Book Review On Life Along The Silk Road History Essay

Book Review On Life Along The Silk Road History Essay The book that I had been given for review is LIFE ALONG THE SILK ROAD. The book offers a glimpse into the character and characters of the Eastern Silk road between AD 750 and 1000. The author of the book Susan Whitfield is the director of the British Library sponsored Dunhuang project, which makes a remarkable collection of Ancient Silk Road manuscripts including those acquired by legendary explorer Sir Auel Stein, available on the internet. Her knowledge of this treasure trove of primary material shows throughout the book. She has written extensively about china and therefore is in a good position to give an account of the all the details regarding the network of roads and paths crossing central Asia and concentrates on the 8th to 10th centuries A.D. The author has an extensive research on the Mongolian Empire and middle kingdom. She has travelled to central Asia several times and has written this novel as reflection of the stories of the Silk Road. She has dedicated the book to prof.Edward Schafer whom she credits for literary excavation of this historic trade route. This book Life along the Silk Road gives a rich account of the varied history of the Silk Road. It is a good read for people with special interest in history. The book recounts the stories, the lives of ten individuals who lived along the Silk Road in different era. The tale of ten different individuals a merchant, a soldier, a horseman, a monk, a nun among others, all form a different walk of life. The author has tried to reconstruct the history of the route through the personal experiences of these characters. The region covered in the book corresponds to modern day eastern Uzbekistan, western China, Mongolia, south to the Himalayas and including Tibet. Today that region is largely occupied by Turkic peoples, mainly the Uighur, as well as Chinese colonists and is more Islamic than not. In the time period covered by the book it was more Indo-European in character, mainly Buddhist, and a great deal more cosmopolitan, with many towns and cities home to Turks, Indians, Chinese, Tibetans, and Mongolians as well as followers of Manicheism, Zoroastrianism, Nestorian Christianity, Islam, Judaism, and shamanism. Many Silk Road towns, once some of the most populous cities in the world, now have largely been reclaimed by the desert sands due to a decline in population and a drop in the water table, a land now rich in archaeology but vulnerable to thieves looking for artifacts to sell on the black market. The major source of information for this book and indeed much of the scholarship done on this region and era comes from the over forty thousand documents uncovered in a Buddhist cave complex outside Dunhuang, now in Gansu province, China. Sealed up in the eleventh century, it was uncovered by accident in 1900. Though many of these precious scrolls, paintings, and sculptures have been lost since then for various reasons (and others tainted by the existence of forgeries), more than enough remained; the importance of the Dunhuang documents cannot be overstated. A whole field of study, Dunhuangology, grew up around the study of the documents. Not only were there many Buddhist texts, but as paper was rare and often recycled (and once Buddhist scripture was written on paper it was considered nearly blasphemous to destroy at that point), many non-Buddhist writings were preserved, unique in providing glimpses into the lives of everyday people. First chapter of the book was the most informative and wide ranging. It takes reader back in 2nd century and helps in understanding what events took place for which during the mid 8th century when the silk route was as its peak. In this chapter reader learns that there was not one Silk road but multiple paths and that also it was not only silk that was traded along it, horses, salt, wool, jade were also major trade items. The distances covered by the merchants around 3000 miles was indeed a challenge for them. Though Silk Road was of major important for centuries by the end of 10th century trade became increasingly maritime in nature. Following the introduction chapter, author begins to tell tale of ten individuals. The author diligently combines information from texts and archaeological discoveries with her own imagination in order to describe her characters and their families and acquaintances to discuss events in which they participated or which they observed and to indicate their feelings and thoughts. Each tale is full of information about material life that is detail of clothing, eating habits and other details gives the reader a feel for the place and time. The stories along with the numerous pictures present in this book give the reader and insightful into the life of ordinary men and women on the Silk Road era. Following the lives and stories of the Merchant, the Soldier, the Monk, the Courtesan, and others, Susan Whitfield brings the dramatic history of pre-Islamic central Asia down to a human scale, describing the battles of conquest and trade with the details of everyday life. One significant point that can be grasped from the book is the history of Central Asia over this period characterized by a complex succession of power struggles. The lives of ten individuals in this book were greatly affected by the rise and fall of Chinese, Tibetan, Uighur, Arab, and the other powers (such as autonomous city-states like Samarkand) that continually fought for control of the eastern Silk Road. when an empire was not defeated on the battlefield it could collapse or fall into chaos due to serious internal disturbances, such as a 755 rebellion led by a general of the Chinese army against the Tang dynasty and when earlier that same year the Tibetan emperor was murdered during a revolt by his ministers. From the 11th century onwards the culture of the region changed and the Road declined as Islamic culture overtook Chinese influence and the sea route grew in importance. With the rise in sea explorations, overland trade routes became sea routes with Marcopolo set sail to trade the Indians, who were renowned for the spices, textiles and ivory products. Towns near present day Afghanistan saw abandoned villages and the trade declined. The author of the book has presented history in a vivacious manner. At some pages reader may feel elated by a time machine: one hears the sounds, smells the smells and hears the multilingual crowds in the capital of Changan or the various desert posts. The major quality of the book is that it is written not only from Chinese point of view but also contains ample information about the people of central Asia. It is not important to have knowledge about the history of china in order savor the stories in the book. Small and fine details about different characters in the books give the reader a feel as if one is itself present there. The wealth of historical data present in the book which one would like to read in one sitting is an inhuman undertaking given the sheer joy and shock of all the little anecdotes, background facts and human insights. For a student like me it is easy to understand history through understanding the lives of the people who lived there and then instead of going through the tedious information regarding the places and dates of battles and monarchial successions. Traditional histories are normally about who won the battle where. But after having read this book I can fittingly say that the author has conscientiously tried to accommodate people like me by showing Central and Eastern Asias history during the prime days of the Silk Road through a series of brief vignettes representing the lives of various types of people who lived then. I found the writing style of the author quite stiff and solid but the technique she used to present the glorious history is quite effective and fascinating. Her depiction of the Silk Road through her unique style drew me in with everyday detail from the period. She presented the greater historical details, like Chinese dynastic changes and which nations gained superiority a t what time, into a context I could understand and enjoy reading. Apart from the above mentioned qualities there are some problems in the book as well though relatively minor is nature. First of all it should be stated at the outset that this book is, in fact, generally unsuitable as a resource for scholars or teachers, for it is in essence a work of historical fiction. While the book may indeed offer the reader images of life along the Silk Road, it cannot be considered an accurate scholarly resource, since it does not make clear to the reader what is imaginative and what is not. Secondly then trouble with names and terminology emerges almost from the outset, after stating that she will employ k rather than q in Turkic terms and names (i.e., Kocho rather than Qocho), since that will make them more accessible to English speakers, the author then informs the reader that she will use Beshbaliq (not Beshbalik, as one would expect) for the city known in Chinese sources as Beiting, this certainly makes understanding more difficult. Thirdly as one would expect in a book intended for the general market, this work contains neither annotation nor a bibliography, although it does offer suggestions for further reading (pp. 226-29). The book has numerous illustrations, including color and black-and-white photographs, and maps. It also has a Table of Rulers, 739-960 (pp. 230-31) that containsfor no clear reason other than much of it has been taken directly (and not without the introduction of errors) from one of Whitfields sources (6)Frankish and Byzantine monarchs as well as those of the Islamic world, China, Tibet, the Turks and Uighurs. There is an index of subjects and proper names as well. In the end I ill just like to conclude by saying that for the general reader, this book may well serve as an engaging and lively evocation of its subject. For the scholar, teacher, or serious student, however, its problems are sufficiently great as to limit its utility.

