What Are the Converse, Contrapositive, and Inverse

Conditional statements make appearances everywhere. In mathematics or elsewhere, it doesn’t take long to run into something of the form â€Å"If P then Q.† Conditional statements are indeed important. What is also important are statements that are related to the original conditional statement by changing the position of P, Q and the negation of a statement. Starting with an original statement, we end up with three new conditional statements that are named the converse, the contrapositive, and the inverse. Negation Before we define the converse, contrapositive, and inverse of a conditional statement, we need to examine the topic of negation. Every statement in logic is either true or false. The negation of a statement simply involves the insertion of the word â€Å"not† at the proper part of the statement. The addition of the word â€Å"not† is done so that it changes the truth status of the statement. It will help to look at an example. The statement â€Å"The right triangle is equilateral† has negation â€Å"The right triangle is not equilateral.† The negation of â€Å"10 is an even number† is the statement â€Å"10 is not an even number.† Of course, for this last example, we could use the definition of an odd number and instead say that â€Å"10 is an odd number.† We note that the truth of a statement is the opposite of that of the negation. We will examine this idea in a more abstract setting. When the statement P is true, the statement â€Å"not P† is false. Similarly, if P is false, its negation â€Å"not ​P† is true. Negations are commonly denoted with a tilde ~. So instead of writing â€Å"not P† we can write ~P. Converse, Contrapositive, and Inverse Now we can define the converse, the contrapositive and the inverse of a conditional statement. We start with the conditional statement â€Å"If P then Q.† The converse of the conditional statement is â€Å"If Q then P.†The contrapositive of the conditional statement is â€Å"If not Q then not P.†The inverse of the conditional statement is â€Å"If not P then not Q.† We will see how these statements work with an example. Suppose we start with the conditional statement â€Å"If it rained last night, then the sidewalk is wet.† The converse of the conditional statement is â€Å"If the sidewalk is wet, then it rained last night.†The contrapositive of the conditional statement is â€Å"If the sidewalk is not wet, then it did not rain last night.†The inverse of the conditional statement is â€Å"If it did not rain last night, then the sidewalk is not wet.† Logical Equivalence We may wonder why it is important to form these other conditional statements from our initial one. A careful look at the above example reveals something. Suppose that the original statement â€Å"If it rained last night, then the sidewalk is wet† is true. Which of the other statements have to be true as well? The converse â€Å"If the sidewalk is wet, then it rained last night† is not necessarily true. The sidewalk could be wet for other reasons.The inverse â€Å"If it did not rain last night, then the sidewalk is not wet† is not necessarily true. Again, just because it did not rain does not mean that the sidewalk is not wet.The contrapositive â€Å"If the sidewalk is not wet, then it did not rain last night† is a true statement. What we see from this example (and what can be proved mathematically) is that a conditional statement has the same truth value as its contrapositive. We say that these two statements are logically equivalent. We also see that a conditional statement is not logically equivalent to its converse and inverse. Since a conditional statement and its contrapositive are logically equivalent, we can use this to our advantage when we are proving mathematical theorems. Rather than prove the truth of a conditional statement directly, we can instead use the indirect proof strategy of proving the truth of that statement’s contrapositive. Contrapositive proofs work because if the contrapositive is true, due to logical equivalence, the original conditional statement is also true. It turns out that even though the converse and inverse are not logically equivalent to the original conditional statement, they are logically equivalent to one another. There is an easy explanation for this. We start with the conditional statement â€Å"If Q then P†. The contrapositive of this statement is â€Å"If not P then not Q.† Since the inverse is the contrapositive of the converse, the converse and inverse are logically equivalent.

