero.

Faith towards the HolyTrinity Adjustments with the help of the spiritual father through repentance, confession and Holy Communion.

Time

There are cases that the adjustments took long time for the individual to adjust and are supported from continuous prayers and fasting from the spiritual father.

Section 6 describes the principles of the Orthodox approach in relation to Boolean algebra

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Boolean algebra will be used within the algebra set theory and binary digits of groupoids to define and form abstract algebra to intersect and complement the set of the Orthodox variables in the business area. The point is to find if there are similarities in terms of isomorphism between the subsets of the Orthodox prayer and the spiritual father. In addition, we are looking for duality in terms of symmetry by interchanging the identity of the elements of the Orthodox variables. Thus, we are trying to find true conditions based on logic to eliminate the wicked thoughts and improve the manipulation of thoughts. Two Boolean algebras B and B’ are isomorphic, if there is matching between them by preserving the three operations. For example, f(c +d ) = f(c) + f(d), f(c*d) = f(c) * f(d) and f(a’) = f(a)’. The duality is obtained by interchanging the operations and their identity elements 0 and 1 in the original statement and getting the same results. For example, the Orthodox variable rejection of personal in the two subsets C and D with the binary values of zero and one could build a truth table. This table will facilitate the manipulation of thoughts based on the results though consecutive time periods. Let’s take for example rejection of personal will and humility of subsets C and D, namely, the Orthodox individual and the spiritual father. Under each of the subsets, we have listed the values of the variables in different time periods. The adjustments are always taking place through repentance and confession. It is worth to mention that the spiritual father is also following the mystery of repentance and confession to adjust and correct himself.

Rejection of personal willNo rejection of personal will =0; rejection of personal will =1

HumilityNo humility =0; humility =1;

The following table illustrates an example of the truth results that could help the Orthodox servant to improve its logic to eliminate the wicked thoughts and achieve a better manipulation of thoughts.

C D C+D C*D1 0 1 1 02 1 1 1 13 1 0 0 04 0 1 1 05 1 1 1 16 1 1 1 17 0 1 1 08 0 1 1 09 0 0 0 010 1 0 1 0Source: author’s illustration.

According to the values of the above tables, the Orthodox individual and the spiritual father make the necessary adjustments to improve the quality of their spiritual integrity and attract the grace of the Holy Spirit.Section 7 summarizes and concludes

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We have used different theorems to explain the Christian Greek Orthodox approach and its variables using set theory, general topology, group theory, and Boolean algebra. Under the Greek Orthodox approach, the set is the Greek Orthodox religion. We denote the elements as x. The elements are strong faith towards the Holy Trinity, Virgin Mary, repentance, confession, Holy Communion, prayers, fast, prostrations, vigil, almsgiving, rejection of personal will, appreciate of Byzantine music, humility and strong belief that Jesus – Christ was resurrected. By following the elements of the set theory the spiritual factors will follow an increasing function. In contrast, by following the elements that are not part of set theory, the spiritual integration of the individual follows a decreasing function.

The set topology was used to illustrate the properties of the metric and normed spaces, the separation axioms, the continuous functions, compactness and connectedness sets. The Orthodox approach will be interrelated with Urysohn’s lemma and metrization theorem. The maximum and minimum association of strength was used to show the degree that the Orthodox individual has strong faith towards the Holy Trinity, the Saints and towards the mother of Jesus – Christ. We have also used topologies induced by continuous functions to show the spiritual falls and the deviations from the initial elements that constitute the set, which is the Greek Christian Orthodox religion.

Under the group theory, permutations or combinations have been used in relation to a set to show how the Orthodox variables are combined with the equations of open economy macroeconomics. The point is to explain why the Orthodox variables have been chosen to achieve symmetry in group theory. The Orthodox theory has been combined with homomorphism, isomorphism and Cayley’s theorem to prove the importance of the repetition of the Orthodox variables to achieve maximum spiritual integrity. We have also discussed the importance of groupoids. The purpose of groupoids is to combine a set of integers Orthodox variables with positive and nonnegative signs and a binary operation. The purpose of binary operation is to assign only two values the Orthodox variables according to their spiritual direction. The interrelation of the binary operation and the positive and nonnegative integers or the carrier forms a groupoid. The purpose of using this repetitive task is to test the spiritual integrity of the Christian Orthodox individual. The objective of the Orthodox religion is to form groupoid of Orthodox individual’s with a positive pair one that shows interest towards keeping the commandments of the Gospel and safeguarding their soul.

Finally, Boolean algebra has been used within the algebra set theory and binary digits to define and form abstract algebra to intersect and complement the set of the Orthodox variables in the business area. The purpose was to construct a truth table that will facilitate the Orthodox individual and the spiritual father to adjust their spiritual efforts and achieve sustainable spiritual integrity with always the grace of the Holy Spirit.

Finally, we have discussed the problem and weakness that is facing the prostitute industry, the gambling industry and the war industry. The prostitute industry, the gambling industry and the war industry face the same spiritual problems from excluding fundamental elements from the subset and insisting in the personal will and the ancestral sin. For example, the soldier depends on dirty strategic plans based on weapons and killing. The female prostitute depends on her body and her capability to

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provide professional sex. The gambler put all his strength to increase fun by trying to win the black jack and the roulette in the Casino of Las Vegas or Macao. The gambler ignores the law of large number in probability theory that his/her chances to win are very limited.

Bibliography

Mperati Theodoros, (2005). The son of tears, (the Sacred Augustine). Edited by the theological brotherhood Sotir.

Saint John Sinaet, (2006). The Ladder. Edited by the Holy monastery Paraklitou.

Saint Nikodemos the Hagiorite, (1819). Consulting Manual. It was first published in 1819, in the first book named interpretation of the fourteen manuscripts of Saint-Paul.

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