Light in August Essay

William Faulkner’s celebrated novel Light in August says the tale of an orphan Joe Christmas, who has an unknown or doubtful ancestry who considers himself to be part-black. The novel includes excellent themes that involve issues of gender, race, community and religion. The capture of the racist identity and the portrayal of Calvinistic religion are the main features of the novel. Light in August is one of the best works of William Faulkner. The story starts with Lena Grove, a helpless pregnant girl (from Alabama) who set out of her house in search of her unborn baby’s father Lucas Burch. Lucas Burch had actually escaped from Lena so that he need not bear the burden of fatherhood. Lena’s hunting for Lucas Burch make her reach another man with the same name Byron Bunch. He was a simple man who makes friendship with Lena and later falls in love with her. Byron was kindhearted and provided Lena with home and security that the actual Byron Bunch (the child’s father) could not provide. After that we can see Byron narrating his feelings for Lena to the former Presbyterian minister Rev. Gail Hightower who was pushed to withdraw from his ministry as his wife committed suicide. In the middle of this strange story between Lena Grove and Lucas Burch, there appears Joe Christmas the roommate of Lucas Burch. He was an orphan working in the planing mill like Byron Bunch. The word ‘Christmas’ was added to his name as it was on the day of Christmas that he reached the orphanage. Christmas was also not sure about his ancestry as he wandered across in search of his lineage; a voyage for self-discovery. Christmas’ journey led him to Jefferson, Mississippi where he falls in a dangerous relation with Joanna Burden spinsterish civil rights activist. This affair led him to grave destruction. The most interesting theme of Light in August is the Southern issue with racial identity. Joe is mistreated and humiliated as people suspect that he has a black ancestry. They never him the way they treat white people (Dondlinger, 98-125). We can find in Light in August that majority of the characters are influenced by the notion of race disparity. For example, Joanna Burden, Joe Christmas, Doc Hines, Nathaniel Burde and lastly Percy Grimm are some or the other way influenced by the concept of race (Towner, 45-65). Some of the characters become victims of the senseless racial classification and suffer. Others who are out of this threat believe that there is nothing wrong in treating blacks inhumanly on the basis of racial difference. Racial discrimination is quite acceptable for them (Dondlinger, 98-125). For example The Jefferson sheriff, Watt Kennedy appears to be a decent man. However we can see him punishing a randomly chosen black person in an unwanted investigation (appeared unwanted at first). Similarly several other characters are found to be under the grip of the notion of race. Gender also plays a crucial role in the novel Light in August. Male-female identity and relations is an important topic. For example, we can find Joe being hostile to women. Lena’s baby’s father Lucas Burch also ran away from his pregnant wife; he was also fleeing away from women. Byron Bunch also did not have any plans to marry and was living a lonely life till Lena comes. He was adjusting his life in such a way that he could continue living alone. Rev. Gail Hightower also do not have a good fellowship with women as he led his wife to suicide. Another character Joanna Burden does not even marry. However we can find that the author never attempts to contrast the lonely life of these unfortunate characters with any happy or satisfied normal family or love relationships until the novel reaches the last chapter. We can see that the marriage of Hines and McEachern was also not a happy one. Moreover the Armstids also do not appear to have any love among them. Finally it appears to come to a happy end when Byron Bunch moves away with Lena Grove. But disappointment suddenly comes in she refuses him to share her bed. That relation also seems to be failure. The only relationship that appears to be happy is the relationship of the furniture dealer and his wife. They only had a relationship that had any trace of happiness or satisfaction.

The Sociology of the Hunger Games Essay - 1026 Words

In a not-too-distant, some 74 years, into the future the United States of America has collapsed, weakened by drought, fire, famine, and war, to be replaced by Panem, a country divided into the Capitol and 13 districts. Each year, two young representatives from each district are selected by lottery to participate in The Hunger Games; these children are referred to as tributes (Collins, 2008). The Games are meant to be viewed as entertainment, but every citizen knows their purpose, as brutal intimidation of the subjugated districts. The televised games are broadcasted throughout Panem as the 24 participants are forced to eradicate their competitors, literally, with all citizens required to watch. The main character†¦show more content†¦It is not said but possibly assumed that this was done to keep the district from rising up together. In order for the government to justify the Hunger Games, it keeps its minor districts in a constant state or poverty, requiring the childre n to take food rations in return for having their name entered in the lottery. Each ration counts your name once and if your name is called, you must enter the Games and fight your fellow district members to the death (2008). These acts while claiming they promote courage, honor, and determination in its citizens and participants do nothing more but enforce the submissiveness and loyalty of the society. Through both of these things, the American values of activity and work and efficiency and practicality (2003) are the only ones that survive to our grandchildren’s generation. That is unless you find racism and group superiority to be a valid type of value. It does not seem that in any state that education, democracy, equality, progress, or freedom (2003) are even thought of let alone key to the running of the society. These values differ from that of the Capitol, which is exempt from the Hunger Games. It appears that their only purpose is to use the goods and services th at the districts create and to do so unscrupulously (2008). The citizens of theShow MoreRelatedSociology of Hunger Games1719 Words   |  7 Pagesï » ¿ Sociology of â€Å"Hunger Games† By. Tom ************ Soc 101 11/17/13 Introduction The nation of Panem has risen out of the ravaged ruins of what was once known as North America. 74 years ago, the poverty-stricken districts of Panem rebelled against the wealthy, controlling the Capitol. After its crushing victory, the Capitol devised the Hunger Games as an annual reminder to the twelve districts of its authority, and as continuing punishment for the rebellionRead MoreHunger Games Sociology Essay1098 Words   |  5 PagesSuzanne Collins’ novel The Hunger Games and its big screen adaptation follow the journey of a teenage girl in the dystopian country of Panem. 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The friends that we make are merely for leeching on when we forget homework or tests dates, and thus we stop caring for each otherRead MoreAlice Pauls Determination in Womens Suffrage833 Words   |  4 PagesIt was January 1 1, 1885 and in Moorestown, New Jersey what I would call a rook in the chess game of women’s suffrage, was born. It’s hard to believe that such an overwhelming infatuation in equality could be so deeply immersed in a woman only twenty-seven years of age. However, when you know that this person is none other than Alice Paul, believing gets easier. It was the defiance caged up inside this fire-cracker of a woman that led her steadily through the great battle of womans suffrage. Read MoreLiterary Scholarship And Criticism Of Shakespeare s The Invention Of The Human 1309 Words   |  6 Pagesthrough a range of individuals from different social constructs in order to contemplate the human condition. Hamlet: The Prince of Denmark is a hallmark example of Shakespeare’s contemplation of social tolerance and equity in the exploration of human sociology. In this play, we see the malignant ambition of newly throned King Claudius, the questionable decision-making of Queen Gertrude, and the emotionally-preying and encompassing force of murdered King Hamlet’s ghost, construct the grounds for contemplation