Essay Classic Mayan Architecture - 1687 Words

Classic Mayan Architecture Mayan culture existed a thousand years ago, in what is now part of Central America. Its ruins were almost entirely abandoned by 600 A.D, and were not rediscovered until the early 1500’s, by Spanish settlers. Mayan architecture astounded the early conquistadors, and continues to be of great interest to modern archeologists as well. These scientists have labeled a certain period of Mayan architectural history as the â€Å"Classic† period. This refers to a period when the Mayan civilization flourished, and extended from 300 to roughly 900 A.D (Miller 52). Because of our image of classical antiquity, the word â€Å"Classic† implies the heights of cultured accomplishment. In the classic period, we envision musicians†¦show more content†¦These people built fake mountains by piling rock, and building a foundation on top to build a temple. The Mayans developed these temples further by adding sculpture, and painted faces onto these pyramids. The whole building would then be covered with plaster, and painted red or other bright colors. The Temples were often small, and had three dark rooms in their interior. One of the rooms would be an inner sanctuary where the king preformed his rituals (Liz 1-6). That temples were, in fact, religious buildings is beyond question; structures of the same shape were still in use when the Spaniards arrived and described with horror the bloody sacrifices that took place within t he temple walls. Some Mayan temples have multiple doorways and suites of interconnected rooms. In some rooms, plaster-and-stone thrones look through doorways onto courtyards where one can easily envision crowds of religious supplicants (Culbert 165). Many temples also contain architecture that curiously corresponds with certain astrological phenomenon. Most temples constructed during the Classic period have a total of 365 stairs, and openings that align with the summer and winter solstices. When the temples were built, the architects aligned the temples with the sun, the moon, and the stars (Liz 1-6). Some of the most distinct features of Classic Mayan culture were the pyramids. The pyramids were either temples orShow MoreRelatedHow Bloodletting And Human Sacrifice1235 Words   |  5 Pagesimportant in the life of the Mayans. The bloodletting allows the ruler the ability to communicate with the Gods and their ancestors. While the human sacrifice is connected to the continuation of the cosmos and the resurrection of the agricultural seasons. A summary and a discussion of the book Popol Vuh is presented, on the topic of reassurance theme of human sacrifice is addressed in this paper. The examination of how bloodletting and human sacrifice is crucial to the Mayan culture, and how these ritualsRead MoreAncient Maya Essay868 Words   |  4 PagesThe Ancient Mayan civilization stands out from other ancient civilizations and is great because of its outstanding achievements and developments. The Mayans are still known to this day for their exceptional architecture, medical discoveries, astronomy, and military tactics which were very successful. In this essay I will explain what the Mayans have achieved to make themselves considered great, why they are great, and why they overcome many of the other ancient civilizations. Mayan astronomy hasRead MoreThe Classic May A Time Of Great Expansion During The Yucatan Peninsula1086 Words   |  5 PagesThe Classic Maya was a time of great expansion in the Yucatan peninsula. Socially impactful art, massive architecture, and thriving complex societies. 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Temple of Frescos Read MoreThe Mayan Civilization And The Maya Empire1677 Words   |  7 Pages The Mayan civilization or the Maya Empire, centered in the tropical lowlands of what is now Guatemala, reached the peak of its power and influence around the sixth century A.D. The Maya excelled at agriculture, pottery, hieroglyph writing, calendar-making and mathematics, and left behind an astonishing amount of impressive architecture and symbolic artwork. Most of the great stone cities of the Maya were abandoned by A.D. 900. The Maya civilization was one of the most dominant indigenous societiesRead MoreThe Fall of the Mayan Empire Essay1517 Words   |  7 Pages The fall of the Mayan Empire The collapse of the Mayan Empire is one of history’s greatest mysteries. It was one of the most advanced and developed civilizations of its time period, reining during the Pre-Classic period and into the Classic and Post-Classic Periods (2000 B.C. – 900 A.D.). The territory stretched from the Yucatan Peninsula in Mexico, down to modern day El Salvador in Central America. Its achievements were monumental for the era, being the first empire communicating with the useRead MoreEssay The Maya Civilization904 Words   |  4 Pagespyramids, highly accurate calendars, mathematics and hieroglyphics, and a complex social and political order (Collapse... 1). Urban centers were important to the Maya during the Classic period; they offered the Mayans a central place to practice religion. The Mayan culture can be traced back to 1500 BC, entering the Classic period about 300 AD and flourishing between 600 and 900 AD. The basis of the culture was farming. They cultivated food crops such as maize (corn), beans, squash, and chili peppersRead MoreAncient Maya Essay763 Words   |  4 Pagesthe basis of the economy of the Mayan and corn was the principal food.(Voorhies 324) Other crops included avocados, tomatoes, and chili peppers. They cultivated an enormous variety of plants.(Foley 20) In hieroglyphic writing, astronomy, and mathematics, the Mayan Indians were far ahead of any other people in the New World.(Foley 20) The Mayan invented a solar â€Å"civil† calendar including three hundred sixty- five days.(Ivanoff 86) The accuracy of the Mayan calculations is all the more extraordinary

The Soviet Union Responsible For The Consolidation Of...