Who Was the Inventor of the iPod

On October 23, 2001, Apple Computers publicly introduced its portable music digital player the iPod. Created under project codename Dulcimer, the iPod was announced several months after the release of iTunes, a program that converted audio CDs into compressed digital audio files and allowed users to organize their digital music collection. The iPod turned out to be one of Apples most successful and popular products. More importantly, it helped enable the company to return to dominance in an industry where it had been losing ground to competitors. And while Steve Jobs has largely been credited with the iPod and the companys subsequent turnaround, it was another employee who is considered to be the father of the iPod.   Who Invented the iPod? Tony Fadell was a former employee of General Magic and Phillips who wanted to invent a better MP3 player. After being turned down by RealNetworks and Phillips, Fadell found support for his project with Apple. He was hired by Apple Computers in 2001 as an independent contractor to lead a team of 30 people to develop the new MP3 player. Fadell partnered with a company called PortalPlayer who had been working on their own MP3 player to design the software for the new Apple music player. Within eight months, Tony Fadells team and PortalPlayer completed a prototype iPod. Apple polished the user interface, adding the famous scroll wheel. In a Wired magazine article titled Inside Look at Birth of the iPod, former senior manager  Ben Knauss at PortalPlayer revealed that Fadell was familiar with PortalPlayers reference designs for a couple of MP3 players, including one about the size of a cigarette packet. And though the design was unfinished, several prototypes had been built and Fadell recognized the designs potential. Jonathan Ive, Senior Vice President of Industrial Design at Apple Computers, took over after Fadells team had finished their contract and kept perfecting the iPod itself. iPod Products The success of the iPod led to several new and upgraded versions of the wildly popular portable music player. In 2004, Apple introduced the iPod Mini — a smaller, more portable music player that featured a 138x110 LCD screen and an easy-to-use interface with click wheel to scroll through playlists and options.In 2005, Steve Jobs debuted the smallest iPod model, called the iPod Shuffle. It was the first iPod to  use faster and more durable flash memory to store music files.  The iPod Mini was replaced in late 2005 by the iPod Nano, which also featured flash memory. Later generations offered a color LCD screen.In 2007, Apple released the sixth-generation iPod, called the iPod Classic, which featured a thinner, metallic design, improved battery life, and  up to 36 hours of music playback and six hours of video playback.  In 2007, Apple also released the iPod Touch, the first iPod product with a touch screen interface similar to the iPhone. Besides playing music, users can play videos, snap photos, and play video games. Fun Facts Apparently, Fadell is quite a character. He was once asked where he would be in life if hed grown up before computers were invented. Fadells response was In jail.What was the first song played using iTunes, Apples proprietary software? It was a house-music dance tune called Groovejet (If This Aint Love).The first generation iPods had scroll wheels that physically rotated. Post-2003 iPods (third generation) have touch-sensitive wheels. Fourth generation (2004) iPods have buttons integrated onto the wheel.The iPods wheel technology can measure changes in position greater than 1/1,000th of an inch. Sources Kahney, Leander. Inside Look at Birth of the iPod. Wired, July 21, 2004. McCracken, Harry. Before iPod and Nest: Fast Company’s 1998 Tony Fadell profile. Fast Company, June 4, 2016.

Code 100 - 1410 Words

Running head: Executive Management Project Code 100 Chamberlain College of Nursing EX630: Executive Practicum Professor Valda Upenieks 6/16/2012 The Master of Science Nursing student with the collaboration of my nurse executive mentor Mrs. Darville created a project that is valuable for the facility and a rewarding experience for me as a graduate student. My personal practicum experience took place at St Elizabeth’s Hospital, which is a small 90 bed community hospital located in Gonzales, La. St Elizabeth’s Hospital is a part of the Franciscan Missionaries of Our Lady Health System which was organized in 1984 to unite with three existing hospitals in Louisiana which were already a part of the system†¦show more content†¦The RRT intervenes before a crisis, which the Code 100 will be to activate the RRT, as a result lives are saved and adverse events are prevented. Data collection from the initiative has proven to be a beneficial tool; it has decreased code blue calls and has produced positive outcomes (Rankins, 2006). The IHI initiative included clinicians in collaboration with hospital and nurse leads to implement best practice protocols to prevent unnecessary deaths. The RRT deployed at first sign of patient decline does save lives (Richard, 2005). Objectives Code 100 implementation will prevent avoidable inpat ient deaths that are not an expected or outcome of a patient’s hospital stay (MHA, 2012). Code 100 which is the first section of a larger project that will require the staff nurses to document whenever they care a physician especially during the day shift and document all the signs and symptoms the patients’ exhibit and the intervention performed. The nurses will have a form to complete and the data will be collected to track the number of incidents and the outcome. The facility will put in place a Rapid Response Team in the coming weeks which will be activated when a nurse on the floor observe signs of deterioration from a patient to prevent the declination of the patient’s status. Mock Code will also be implemented to ensure the staff is ready when a Code 100 is calledShow MoreRelatedProcedures And Procedures Of Medical Coding871 Words   |  4 Pagesdetailed process in which health care professionals use to code an insurance claim, to submit for reimburseme nt of services. In the following scenario, my task would be to explain the process and steps I took, to code patients cases using ICD-9. When coding it’s very important to code correctly to avoid delays in reimbursement and proper reimbursement. The following will explain the steps I took to perform proper medical coding. When I code with ICD-9 I generally weed out the diagnosis that has beenRead MoreFinancial Analysis and Market Updates Essay1492 Words   |  6 Pagesinclude distribution of petroleum products, transportation, refinery, sale, Water projects , Lands reform, and distribution of crude oil and petroleum products in the Kingdom of Saudi Arabia and internationally. KINGDOM key parameters Bloomberg code SARCO:AB Target price (SAR)* Market Cap (Millions) 1,815.000 Price/Book Value 1.94 P/E 210.108 ROE 0.566 Beta* 0.963 EPS TTM SR0.23 Purchase price SAR * Beta vs. SASEIDX taken from Bloomberg.com Sector cooperative returns SARCO:ABRead MoreQuestions On Data, Computing And Information Essay1150 Words   |  5 Pages Faculty of Computer Studies Course Code: M150A TMA Course Title: DATA, COMPUTING and INFORMATION – A Student Name: Mohamed Khaled Ahmed Mohamed. Student ID : 1551310036. Date: â€Æ' Question 1) What is the difference between Data and Information? Data is raw. Computers need data. 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Due to the recent scandals in corporate America many companies have acted quickly in actually implementing rules and regulations. They are sometimes referred to as the ethical codes. In this paper I will discuss the importance of corporate governance and ethical codes within a corporation. How has the recent crackdown changed corporate America? Has there been an improvement? Does the gender of top management have an impact on the company’s performance? Are we able to

Simplification of Switching Function Free Essays

EEN1036 Digital Logic Design Chapter 4 part I Simplification of Switching Function 1 Objective s s s s Simplifying logic circuit Minimization using Karnaugh map Using Karnaugh map to obtain simplified SOP and POS expression Five-variable Karnaugh map 2 Simplifying Logic Circuits †¢ †¢ †¢ A A Boolean expression for a logic circuit may be reduced to a simpler form The simplified expression can then be used to implement a circuit equivalent to the original circuit Consider the following example: B C A B C + A BC Y AB C + AB C Y = A B C + A BC + AB C + AB C 3 Continue †¦ Checking for common factor: Y = A B C + A BC + AB C + AB C = A C ( B + B ) + AB (C + C ) Reduce the complement pairs to ‘1’ Y = A C ( B + B ) + AB (C + C ) = A C + AB Draw the circuit based on the simplified expression A B C Y 4 Continue †¦ †¢ A Consider another logic circuit: B C Y Y = C( A + B + C ) + A + C Convert to SOP expression: Y = C( A + B + C ) + A + C = AC + B C + AC Checking for common factor: Y = A(C + C ) + B C = A + BC 5 Continue †¦ †¢ †¢ Simplification of logic circuit algebraically is not always an easy task The following two steps might be useful: i. The original expression is convert into the SOP form by repeated application of DeMorgan’s theorems and multiplication of terms ii. The product terms are then checked for common factors, and factoring is performed wherever possible 6 Continue †¦ †¢ Consider the truth table below: A 0 0 0 0 1 B 0 0 1 1 0 C 0 1 0 1 0 Y 0 0 1 0 0 Minterm Boolean expression: Simplify to yield: Y = A BC + ABC + AB C Y = BC ( A + A) + AB C = BC + AB C 1 0 1 1 1 1 0 1 1 1 1 0 †¢ If minterms are only differed by one bit, they can be simplified, e. We will write a custom essay sample on Simplification of Switching Function or any similar topic only for you Order Now g. A BC ABC 7 Continue †¦ †¢ More example: A 0 0 0 0 1 1 1 1 B 0 0 1 1 0 0 1 1 C 0 1 0 1 0 1 0 1 Y 0 1 1 0 0 1 1 0 Minterm Boolean expression: Y = A B C + A BC + AB C + ABC Minterms 1 and 5, 2 and 6 are only differ by one bit: Y = B C ( A + A) + BC ( A + A) = BC + B C A B C Y 0 0 0 1 0 0 1 0 0 0 1 1 1 1 1 1 0 0 1 1 0 1 0 1 0 1 1 0 1 0 1 0 Minterm Boolean expression: Y = A B C + A BC + AB C + ABC Checking and factoring minterms differed by only by one bit: Y = A C ( B + B ) + AC ( B + B ) = A C + AC = C ( A + A) =C 8 Continue †¦ †¢ †¢ †¢ Though truth table can help us to detect minterms which are only differed by one bit, it is not arranged in a proper way A Karnaugh map (K-map) is a tool, which help us to detect and simplify minterms graphically It is a rearrangement of the truth table where each adjacent cell is only differed by one bit By looping adjacent minterms, it is similar to grouping the minterms with a single bit difference on the truth table 9 K arnaugh Map †¢ †¢ A K-map is just a rearrangement of truth table, so that minterms with a single-bit difference can be detected easily Figure below shows 4 possible arrangement of 3-variable K-map A BC 0 0 01 1 11 3 10 2 C AB 00 0 01 2 11 6 10 4 0 1 4 5 7 6 0 1 1 3 7 5 AB C 0 0 1 1 BC A 0 0 1 4 00 01 2 3 00 01 1 5 11 6 7 11 3 7 10 4 5 10 2 6 10 Continue †¦ †¢ Figure below show two possible arrangement of 4variable K-map CD AB 00 0 01 1 11 3 10 2 AB 00 CD 01 4 11 12 10 8 00 01 4 5 7 6 00 0 01 1 5 13 9 11 12 13 15 14 11 3 7 15 11 10 8 9 11 10 10 2 6 14 10 †¢ Notice that the K-map is labeled so that horizontally and vertically adjacent cells differ only by one bit. 11 Continue †¦ †¢ The K-map for both SOP and POS form are shown below: C D C D CD C D AB AB AB 0 1 3 2 C+D C+ D C + D C +D A +B 0 1 3 2 4 5 7 6 A+B A+B A +B 4 5 7 6 12 13 15 14 12 13 15 14 AB 8 9 11 10 8 9 11 10 SOP form (minterm) POS form (maxterm) †¢ †¢ The simplified SOP expression can be obtained by properly combining those adjacent cells which contains ‘1’ This process of combining adjacent minterms is known as 12 looping Continue †¦ †¢ †¢ Each loop of minterms will form a group which can be represented by a product term When a variable appears in both complemented and uncomplemented form within a group, that variable is eliminated from the product term C D C D CD C D AB AB AB AB 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 group 2 group 1: C D( AB + AB ) = AC D group 2: AB(C D + CD ) = ABD Simplified SOP expression: Y = AC D + ABD 13 group 1 Continue †¦ †¢ Consider another K-map: C D C D CD C D AB AB AB AB 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 group 1 C D C D CD C D AB AB AB AB 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 14 group 1: ( A B + AB )(C D + CD ) = BD Simplified SOP expression: Y = BD group 1: C D ( A B + A B + AB + AB ) = C D Simplified SOP expression: Y = CD group 1 From truth table to K-map †¢ The content of each cell can be directly plot on the Kmap according to the truth table Consider the following example: 0 1 2 3 4 5 6 7 A 0 0 0 0 1 1 1 1 B C Y 0 0 1 0 1 1 1 0 1 1 1 0 0 0 0 0 1 0 1 0 1 1 1 0 B C B C BC B C A A 1 0 1 1 0 3 1 2 0 4 0 5 0 7 1 6 AB BC Simplified SOP expression: Y = A B + BC 15 Continue †¦ †¢ Consider the following 4-variable K-map A 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 B 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 C D Y 0 0 0 0 1 1 1 0 0 1 1 0 0 0 0 0 1 1 1 0 0 1 1 0 0 0 0 0 1 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 0 1 1 1 C D C D CD C D AB AB AB 0 0 0 1 1 1 0 1 0 0 1 0 3 0 0 0 ACD 2 4 5 7 6 12 13 15 14 AB 0 8 9 11 0 10 ABD Simplified SOP expression: Y = A C D + ABD 16 Continue †¦ †¢ Some guidelines: i. Construct K-map and fill it according to the truth table ii. Only loop cells in the power of 2, i. e. 2 cells, 4 cells, 8 cells and so on iii. Always start by looping the isolated minterms iv. Look for minterms which are