ESSAY PLAN To what extent was the Soviet Union responsible for the consolidation of communism in Eastern Europe in the period 1945-1953? Introduction During the aftermath of World War II, the Soviet Union was primarily responsible for the consolidation of communism in Eastern Europe. †¢ It was in the spring of 1948 that the Soviet Union had aggressively pushed for the imposition of Communist rule in most East European nations o Eastern Europe under Communist rule was comprised of Czechoslovakia, Hungary, Poland, Albania, Bulgaria, Romania and Yugoslavia †¢ East European nations started to adopt USSR foreign policy agreements and embarked on Stalinist transformations of their political, social and economic systems †¢ The Soviet Union’s domestic policies combined with Stalin’s external ambitions were key factors in the consolidation of Communism in Eastern Europe The Soviet Union as an emerging global power in the post-war context †¢ After WWII, Stalin and other Soviet officials were determined that Eastern Europe be converted into a buffer zone against future invasions from other nations o Many Eastern European countries began to support the Soviet Union †¢ Stalin viewed the creation of a buffer zone in Eastern Europe as the best way to obtain economic benefits from the region o From eastern Germany, the Soviet Union extracted 3 500 factories and 1.15 million pieces of industrial equipment in 1945 and 1946 o There was an increase in Soviet military power in Eastern EuropeShow MoreRelatedDeclaration of Independence from My Parents1033 Words   |  5 Pagesthe Soviet Union Animal Farm is most famous in the West as a stinging critique of the history and rhetoric of the Russian Revolution. 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In the years after World War II, German scientists and engineers who had been responsible for the development of theRead MoreThe Bolshevik Consolidation of Power 1918-21 Essay2585 Words   |  11 PagesThe Bolshevik Consolidation of Power 1918-21 The Bolsheviks under Lenin, when they came into power in October 1917, faced immense problems in trying to consolidate their hold over the ex-tsarist empire. Firstly, how were the Bolsheviks, in view of their military resources, to extend their hold over the nation at large? The second, was how could they achieve a speedy end to the war and effect a rapid withdrawal of the German army, which was currently occupying theRead MoreDiscussion on Whether Stalin Was a Necessary Evil Essay2325 Words   |  10 Pageswere influenced by economic forces and that each stage had to be completed before the next could begin. 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Unique social aspects that needed changing marked each stage of the CubanRead MoreTo What Extent Were the Purges Caused by Stalins Paranoia?4348 Words   |  18 PagesStalin’s paranoia? Daniel Johnston Stromness Academy SCN: O62114657 Introduction J.Arch Getty says that â€Å"The Great Terror of the 1930s in the Soviet Union was one of the most horrible cases of political violence in modern history†[i] but was this political terror a result of Stalin’s own paranoia or a necessity to maintain control in Soviet Russia? Robert Service argues that â€Å"Nowadays, virtually all writers accept that he [Stalin] initiated the Great Terror†[ii] however historians are careful

Strategic Marketing Plan For A Company s Distribution...

1. The Costs of operating a company’s distribution channels and the ways to manage such costs. Decisions relating to the channels of distribution for a product or service are part of the strategic marketing plan. In that plan, the target market will have been specified along with target levels of market share, market coverage, customer service and so on. The channels of distribution used by an organisation must be capable of assisting in reaching these targets. Moreover, the establishment of a distribution system can take a long time, perhaps several years, and so decisions about the channels of distribution cannot be taken lightly, and have to be taken with a view to the longer term since it is not usually that easy to switch between channels. Strategic planning: Distribution channels must be compatible with the strategic marketing plan. If, for instance, a skimming strategy has been adopted or the product requires technical sales support, then mass marketing is probably inappropriate. Alternatively, if large volume sales are required in order to achieve particular profit targets, then selective distribution would be inappropriate. As new products are introduced, existing channels have to be reassessed since they may not be the right channels for the new product. In some cases, a company will decide not to launch a new product because it does not fit in with existing distribution channels and existing strategy. A few years ago Ciba, the Swiss chemicals company, was lookingShow MoreRelatedMarketing Plan1347 Words   |  6 PagesA marketing plan is the key to business. Its purpose is to maximize the business profits. As opportunities crop up or the business environment changes, the objective and marketing strategies in the plan will aim toward the best action. 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Business Mission 4. External Marketing Audit Macroenvironment The Market Competition 5. Internal Marketing Audit Operating Results Strategic Issues Analysis Marketing Mix Effectiveness Marketing Structures and Systems 6. SWOT Analysis 7. Marketing Objectives Strategic Thrust Strategic Objectives 8. Core Strategy Read MoreMarketing Plan1270 Words   |  6 PagesIndividual Assignments - Marketing Plan. Review individual marketing plan assignment. The major written assignment for this course is to write a marketing plan, utilizing the resources and examples of The Marketing Plan Book. Students will work on this project individually. Students will develop a hypothetical company that operates in the US and one or more foreign markets. The marketing plan needs to involve a product that has a service component to the product. As students determine whether

Dance Hall Of The Dead Book Report free essay sample