adjacent to only one minterm and loop them together v. Proceed on to loop the largest possible groups, from eight minterms (octet), 4 minterms (quad) to 2 minterms (pair) vi. Obtain the product term for each group vii. The sum of these product terms will be the simplified SOP expression 17 Continue †¦ Example: a. Obtain the simplify SOP expression for the truth table: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 A 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 B 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 C 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 D 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 Y 0 0 1 0 0 1 0 1 0 0 0 1 0 1 0 1 C D C D CD C D AB AB AB AB A B CD 0 0 0 0 0 0 1 1 0 1 0 1 1 3 1 0 0 0 2 4 5 7 6 12 13 15 14 8 9 1 11 10 BD ACD Simplified SOP expression: Y = A B CD + ACD + BD 18 Continue †¦ b. Obtain the simplify SOP expression from the K-map: ACD C D C D CD C D AB AB ABC 0 0 1 0 1 1 1 0 0 1 1 1 ACD 0 1 0 0 A BC AB AB Simplified SOP expression: Y = A C D + A BC + ACD + ABC 19 Continue †¦ c. Obtain the simplify SOP expression from the K-map: alternative solution: C D C D CD C D AB AB AB C D C D CD C D AB A CD 0 0 0 0 AC D 0 0 1 1 1 1 0 1 0 0 0 0 AB D 0 0 0 0 AC D 0 0 1 1 1 1 0 1 0 0 0 0 B CD A CD AB AB AB AB Y = A CD + AC D + AB D Y = A CD + AC D + B CD 20 General Terminology for Logic Minimization †¢ †¢ Here, we define four terms to provide the basis for general function minimization techniques These terms are implicant, prime implicant, essential prime implicant and cover We refer to the K-map below in explaining each term B C B C BC B C A A 1 0 1 1 3 2 1 4 1 5 1 7 6 †¢ †¢ An implicant is a product term that could be used to cover minterms of the function In the K-map above, there are 11 implicants: 5 m interms: {A B C , A BC , AB C , AB C , ABC} 5 group of two adjacent minterms: {AB , AC , A C , B C , BC} 1 group of four adjacent minterms:{C} 21 Continue †¦ †¢ †¢ †¢ A prime implicant is an implicant that is not part of any other mplicant In the K-map, there are two prime implicant: C and AB An essential prime implicant is a prime implicant that covers at least one minterm that is not covered by any other prime implicants Prime implicant AB is essential as it is the only prime implicant that covers minterm 4 Prime implicant C is also essential as it is the only prime implicant that covers minterm 1, 3 and 7 A cover of a function is a set of prime implicants for which each minterm of the function is contained in (covered by) at least one prime implicant All essential prime implicants must be used in any cover of a function 22 †¢ †¢ †¢ Continue †¦ †¢ †¢ For the K-map above, the set of implicants { AB , C} represents a cover of the fun ction A minimum cover contains the minimum number of prime implicants which contains all minterm in the function Consider the 4-variable K-map below: C D C D CD C D AB AB AB AB 1 1 Prime implicants †¢ C D C D CD C D AB AB AB 1 1 1 1 1 1 1 1 1 AB AB AB AB C D C D CD C D 1 1 1 1 Minimum cover 1 1 1 1 1 1 1 1 1 1 1 1 AB Essential prime implicants 23 Continue †¦ †¢ Consider another K-map C D C D CD C D AB AB AB AB 1 1 1 1 1 1 1 Prime implicants C D C D CD C D AB 1 1 1 1 1 1 1 1 1 1 AB AB 1 AB Essential prime implicants (minimum cover) 24 Don’t Care Conditions †¢ †¢ †¢ †¢ Some logic circuit will have certain input conditions whereby the output is unspecified This is usually because these input conditions would never occur In other words, we â€Å"don’t care† whether the output is HIGH or LOW Consider the following example: An air conditioning system has two inputs, C and H: – C will be ‘1’ if temperature is too cold (below 15 °C) Otherwise, it will be ‘0’ – H will be ‘1’ if temperature is too hot (above 25 °C) Otherwise, it will be ‘0’ – Output Y will be ‘1’ if temperature is too cold or too hot. If the temperature is acceptable, Y will be ‘0’ 25 Continue †¦ As there are two inputs, there are 4 possible logical conditions: C 0 0 1 1 H 0 1 0 1 Y 0 1 1 X meaning just nice too hot too cold ? Input condition C = 1, H = 1 has no real meaning, as it is impossible to be too hot and too cold at the same time We put a ‘X’ at the output corresponds to this input condition as this input condition cannot occur 26 K-map and Don’t Care Term †¢ Don’t care term, ‘X’ can be treated as ‘0’ or ‘1’ since they cannot occur In K-map, we can choose the don’t care term as ‘0’ or ‘1’ to our advantage A B C D Y 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 1 1 X 0 1 0 0 1 0 1 0 1 X 0 1 1 0 0 0 1 1 1 X 1 0 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 1 1 X 1 1 0 0 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 X C D C D CD C D AB AB AB 0 1 1 0 1 X 1 0 X X X X 0 0 1 0 AB Simplified Boolean expression: Y = AB + BC + A D 27 More examples †¦ C D C D CD C D AB AB AB AB C D C D CD C D AB AB AB AB 1 1 X 1 0 1 X 1 0 0 X X 0 1 X X 1 0 X 1 0 0 X 0 0 0 X X 1 X X Y = C D + BC + BD + A C D C D CD C D AB AB AB Y = B D + CD C D C D CD C D AB AB AB 0 0 1 0 1 X 1 1 0 1 X 0 0 0 0 0 1 1 X 0 1 X X 1 0 1 X X 0 0 X X 28 AB AB Y = ABC + C D + BD Y = A C + BD + AD Plotting function in Canonical Form †¢ Logic function may be expressed in many forms, ranging from simple SOP/POS expression to more complex expressions However, each of them has a unique canonical SOP/POS form If a Boolean expression is expressed in canonical form, it can be readily plotted on the K-map Consider the following Boolean expression: †¢ †¢ †¢ Y = ABC + B C Convert to canonical SOP expression: Y = ABC + B C ( A + A) = ABC + A B C + AB C 29 Continue †¦ Y = ABC + A B C + AB C Plotting the canonical SOP expression onto K-map B C B C BC B C A A 1 1 0 0 BC 0 0 0 1 AC Simplified SOP expression: Y = B C + AC †¢ Consider plotting the following Boolean expression on K-map: Y = C ( A ? B) + A + B 30 Continue †¦ First, convert to SOP expression Y = C ( A ? B) + A + B = C ( AB + A B) + A B = AB C + A BC + A B (C + C ) = AB C + A BC + A B C + A B C B C B C BC B C A A 1 0 AB 1 1 1 0 BC 0 0 AC ?Y = A B + B C + A C 31 Plotting K-map from SOP expression †¢ †¢ It is sometime too tedious to convert a Boolean expression to its canonical SOP form Consider the following Boolean expression: Y = AB (C + D )(C + D ) + A + B Convert to SOP form: Y = ( AB C + AB D )(C + D ) + A B = AB C D + AB CD + A B Convert to canonical form: Y = AB C D + AB CD + A B (C + C )( D + D) = AB C D + AB CD + ( A B C + A B C )( D + D) = AB C D + AB CD + A B C D + A B C D + A B CD + A B CD 32 Continue †¦ Y = AB C D + AB CD + A B C D + A B C D + A B CD + A B CD Plot the minterm on K-map: C D C D CD C D AB AB AB 1 0 0 1 1 0 0 0 1 0 0 1 1 0 0 0 AB AB B CD BC D Simplified SOP expression: Y = B C D + B CD + A B 33 Continue †¦ †¢ †¢ †¢ †¢ †¢ Boolean expression can be plotted on to the K-map from its SOP form Product terms with four variables are the minterms and correspond to a single cell on the K-map Product term with three variables corresponds to a loop of two adjacent minterms Product term with only two variables is a quad (a loop of four adjacent minterms) Product term with a single variable is an octet (a loop of eight adjacent minterms) 1 cell 2 cells Y = A + BC + B CD + ABCD 4 cells 8 cells 34 Continue †¦ †¢ Consider the previous example: Y = AB C D + AB CD + A B minterms 4 cells †¢ †¢ †¢ Both minterms are directly plotted on the K-map The loop which corresponds to A B is drawn on the K-map The cells inside the loops are filled with ‘1’ C D C D CD C D AB AB AB AB 1 0 0 1 1 0 0 0 1 0 0 1 1 0 0 0 AB AB C D A B CD 35 Continue †¦ †¢ Consider the following Boolean expression: Y = ( A + B )( AC + D ) Convert to SOP form: Y = AC + AD + ABC + BD Plot the SOP onto K-map C D C D CD C D AB AB AB AB AC BD C D C D CD C D AB AB ill cells in loops with ‘1’ 0 0 0 0 0 1 1 1 0 1 1 1 0 0 1 1 36 ABC AB AB AD Continue †¦ Obtain the simplified SOP expression from K-map: C D C D CD C D AB AB AB AB 0 0 0 0 0 1 1 1 0 1 1 1 0 0 1 1 Simplified SOP expression: Y = AC + AD + BD 37 Continue †¦ Example: Redesign the logic circuit below from its simplified SOP expression: A B C D Z Z = ( B + D )( B + D ) + B(CD + A D ) 38 Continue †¦ Z = ( B + D )( B + D ) + B(CD + A D ) = B + D + B + D + BCD + A BD = BD + B D + BCD + A BD C D C D CD C D AB AB AB 1 1 0 1 0 1 1 0 0 1 1 0 1 1 0 1 AB Z = BD + B D + A B 39 How to cite Simplification of Switching Function, Papers Simplification of Switching Function Free Essays EEN1036 Digital Logic Design Chapter 4 part I Simplification of Switching Function 1 Objective s s s s Simplifying logic circuit Minimization using Karnaugh map Using Karnaugh map to obtain simplified SOP and POS expression Five-variable Karnaugh map 2 Simplifying Logic Circuits †¢ †¢ †¢ A A Boolean expression for a logic circuit may be reduced to a simpler form The simplified expression can then be used to implement a circuit equivalent to the original circuit Consider the following example: B C A B C + A BC Y AB C + AB C Y = A B C + A BC + AB C + AB C 3 Continue †¦ Checking for common factor: Y = A B C + A BC + AB C + AB C = A C ( B + B ) + AB (C + C ) Reduce the complement pairs to ‘1’ Y = A C ( B + B ) + AB (C + C ) = A C + AB Draw the circuit based on the simplified expression A B C Y 4 Continue †¦ †¢ A Consider another logic circuit: B C Y Y = C( A + B + C ) + A + C Convert to SOP expression: Y = C( A + B + C ) + A + C = AC + B C + AC Checking for common factor: Y = A(C + C ) + B C = A + BC 5 Continue †¦ †¢ †¢ Simplification of logic circuit algebraically is not always an easy task The following two steps might be useful: i. The original expression is convert into the SOP form by repeated application of DeMorgan’s theorems and multiplication of terms ii. The product terms are then checked for common factors, and factoring is performed wherever possible 6 Continue †¦ †¢ Consider the truth table below: A 0 0 0 0 1 B 0 0 1 1 0 C 0 1 0 1 0 Y 0 0 1 0 0 Minterm Boolean expression: Simplify to yield: Y = A BC + ABC + AB C Y = BC ( A + A) + AB C = BC + AB C 1 0 1 1 1 1 0 1 1 1 1 0 †¢ If minterms are only differed by one bit, they can be simplified, e. We will write a custom essay sample on Simplification of Switching Function or any similar topic only for you Order Now g. A BC ABC 7 Continue †¦ †¢ More example: A 0 0 0 0 1 1 1 1 B 0 0 1 1 0 0 1 1 C 0 1 0 1 0 1 0 1 Y 0 1 1 0 0 1 1 0 Minterm Boolean expression: Y = A B C + A BC + AB C + ABC Minterms 1 and 5, 2 and 6 are only differ by one bit: Y = B C ( A + A) + BC ( A + A) = BC + B C A B C Y 0 0 0 1 0 0 1 0 0 0 1 1 1 1 1 1 0 0 1 1 0 1 0 1 0 1 1 0 1 0 1 0 Minterm Boolean expression: Y = A B C + A BC + AB C + ABC Checking and factoring minterms differed by only by one bit: Y = A C ( B + B ) + AC ( B + B ) = A C + AC = C ( A + A) =C 8 Continue †¦ †¢ †¢ †¢ Though truth table can help us to detect minterms which are only differed by one bit, it is not arranged in a proper way A Karnaugh map (K-map) is a tool, which help us to detect and simplify minterms graphically It is a rearrangement of the truth table where each adjacent cell is only differed by one bit By looping adjacent minterms, it is similar to grouping the minterms with a single bit difference on the truth table 9 K arnaugh Map †¢ †¢ A K-map is just a rearrangement of truth table, so that minterms with a single-bit difference can be detected easily Figure below shows 4 possible arrangement of 3-variable K-map A BC 0 0 01 1 11 3 10 2 C AB 00 0 01 2 11 6 10 4 0 1 4 5 7 6 0 1 1 3 7 5 AB C 0 0 1 1 BC A 0 0 1 4 00 01 2 3 00 01 1 5 11 6 7 11 3 7 10 4 5 10 2 6 10 Continue †¦ †¢ Figure below show two possible arrangement of 4variable K-map CD AB 00 0 01 1 11 3 10 2 AB 00 CD 01 4 11 12 10 8 00 01 4 5 7 6 00 0 01 1 5 13 9 11 12 13 15 14 11 3 7 15 11 10 8 9 11 10 10 2 6 14 10 †¢ Notice that the K-map is labeled so that horizontally and vertically adjacent cells differ only by one bit. 11 Continue †¦ †¢ The K-map for both SOP and POS form are shown below: C D C D CD C D AB AB AB 0 1 3 2 C+D C+ D C + D C +D A +B 0 1 3 2 4 5 7 6 A+B A+B A +B 4 5 7 6 12 13 15 14 12 13 15 14 AB 8 9 11 10 8 9 11 10 SOP form (minterm) POS form (maxterm) †¢ †¢ The simplified SOP expression can be obtained by properly combining those adjacent cells which contains ‘1’ This process of combining adjacent minterms is known as 12 looping Continue †¦ †¢ †¢ Each loop of minterms will form a group which can be represented by a product term When a variable appears in both complemented and uncomplemented form within a group, that variable is eliminated from the product term C D C D CD C D AB AB AB AB 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 group 2 group 1: C D( AB + AB ) = AC D group 2: AB(C D + CD ) = ABD Simplified SOP expression: Y = AC D + ABD 13 group 1 Continue †¦ †¢ Consider another K-map: C D C D CD C D AB AB AB AB 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 group 1 C D C D CD C D AB AB AB AB 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 14 group 1: ( A B + AB )(C D + CD ) = BD Simplified SOP expression: Y = BD group 1: C D ( A B + A B + AB + AB ) = C D Simplified SOP expression: Y = CD group 1 From truth table to K-map †¢ The content of each cell can be directly plot on the Kmap according to the truth table Consider the following example: 0 1 2 3 4 5 6 7 A 0 0 0 0 1 1 1 1 B C Y 0 0 1 0 1 1 1 0 1 1 1 0 0 0 0 0 1 0 1 0 1 1 1 0 B C B C BC B C A A 1 0 1 1 0 3 1 2 0 4 0 5 0 7 1 6 AB BC Simplified SOP expression: Y = A B + BC 15 Continue †¦ †¢ Consider the following 4-variable K-map A 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 B 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 C D Y 0 0 0 0 1 1 1 0 0 1 1 0 0 0 0 0 1 1 1 0 0 1 1 0 0 0 0 0 1 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 0 1 1 1 C D C D CD C D AB AB AB 0 0 0 1 1 1 0 1 0 0 1 0 3 0 0 0 ACD 2 4 5 7 6 12 13 15 14 AB 0 8 9 11 0 10 ABD Simplified SOP expression: Y = A C D + ABD 16 Continue †¦ †¢ Some guidelines: i. Construct K-map and fill it according to the truth table ii. Only loop cells in the power of 2, i. e. 2 cells, 4 cells, 8 cells and so on iii. Always start by looping the isolated minterms iv. Look for minterms which are adjacent to only one minterm and loop them together v. Proceed on to loop the largest possible groups, from eight minterms (octet), 4 minterms (quad) to 2 minterms (pair) vi. Obtain the product term for each group vii. The sum of these product terms will be the simplified SOP expression 17 Continue †¦ Example: a. Obtain the simplify SOP expression for the truth table: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 A 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 B 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 C 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 D 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 Y 0 0 1 0 0 1 0 1 0 0 0 1 0 1 0 1 C D C D CD C D AB AB AB AB A B CD 0 0 0 