Throughout the book, many detectives were given false information that loud attempt to point them in the wrong direction. The next important character of this book is the twelve year old Gun boy that was murdered. His name is Ernest ACTA, who was a very important child to the Badger Clan, who all thought he would grow up to be a natural leader in the community. Ernest was selected by his community for the honor of being the Little Fire God at the Shalom festival. Ernest was apparently open minded, and the least prejudice person of all the Gun community.Next important people are the Bowlegs family. There is the father Shorts Bowlegs, the youngest brother Cecil Bowlegs, and finally the other of the rooters George Bowlegs. According to Loophole, Shorts was a major alcoholic. Loophole also said he seemed to be a loving father, but his alcohol deterred often deterred most of his great actions. He eventually was killed during sometime in the book. Cecil Bowlegs according to Loophole was a very young but brave Navajo boy that he misinterpreted to being George Bowlegs, Cells brother.Cecil was a very cautious person when it came to anyone who worked for the authorities, which made it a really hard challenge for Loophole to get information about the case. Throughout the book, Loophole had to essentially lie to Cecil to try ND gain his trust. Cecil caught on to his tactics, which lead to him having to apologize for lying to the boy and lead to him treating him with much more respect. After Cells father died he truly entrusted himself in Loophole. George Bowlegs is best known as Reenters best friend who was a Navajo. George seems to be growing up with a tough life, knowing that his mother ran away and his father being a raging alcoholic. To further this problematic situation, George is attending a Gun school where is suffering from very discriminatory actions from his fellow students. George was seen as a role del for Cecil. George was hunting for food for both his brother and father, and became a successful hunter Of both styles Of the Gun and the Navajo. The next character of importance is the man who Lieutenant Loophole determined through his investigation that murdered Shorts Bowlegs, George Bowlegs, and Ernest ACTA His name is DRP.Reynolds. Loophole has uncovered through his investigations that DRP. Reynolds was a well-known anthropologist that had planted fake arrowheads at an archeological dig site that he was supposedly funding for his upcoming protog, Ted Isaacs. With hat being said, Loophole believes this the reason why DRP. Reynolds killed Ernest, George, and Shorts, to prevent anyone from finding out the truth behind what he had done. The last important person to the story is the Gun Chief of Police De Piquant. When Loophole is dead certain that DRP.Reynolds was the killer, he went right to the chief. De then doesnt even make an attempt to arrest DRP. Reynolds. Coincidentally this leads Loophole to tell the Gun of the murder of Ernest ACTA, George Bowlegs, and Shorts Bowlegs. This is when the Gun found DRP. Reynolds, and killed him for killing one of their own, Ernest ACTA. In the beginning of the book, we learn of who Ernest ACTA was and how he ended up being murdered. Ernest was a young boy that was selected by his people, the Gun, to be the Qualities, also known as the little fire god, at the festival of Shalom.Then the writer goes on to explain that the fire god must be in shape, if he isnt Gun people would beat him with their wands. We also meet George here, we find out that he wanted to know the secrets of the Gun people and this tended to make Ernest angry. Then one night that Ernest running along one of his routes, he thought George would be waiting for him, UT to his dismay it was an ancestor spirit that only one sees when they are about to die. Finally we get to meet the Lieutenant that is going to be investigating the murder of Ernest ACTA.His name is Lieutenant Joe Loophole who is a Navajo Dine policeman. One of the first things that Loophole did was he went to try and investigate Reenters best friend George. But in return this is when he ended up meeting Cecil, Georges younger brother, who was a Navajo also. Loophole then tries to get some answers our of Cecil, but Cecil is very skeptical of law enforcement figures, especially since he thinks that he s trying to find his brother George who they may think killed Ernest. Cecil tends to think otherwise from Loopholes investigation.Loophole then goes to where Cecil has pointed out that Ernest and George had stolen some artifacts from, which took him to an archeological dig site. He meets Ted Isaacs and DRP. Reynolds, the two who want to prove DRP. Reynolds theory that some Folsom men have artistic arrowhead styles that were credited to other such cultures in a form of error. After he hears about the theory, Loophole then asks about if he has seen Ernest and George. DRP. Reynolds then tells Loophole that he had chased the two boys out of his dig site so that they wouldnt destroy or tamper with anything.Loophole asked if they had stolen anything and he claimed that they did not, which doesnt add up to what Cecil had told him before he went to the dig. A couple days after the death of Ernest Cater authorities have determined that it was indeed a homicide, Ernest was murdered. Loophole then started to investigate where he was murdered. While he was there, Loophole had used his excellent tracking skills and discovered the many tracks he found. He found Reenters uncles, who found Ernest, the chief of polices, Georges boots, and finally an unknown track made from a moccasin.According to one of the details Loophole found, it is to his understanding that the killer and Ernest had a stare down before he was eventually murdered. Then to finalize his investigation, he doesnt think that George killed Ernest. Loophole then reflected back on what his grandfather Instabilities had taught him. He told him that the only goal for man was beauty, and that beauty was found only in harmony, and that this harmony of nature was a matter of dazzling complexity. His grandfather told him that all patterns are caused by action and reaction and by cause and effect.The term he told him was in all things a pattern, and in this pattern, the beauty of harmony. Thus one learned to live with evil, by understanding it, by reading its cause. Thus one learned, gradually and methodically, if one Was lucky, to always go in beauty, to always look for the pattern, and to find it. All of these terms led Loophole to think back on the evidence that he had and try to connect someone with what happened to Ernest. Near the middle of the book, Loophole went back to interview Shorts Bowlegs. When he went inside, he was trying to remember what he was wearing and how he was acting.When Loophole opened the door, the only thing he saw was Shorts lying face down with a knife wound to his back. This is also when he noticed that two horses had been missing, but one returned and Cecil had arrived back to the Hogan. The other he assumed was missing from George taking it to wherever he may be hiding. The only thing left for Loophole to do is to start interrogating the students that George went to school with to see if he can find his whereabouts. Looking back on all the information Loophole has, he now has a double minced, and the only person that may truly know what is going on is missing.He reflects that he doesnt think that George is the killer because of the mysterious moccasins he seen at the scene of the murder of Ernest. If it were the case of George, he would have had to switch shoes which didnt really make sense to Loophole. The only key part of this investigation is he needs to either find George the murderer, or George the future victim of a double homicide. About three-fourths the way through the book, Loophole finally has a lead from Father Mingles that makes him believe he has headed to the Dance Hall of he Dead.Amazingly on the way he finds Susanne hitchhiking down the road, which he uses to his advantage because he knows George, would talk to her if they were to find him. After they arrive Loophole uses his excellent tracking skills to find the horse tracks from the missing horse at the Bowlegs Hogan, which h e ends up finding. Then after he found the tracks, he knew that George was an excellent hunter and that he needed food, so Loophole was to track the nearest deer tracks and hopefully those will lead to Georges hopeful location.Finally near the end of the book, Loophole was at the festival, trying to let people know that he has given up looking for George. Just as he was about to call it quits, he sees George standing on a balcony across from him. Loophole obviously went into pursuit to try and catch him, but by the time he got to the alley he saw George go in, he heard a loud shot. When he found George, he was dead, and all he seen was his body and a feather from a Colombia mask. He then followed the tracks from the killer, and saw two moccasin tracks had captured the killer and this is when he realized that the investigation was all over.Loophole never told Maloney who the killer was, but thought maybe nee day he will send him a note explaining who the killer was. This was a very interesting and very well illustrated and detailed book. I am not one to read books but I am glad you assigned these to us. Was surprised to See that the killer wasnt George to say the least. He lasted how long out on the run from a professional tracker and from a man who wanted to make sure he wouldnt unfold his fake important dig. It was also very intriguing to see it was DRP. Reynolds, and how they incorporated him into the story.It all revolved around the Gun/Navajo ways and it showed me a lot about how hey all will do anything for their own culture. The only thing I thing that made it difficult for police to help solve a case is when they dont give any details or the will to work with the police. For example, Cecil would not help Loophole because he worked for the police and he was even a fellow Navajo. I also see this being an issue anywhere around us today. You have those who still own land and they cherish that within their community. They have their own ways of policing and don t really let others interfere with anything on their land.

Electrolux free essay sample

How did Electrolux Chief Executive Strasberg break down barriers (and increase communications) between departments? Why did he do that? Explain Some of the measures he did were: a. He forces all his personal from different departments to see the objectives of the company as the main of objective of everybody. b. The main focus or the principal objectives was to find out what were the need of the customer, and the way to satisfied them. c. He recruited top level managers with vast experiences facing tough situation. d. Those executive come on from other famous firms which have face hard time and had succeed. . He motivated the employees to think as the customers as a way to discover what they need and the way to satisfy that need with an invention from them. He did that to break the communication barriers that could exist between each department, and to empower each person of the company to decide and make the right choice on their own. We will write a custom essay sample on Electrolux or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page He also did that as a way to reduce cost, increase speed and innovation, plus to diminish complexity among hierarchy levels. By doing theses, the employees change the view of the customers as someone aside them.He breaks also any possible barriers that could exist between departmental units. He made them interrelate and communicate more one another, and shows them to pursuit a common goal. He also motivates them to create cross functional team. 2. What are the advantages for Electrolux of having individuals from different departments and functional areas work together on product designs? Describe 1. The bureaucracy process that takes to create a product, and wait for the next department in line to complete their part was eliminated.Since inter-departmental personal were in the same team, possible problem were caught or avoid by the first previous departmental employee. 2. He avoids the waiting process of inter-departmental inputs. The opinion of personal from different department was expressed immediately and taken in consideration. 3. The first team to start the design of the new product, gets the right information even before to start. In that way, any possible mistake was eliminated even before it could be originated. 4. The friendship, trust, and enhance performance of each team was raised to another level.Compenetration, trusted feeling, and partnership among all team members were expressed. 5. Different perspectives from a second, third or any level departmental view were taking in consideration since the creation and development of a future product. Everybody opinions was evaluated and taking in consideration regardless the department heshe belongs or hisher job position 6. The interaction among members of different departments increases exponentially. Partnership among members of different department climbs to a higher level. 7. Departmental differences were reduced at his minimal expression.It didn’t matter if you belong to another department; you were a member of my team now. That was the general view for the employees. 8. A clear purpose to achieve the firm‘s goal is a generic task for everybody. 9. The expertise of different level is mixed in each team. 10. The fear to speak out in front of a higher ranking supervisor is not taking in consideration. 11. The unity’s concept binds the team’s purpose. 3. In a era with intense competition and several low-cost products on the market, how can Electrolux use teamwork and groups to succeed? Describe. The concept of teamwork is not control by time or economy.Teamwork is the best develop tool a group can use at any time. The team also must feel security by the firm CEOs, that their job will be maintain regardless the economy downturn. Also incentives and rewards must be given to all teams to keep the motivation flowing among all members. Bibliography Ivancevich, J. , et al. (2011). Organizational behaivor and management. New York, NY: Mcgraw Hill. Ninth Edition Parker, G. (2003). Cross functional teams: working with allies, enemies and other strangers. 6. Retrieved from http://extension. unh. edu/Intranet/teamefct/XfuncTms. pdf