0 0 0 1 1 0 1 0 1 1 3 1 0 0 0 2 4 5 7 6 12 13 15 14 8 9 1 11 10 BD ACD Simplified SOP expression: Y = A B CD + ACD + BD 18 Continue †¦ b. Obtain the simplify SOP expression from the K-map: ACD C D C D CD C D AB AB ABC 0 0 1 0 1 1 1 0 0 1 1 1 ACD 0 1 0 0 A BC AB AB Simplified SOP expression: Y = A C D + A BC + ACD + ABC 19 Continue †¦ c. Obtain the simplify SOP expression from the K-map: alternative solution: C D C D CD C D AB AB AB C D C D CD C D AB A CD 0 0 0 0 AC D 0 0 1 1 1 1 0 1 0 0 0 0 AB D 0 0 0 0 AC D 0 0 1 1 1 1 0 1 0 0 0 0 B CD A CD AB AB AB AB Y = A CD + AC D + AB D Y = A CD + AC D + B CD 20 General Terminology for Logic Minimization †¢ †¢ Here, we define four terms to provide the basis for general function minimization techniques These terms are implicant, prime implicant, essential prime implicant and cover We refer to the K-map below in explaining each term B C B C BC B C A A 1 0 1 1 3 2 1 4 1 5 1 7 6 †¢ †¢ An implicant is a product term that could be used to cover minterms of the function In the K-map above, there are 11 implicants: 5 m interms: {A B C , A BC , AB C , AB C , ABC} 5 group of two adjacent minterms: {AB , AC , A C , B C , BC} 1 group of four adjacent minterms:{C} 21 Continue †¦ †¢ †¢ †¢ A prime implicant is an implicant that is not part of any other mplicant In the K-map, there are two prime implicant: C and AB An essential prime implicant is a prime implicant that covers at least one minterm that is not covered by any other prime implicants Prime implicant AB is essential as it is the only prime implicant that covers minterm 4 Prime implicant C is also essential as it is the only prime implicant that covers minterm 1, 3 and 7 A cover of a function is a set of prime implicants for which each minterm of the function is contained in (covered by) at least one prime implicant All essential prime implicants must be used in any cover of a function 22 †¢ †¢ †¢ Continue †¦ †¢ †¢ For the K-map above, the set of implicants { AB , C} represents a cover of the fun ction A minimum cover contains the minimum number of prime implicants which contains all minterm in the function Consider the 4-variable K-map below: C D C D CD C D AB AB AB AB 1 1 Prime implicants †¢ C D C D CD C D AB AB AB 1 1 1 1 1 1 1 1 1 AB AB AB AB C D C D CD C D 1 1 1 1 Minimum cover 1 1 1 1 1 1 1 1 1 1 1 1 AB Essential prime implicants 23 Continue †¦ †¢ Consider another K-map C D C D CD C D AB AB AB AB 1 1 1 1 1 1 1 Prime implicants C D C D CD C D AB 1 1 1 1 1 1 1 1 1 1 AB AB 1 AB Essential prime implicants (minimum cover) 24 Don’t Care Conditions †¢ †¢ †¢ †¢ Some logic circuit will have certain input conditions whereby the output is unspecified This is usually because these input conditions would never occur In other words, we â€Å"don’t care† whether the output is HIGH or LOW Consider the following example: An air conditioning system has two inputs, C and H: – C will be ‘1’ if temperature is too cold (below 15 °C) Otherwise, it will be ‘0’ – H will be ‘1’ if temperature is too hot (above 25 °C) Otherwise, it will be ‘0’ – Output Y will be ‘1’ if temperature is too cold or too hot. If the temperature is acceptable, Y will be ‘0’ 25 Continue †¦ As there are two inputs, there are 4 possible logical conditions: C 0 0 1 1 H 0 1 0 1 Y 0 1 1 X meaning just nice too hot too cold ? Input condition C = 1, H = 1 has no real meaning, as it is impossible to be too hot and too cold at the same time We put a ‘X’ at the output corresponds to this input condition as this input condition cannot occur 26 K-map and Don’t Care Term †¢ Don’t care term, ‘X’ can be treated as ‘0’ or ‘1’ since they cannot occur In K-map, we can choose the don’t care term as ‘0’ or ‘1’ to our advantage A B C D Y 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 1 1 X 0 1 0 0 1 0 1 0 1 X 0 1 1 0 0 0 1 1 1 X 1 0 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 1 1 X 1 1 0 0 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 X C D C D CD C D AB AB AB 0 1 1 0 1 X 1 0 X X X X 0 0 1 0 AB Simplified Boolean expression: Y = AB + BC + A D 27 More examples †¦ C D C D CD C D AB AB AB AB C D C D CD C D AB AB AB AB 1 1 X 1 0 1 X 1 0 0 X X 0 1 X X 1 0 X 1 0 0 X 0 0 0 X X 1 X X Y = C D + BC + BD + A C D C D CD C D AB AB AB Y = B D + CD C D C D CD C D AB AB AB 0 0 1 0 1 X 1 1 0 1 X 0 0 0 0 0 1 1 X 0 1 X X 1 0 1 X X 0 0 X X 28 AB AB Y = ABC + C D + BD Y = A C + BD + AD Plotting function in Canonical Form †¢ Logic function may be expressed in many forms, ranging from simple SOP/POS expression to more complex expressions However, each of them has a unique canonical SOP/POS form If a Boolean expression is expressed in canonical form, it can be readily plotted on the K-map Consider the following Boolean expression: †¢ †¢ †¢ Y = ABC + B C Convert to canonical SOP expression: Y = ABC + B C ( A + A) = ABC + A B C + AB C 29 Continue †¦ Y = ABC + A B C + AB C Plotting the canonical SOP expression onto K-map B C B C BC B C A A 1 1 0 0 BC 0 0 0 1 AC Simplified SOP expression: Y = B C + AC †¢ Consider plotting the following Boolean expression on K-map: Y = C ( A ? B) + A + B 30 Continue †¦ First, convert to SOP expression Y = C ( A ? B) + A + B = C ( AB + A B) + A B = AB C + A BC + A B (C + C ) = AB C + A BC + A B C + A B C B C B C BC B C A A 1 0 AB 1 1 1 0 BC 0 0 AC ?Y = A B + B C + A C 31 Plotting K-map from SOP expression †¢ †¢ It is sometime too tedious to convert a Boolean expression to its canonical SOP form Consider the following Boolean expression: Y = AB (C + D )(C + D ) + A + B Convert to SOP form: Y = ( AB C + AB D )(C + D ) + A B = AB C D + AB CD + A B Convert to canonical form: Y = AB C D + AB CD + A B (C + C )( D + D) = AB C D + AB CD + ( A B C + A B C )( D + D) = AB C D + AB CD + A B C D + A B C D + A B CD + A B CD 32 Continue †¦ Y = AB C D + AB CD + A B C D + A B C D + A B CD + A B CD Plot the minterm on K-map: C D C D CD C D AB AB AB 1 0 0 1 1 0 0 0 1 0 0 1 1 0 0 0 AB AB B CD BC D Simplified SOP expression: Y = B C D + B CD + A B 33 Continue †¦ †¢ †¢ †¢ †¢ †¢ Boolean expression can be plotted on to the K-map from its SOP form Product terms with four variables are the minterms and correspond to a single cell on the K-map Product term with three variables corresponds to a loop of two adjacent minterms Product term with only two variables is a quad (a loop of four adjacent minterms) Product term with a single variable is an octet (a loop of eight adjacent minterms) 1 cell 2 cells Y = A + BC + B CD + ABCD 4 cells 8 cells 34 Continue †¦ †¢ Consider the previous example: Y = AB C D + AB CD + A B minterms 4 cells †¢ †¢ †¢ Both minterms are directly plotted on the K-map The loop which corresponds to A B is drawn on the K-map The cells inside the loops are filled with ‘1’ C D C D CD C D AB AB AB AB 1 0 0 1 1 0 0 0 1 0 0 1 1 0 0 0 AB AB C D A B CD 35 Continue †¦ †¢ Consider the following Boolean expression: Y = ( A + B )( AC + D ) Convert to SOP form: Y = AC + AD + ABC + BD Plot the SOP onto K-map C D C D CD C D AB AB AB AB AC BD C D C D CD C D AB AB ill cells in loops with ‘1’ 0 0 0 0 0 1 1 1 0 1 1 1 0 0 1 1 36 ABC AB AB AD Continue †¦ Obtain the simplified SOP expression from K-map: C D C D CD C D AB AB AB AB 0 0 0 0 0 1 1 1 0 1 1 1 0 0 1 1 Simplified SOP expression: Y = AC + AD + BD 37 Continue †¦ Example: Redesign the logic circuit below from its simplified SOP expression: A B C D Z Z = ( B + D )( B + D ) + B(CD + A D ) 38 Continue †¦ Z = ( B + D )( B + D ) + B(CD + A D ) = B + D + B + D + BCD + A BD = BD + B D + BCD + A BD C D C D CD C D AB AB AB 1 1 0 1 0 1 1 0 0 1 1 0 1 1 0 1 AB Z = BD + B D + A B 39 How to cite Simplification of Switching Function, Essay examples