Leadership Underpinning Proposals

Question: Describe about the Leadership for Underpinning Proposals. Answer: Introduction There are various ways and means that leaders can use to lead the people. In this case, the following leaders are going to be used in the analysis (Mahatma Gandhi, Martin Luther King and Steve Jobs). It is considered vital to learn the impact of leadership styles indicated by the above leaders in order to develop individual approach on leadership, this will make one a productive and effective leader. Leadership is considered as one of the success stories that had made significant changes for the betterment of the society. Various scholars have deduced that leadership is a learned behavior, while others have dismissed such claims and considered leadership as a born behavioral trait. Chaudhry and Javed (2012) on the other hand depicts that, the failures of similar organizations are also attributed to poor leadership. Leadership is a behavioral practice that has to be cultured in order to meet the needs and demands of the people. We cannot shy away from good leadership, but we can corre ct or amend our bad leadership styles to conform to better and well-known leadership traits that meets modern or contemporary needs and demands (Chemers 2014). The purpose of this paper is to discuss leadership theories as well as factors that attributed to bad and good leadership (limitations of theories). The above three leaders are known to offer the best in leadership hence able to influence their people to the best. For example, Mahatma Gandhi was a great leader from India was leading concept was based on situational leadership concepts and ideology (John and Taylor 2014). According to situational leadership theory, leaders in this type of category are viewed as changing their leadership tactics based on the level of task ahead of them. This type of leadership can be attained by using participatory style whereby ideas are shared among followers. This leadership style also allows delegation of duties (Shriberg, A, Shriberg and Lloyd 2002). According to Mahatma Gandhi, his leadership was full of selling style-all the decisions taken by the leader are explained to others. Moreover, Preston (2012) illustrated that Mahatma Gandhi used telling style-specific concerns are shared with specific individuals while undertaking their chores and responsibilities. The other leader is Steve Jobs. He is considered as a transformational leader. Under this type of leadership, he was able to use creativity and innovation amid his employees at Apple to drive his leadership style (Susanj and Jakopec 2012). In addition, he used rewarding system to attain the best performance among his employees or followers. Lastly, Martin Luther King was a transformational leader that incorporated charismatic and inspirational values to manage his followers. The key ideology behind this type of leadership is to foster motivation to create or boost performance. It can be deduced that leaders such as the ones discussed above used several leadership style, hence their success in leadership (Preston, 2012) Leader Analysis Steve Jobs To start we are going to examine the leadership qualities, styles, and achievements comparatively and analytically of a man named Steve Jobs, who started small in a garage, in 1976, become separated in 1985 but returned to find it almost bankrupt and in ruins in 1997. By the time he passed away in October 2011 the company became the globes most valuable business entity, and in his life cycle at the helm of apple he changed the world with tablet computing, personal computing, animated movies, mobile phones, digital publishing not to mention retail stores and the huge success they derived. This revitalizing seven industries along the way, IE, communication, software and computer rendering, Pixar to name a few. This places him in midst of giants like Henry Ford, Thomas Edison and Walt Disney, all using their innovation and intelligence to make a better world. The value of interpersonal connections in solidifying a relationship, organizational culture, or business deals, Social capital conveys the notion that strong relationships have a tremendous value of goodwill. Great multicultural leaders are able to generate this goodwill across cultures and inspire strong interpersonal relationships, not only between people very much like themselves, but also among people who may be quite different from them. Steve jobs achieved this through vision and inventiveness of products and systems, which made the world more connected and dissimilar. This created enhanced understandings of cultures and their different ways of seeing each other, according to (Hamm 2006), CEOs who fail to communicate, clarify and relate their vision, will fail to impress upon people the importance of this and create meaningless cultures. Some keys to his success according to Walter Isaacson (2012) viewpoint were that in 1997 taking back an array of computers and peripherals. Moreover, at least a dozen differing Macintosh clones, screaming stop and it is explained how they were going to concentrate on just four computers, by doing this saved the company from bankruptcy, stating what to do is as important a decision as what not to do, thats true for products as well for companies. Simplification strategies like, taking responsibility end to end, whereby he knew the best method to gain simplicity was to guarantee seamless integration of peripheral devices, hardware and software. IE, a Mac with iTunes connected to an iPod allowing problem free, smoother and simpler synchronizing between the devices. Analyzing Steve Jobs effectiveness, Steves early mentor, wrote him a memo urging three basic principles, empathy, focus, and impute. The last being a little awkward but became a key for jobs, he knew people form opinions on companies and products on structures of how it has presented and packaged, that taught him people do judge a book by its cover. These and the hunger for integrating a world with the success of his ingenuity, and vision while being very successful made him a very effective leader that unified and shrunk the world a little more, also saving lives through knowledge sharing genius and inventiveness. Leader Analysis Martin Luther King An example of a modern leader who transformed life, as we know it for many in the USA and for many around the world was Dr Martin Luther King Jr (MLK). Dr King in his journey to racial equality and justice displayed many of the traits the modern study of leadership deems necessary in order to lead a company or be successful in leadership in everyday life, these traits include but are not limited to ethics, multicultural leadership, the psychology of leadership. Through Martin Luther Kings leadership we can also see an example of the four characteristics of a transformational leader (Burns 1978 ), similar to that of Gandhi who transformed the way people thought and acted about social justice and race in India years earlier. In order to properly determine if Dr Martin Luther King would be would be the ideal candidate from Mr Jobs and Gandhi we must first analyses these key traits and behaviors. Trust is the strongest form of currency in leadership (Brunicardi et al. 2007, Shriberg and Shriberg 2011), you cannot have trust without ethics, therefore it is the strength of MLKs ethics that when examined can help explain the strength of his transformational abilities. MLK born into a family of Baptist ministers in Atlanta Georgia was always certain of his convictions, he was a strong moral person and a strong moral manager (Shriberg and Shriberg 2011), this birthed his understanding of life and his value system. The most outstanding of which was to love your brother, MLK believed that he must match hate with love in order to change the opinions of white Americans. On top of this unwavering belief MLK opposed all the trappings of wealth and success, he lived a modest life in order to stay true to himself and stay a part of the community he was fighting for (Clayborne 2005). Such strong morals and ethical convictions made believers out of many people, both black and white. MLKs power to change the opinions and gain leadership over many was displayed most prominently when he matched on the nations capital, he enlisted the help of Caucasian to stand with him and fight the injustice they see around them. This ability to understand other cultures makes MLK, by definition, a multiculturalist; he has transitioned through the stages of acceptance to be able to understand all fellow men. The only other prominently documented example of this is Dr Kings respect for Ghandi (McGuire and Hutchings 2007) and his teachings; he drew from men of all cultures and faiths because of his respect for them, therefore, if MLK was a leader today he would thrive as a leader in a globalized world. Further evidence of MLKs ability to outperform any other leader in todays globalised world is shown when we look at MLKs self doubt and questioning character, he was keenly aware of his limitations. This level of self-understanding along with his ability to understand all other manner of people makes him a person with incredibly high EI, something that is more valuable than intelligence in the work place. Dr King is perhaps one of the best examples of a Transformational leader alongside Ghandi. The philosophy of non-violent protests for social change uplifted people to a higher motivational and moral level in order to create real and lasting change. He did this by utilizing the four key characteristics of transformational leadership (Burns 1978), namely: -Idealized influence: MLK was an upstanding person and someone to model yourself by, because of this influence he achieved this first characteristic (Clayborne 2005). -Inspirational motivation: MLK had the enviable skill of being able to eloquently articulate the change he wanted to see in a clear and systematic way, this allowed for a clear mission statement for people to get behind (Stone et al. 2004). -Intellectual stimulation: MLK held services and talks to inform African Americans of their oppressions; however, he also encouraged his community to take hold of the information, educate themselves to expand their own understanding (McGuire and Hutchings 2007). -Individualised consideration: MLK cared and valued his followers, he personally attended funerals of those who he lost and grieved as one of the community, this personal attachment to his clan allowed for a deep emotional connection to his devotees (Clayborne 2005). It is all the qualities, traits and behaviors combined in the one entity of Dr Martin Luther King Jr that made him one of the greatest transformational leaders in the western world. Those traits would also make him one of the greatest leaders of the globalised business world today. Leader Analysis Mahatma Gandhi Mahatma Gandhi is commonly credited with leading India to its independence from British occupation (Shriberg and Shriberg), (Barnabas and Clifford). However, he was not necessarily a natural leader in his younger years and only shaped his true leadership ambitions following successful litigation in South Africa where he practiced law. It was also, where he experienced personal discrimination and perhaps more importantly fashioned his own personal philosophies. Of particular interest in terms of leadership is the style of opposition Gandhi employed against the British. Prior to Gandhi joining the Indian National Congress Indians had several violent clashes with British soldiers resulting in bloodshed and little progress. Gandhi looked to adopt a strategy of non-violence thus empowering all Indians in their struggle against occupation. In describing Gandhis form of leadership the literature is varied and considers him an ideal representative of servant (Barnabas and Clifford 2012), transformational (Shriberg and Shriberg 2011) and democratic (Ray and Ray 2012) leadership. Servant leadership refers to the concept where a leader has to have experience of acting as a servant (Spears 1996) or experience of his followers challenges. Barnabas and Clifford (2012) argue that Mahatma Gandhi personified the model of servant leadership as a result of him exhibiting qualities such as seeing himself as a servant of the people; acting in service of his followers; displaying integrity and interestingly using moral reasoning to influence others (Barnabas and Clifford 2012). As Shriberg and Shriberg note (2011) transformational leadership results in mutually beneficial relationships between leaders and followers whereby the leader is seeking to promote growth and real change. Thus, transformational leadership is focussed on the growth of both leaders and followers. Of particular interest is the crossover between transformational and servant style leadership styles as described by Shriberg and Shriberg (2011) stating that both transformational and servant leadership are about power to rather than power over. The term democratic leadership is also used to describe Gandhi and refers to a very open and collegial style of running a team (Ray and Ray 2012) and describe Gandhi as an ideal model of democratic leadership in democratic movements (Ray Ray 2012). This assertion is made on the basis that Gandhi worked hard to deliver democracy to India and was focussed on human rights. It can be argued that it is appropriate to describe Gandhis leadership style as servant, transformational and democratic. In terms of servant leadership Gandhi acted like a common man and was seen as a common man by his followers. They were able to relate to him. He acted in service of them, understood the difficulties of the common person and as such can be described as a servant leader. Gandhis concept of non-violent opposition and ability to motivate millions of Indians in this action represents true elevation of both follower and leader. It also required sacrifice by both follower and leader and is a powerful example of transformational leadership. Both India and Gandhi grew and matured as Gandhi led and India followed. The argument for Gandhi as a democratic leader appears less compelling and whilst it is true Gandhi truly symbolizes servant and transformational leadership all three can be used to describe his leadership. Individual Feedback 1 There is nothing good like undertaking a task in a group. We were able to integrate the course materials as well as come with better understanding of the assignment because of the efforts that each member of the group contributed. It is also evident that the group was split into further groups, this made efforts towards undertaking the task an easy one. I was given a task to undertake in the group. This is one of the added advantage that come with group work. I can deduce that there were various merits related to group work, which I was part of. The below are example of the merits: All the work that was done individually was evaluated by other members of the group, as a result, the group was attained the best results ever. During the group was an added advantage in several ways such as timely delivery of the assignment done and that the work did meet the needs and the demands expected. I consider that group work was the best thing that ever happened in my academic life. I was able to understand all the coursework as well as undertake our group work assignment as a team. Lastly, I conclude that if it were not of the teamwork sprit vested in us, we could not have completed the assignment on time, not to say even doing it better than we did as a team. Individual Feedback 2 I consider leadership is the ability of a superior person to introduce to its subordinates the roles and responsibility in confidence and power to handle their chores (Chemers 2014). The group process was able to guide and influence the work of others in order to meet the needs and the demands or to attain certain goals (Herrmann and Felfe 2014). It may also be defined as the position held by group of people or an individual that offers him or her a chance to participate interpersonal influence on others members in order to direct and mobilize their efforts towards attaining certain goals and objectives. The leader is supposed to manage the group, promote group cohesion and move others to attain the goals of the group and to maintain the pace or momentum of the group or team. The leader may arise in the team or group because of his or her personal values and qualities or by the virtue of common consent by other group members. In this case, the leader is able to drive his or her will and power from others team or group members. He is considered to go ahead with the leadership position at will or the team in a collective manner (Herrmann and Felfe 2014). It is considered that leadership exists in most team or group setting regardless of the size of the group. Individual Feedback 3 I consider that there are various theories states that good leaders or effective leaders have a common interest on some of the personalities such as traits and characteristics. According to early trait theories, leadership is innate, it has instinctive qualities that you may or may not have. Thanks to the new values of leadership that denote leadership, qualities can be learned. It is clear that trait theories help to identify the best leadership traits and qualities. However, it is clear that none of the traits or even a combination of the theories will be able to offer or support the success of leadership. The theories also state that traits are external behaviours that come up from our minds. It is clear that the theories are considered as an internal and external processes and beliefs that are vital of effective leadership (Lam and O'Higgins 2012). Therefore, it is ideal to note that the facets of trait theories are there to support of clarify the traits that are related to the ethical systems of a group or an individual when it comes to support of better leadership. It is ideal to note that trait theories have offered better understanding of the values that have to be on board to create a better leader. Leadership is considered as a behavior or values that is developed or created in order to support leadership success (Preston, 2012). Likewise, behavioral theories have limitations as trait theories. First it does not explain the causes of bad leadership especially in the contemporary world, knowing that most of the leaders are known to have better leadership. Lastly, it does not pin the values that are considered important when it comes to provision of best leadership traits (Roueche, Baker III and Rose 2014). 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