Comparing poems Essay Example For Students

Comparing poems Essay The poem Presents from my aunts in Pakistan is about a half English, half Pakistani Girl who was born in Pakistan and brought up in England. Her Aunts in Pakistan sent her some traditional Pakistani clothes. In private she loves the clothes but she doest what to show her friends because she thinks that her friend wont like them.  The poets thoughts and feelings change throughout the poem. In lines 1-19 the girl is happy and grateful glistening like an orange split open (line 4) this is an optimistic image, the presents seem to be exhilarating, stunning and full of assurance. In lines 20-26 the girl feels she cant wear these clothes because she is half-English, unlike Aunt Jamila.(Lines 25 and 26). Lines 27-38 show two sides to her. One side by her saying I wanted my parents camel skin lamp. (Line 27) where she is attracted to but at the same time repelled from to consider the cruelty (line 29) but then in line 32 back to attracted to the lamp to marvel at the colours (line 32).  Lines 40-44 is about her in her bed room trying to show off her clothes but her friend isnt impressed asked to see my weekend clothes. (Line 42-43)  Lines 50-end are meant to be about the girl searching from somewhere she belongs And I was there with no fixed nationality she doesnt believe that she belongs to any fixed nationality. Staring through the fret work (line 68) she is too far away from the re-establishing Pakistan. There seems to be no set structural form to this poem. The stanzas are split up in which ever way the poet feels like in broken rhythm. The poem is written in free verse so there is no rhyme.  The use of similes and metaphors are mostly used in lines (lines 1-39) where the content is describing how beautiful the clothes are. The poet uses colours and fruit to enforce a mood across to the reader of the poem.  I liked this poem because of the way she used colours and fruit to show the way in which the girl in the poem is feeling. I also liked how she describes the all the clothes. Charlotte ONeils Song  The poem Charlotte ONeils Song is about a girl who is a servant in a mansion. She talks about her life scrubbedscrappedwashed and how she is getting bored of doing everything. The girl Charlotte ONeil, according to ships records, is a 17-year-old girl sailing to New Zealand to be a general servant in a rich mans home.  At the end she says I wont be there anymore. and it seems she slams the door and shes gone.  The mood of this poem is quite anger at the beginning and tired of doing the same job day in day out and at the end she glad she gone but know youre on your own my dear. The structure to this poem is not set up in stanzas with set syllables and number of lines. The rhyme starts of steady but the further you go into the poem the angrier she gets and the more lose the rhymes scheme goes. There are no similies or metaphors because charlotte is trying to keep you focused on what shes saying so you wont be distracted by the comparisons making what she has gone through less graphic.  I like this poem because there are no adjectives in the poem so it looks a little bare in places but she keeps it tighter by increasing the anger and letting the rhyme scheme slip to show this. I also liked the way in which she concluded the piece by saying and you can open your own front door.  Nothings changed  This poem is about the apartheid in district six, Cape Town, Africa. A man is walking through the field the fields just out side of dictrict six. He then walks into district six no board says it is, but my feet know he walks up to a white s only Inn and says theres a guard at the gatepost to hold the true Africans out. .ub8501c962e615083cf19e8c7356dab35 , .ub8501c962e615083cf19e8c7356dab35 .postImageUrl , .ub8501c962e615083cf19e8c7356dab35 .centered-text-area { min-height: 80px; position: relative; } .ub8501c962e615083cf19e8c7356dab35 , .ub8501c962e615083cf19e8c7356dab35:hover , .ub8501c962e615083cf19e8c7356dab35:visited , .ub8501c962e615083cf19e8c7356dab35:active { border:0!important; } .ub8501c962e615083cf19e8c7356dab35 .clearfix:after { content: ""; display: table; clear: both; } .ub8501c962e615083cf19e8c7356dab35 { display: block; transition: background-color 250ms; webkit-transition: background-color 250ms; width: 100%; opacity: 1; transition: opacity 250ms; webkit-transition: opacity 250ms; background-color: #95A5A6; } .ub8501c962e615083cf19e8c7356dab35:active , .ub8501c962e615083cf19e8c7356dab35:hover { opacity: 1; transition: opacity 250ms; webkit-transition: opacity 250ms; background-color: #2C3E50; } .ub8501c962e615083cf19e8c7356dab35 .centered-text-area { width: 100%; position: relative ; } .ub8501c962e615083cf19e8c7356dab35 .ctaText { border-bottom: 0 solid #fff; color: #2980B9; font-size: 16px; font-weight: bold; margin: 0; padding: 0; text-decoration: underline; } .ub8501c962e615083cf19e8c7356dab35 .postTitle { color: #FFFFFF; font-size: 16px; font-weight: 600; margin: 0; padding: 0; width: 100%; } .ub8501c962e615083cf19e8c7356dab35 .ctaButton { background-color: #7F8C8D!important; color: #2980B9; border: none; border-radius: 3px; box-shadow: none; font-size: 14px; font-weight: bold; line-height: 26px; moz-border-radius: 3px; text-align: center; text-decoration: none; text-shadow: none; width: 80px; min-height: 80px; background: url(https://artscolumbia.org/wp-content/plugins/intelly-related-posts/assets/images/simple-arrow.png)no-repeat; position: absolute; right: 0; top: 0; } .ub8501c962e615083cf19e8c7356dab35:hover .ctaButton { background-color: #34495E!important; } .ub8501c962e615083cf19e8c7356dab35 .centered-text { display: table; height: 80px; padding-left : 18px; top: 0; } .ub8501c962e615083cf19e8c7356dab35 .ub8501c962e615083cf19e8c7356dab35-content { display: table-cell; margin: 0; padding: 0; padding-right: 108px; position: relative; vertical-align: middle; width: 100%; } .ub8501c962e615083cf19e8c7356dab35:after { content: ""; display: block; clear: both; } READ: Rhoda"s Diary The Withered Arm by Thomas Hardy EssayThe second half of the poem is describes the way hes feeling about the whites that are better than him .he walks up to the window and without out looking already knows what hes going to see. Before I see them, there will be crush ice white glass then he takes us down the street to working mans he backs away from the window and has a burning desire to smash the clear panes in any way which he can.  The poet feeling and thoughts develop as the significantly during the course of the poem. From when the poem begins hes walking thought the fields and although the terrain is harsh and hard he feels comfortable there. He then walks down into district six and his anger begins to be revealed. Talks about his hands, bones, lungs and hot, white inwards turning anger of my eyes. This surggests that he is beginning he is angry a bit but as the poem progresses the anger enhances with it.  The three poems are about different cultures. I liked the nothings changed poem because it has the most effect on me. The way in which the poet unravels the plot gradually shows how much nothing has changed.