The Chicago Tribune is a daily newspaper based in Chicago, United States, owned by Tribune Publishing. Founded in 1847, self-styled as the "World's Greatest Newspaper", it remains the most-read daily newspaper of the Chicago metropolitan area and the Great Lakes region, it is the eighth-largest newspaper in the United States by circulation. Traditionally published as a broadsheet, on January 13, 2009, the Tribune announced it would continue publishing as a broadsheet for home delivery, but would publish in tabloid format for newsstand, news box, commuter station sales; this change, proved to be unpopular with readers and in August 2011, the Tribune discontinued the tabloid edition, returning to its traditional broadsheet edition through all distribution channels. The Tribune's masthead is notable for displaying the American flag, in reference to the paper's motto, "An American Paper for Americans"; the motto is no longer displayed on the masthead. The Tribune was founded by James Kelly, John E. Wheeler, Joseph K. C.
Forrest, publishing the first edition on June 10, 1847. Numerous changes in ownership and editorship took place over the next eight years; the Tribune was not politically affiliated, but tended to support either the Whig or Free Soil parties against the Democrats in elections. By late 1853, it was running xenophobic editorials that criticized foreigners and Roman Catholics. About this time it became a strong proponent of temperance; however nativist its editorials may have been, it was not until February 10, 1855 that the Tribune formally affiliated itself with the nativist American or Know Nothing party, whose candidate Levi Boone was elected Mayor of Chicago the following month. By about 1854, part-owner Capt. J. D. Webster General Webster and chief of staff at the Battle of Shiloh, Dr. Charles H. Ray of Galena, through Horace Greeley, convinced Joseph Medill of Cleveland's Leader to become managing editor. Ray became editor-in-chief, Medill became the managing editor, Alfred Cowles, Sr. brother of Edwin Cowles was the bookkeeper.
Each purchased one third of the Tribune. Under their leadership, the Tribune distanced itself from the Know Nothings, became the main Chicago organ of the Republican Party. However, the paper continued to print anti-Catholic and anti-Irish editorials, in the wake of the massive Famine immigration from Ireland; the Tribune absorbed three other Chicago publications under the new editors: the Free West in 1855, the Democratic Press of William Bross in 1858, the Chicago Democrat in 1861, whose editor, John Wentworth, left his position when elected as Mayor of Chicago. Between 1858 and 1860, the paper was known as the Chicago Tribune. On October 25, 1860, it became the Chicago Daily Tribune. Before and during the American Civil War, the new editors supported Abraham Lincoln, whom Medill helped secure the presidency in 1860, pushed an abolitionist agenda; the paper remained a force in Republican politics for years afterwards. In 1861, the Tribune published new lyrics by William W. Patton for the song "John Brown's Body".
These rivaled the lyrics published two months by Julia Ward Howe. Medill served as mayor of Chicago for one term after the Great Chicago Fire of 1871. Under the 20th-century editorship of Colonel Robert R. McCormick, who took control in the 1920s, the paper was isolationist and aligned with the Old Right in its coverage of political news and social trends, it used the motto "The American Paper for Americans". Through the 1930s to the 1950s, it excoriated the Democrats and the New Deal of Franklin D. Roosevelt, was resolutely disdainful of the British and French, enthusiastic for Chiang Kai-shek and Sen. Joseph McCarthy; when McCormick assumed the position of co-editor in 1910, the Tribune was the third-best-selling paper among Chicago's eight dailies, with a circulation of only 188,000. The young cousins added features such as advice columns and homegrown comic strips such as Little Orphan Annie and Moon Mullins, they promoted political "crusades", with their first success coming with the ouster of the Republican political boss of Illinois, Sen. William Lorimer.
At the same time, the Tribune competed with the Hearst paper, the Chicago Examiner, in a circulation war. By 1914, the cousins succeeded in forcing out Managing Editor William Keeley. By 1918, the Examiner was forced to merge with the Chicago Herald. In 1919, Patterson left the Tribune and moved to New York to launch his own newspaper, the New York Daily News. In a renewed circulation war with Hearst's Herald-Examiner, McCormick and Hearst ran rival lotteries in 1922; the Tribune won the battle. In 1922, the Chicago Tribune hosted an international design competition for its new headquarters, the Tribune Tower; the competition worked brilliantly as a publicity stunt, more than 260 entries were received. The winner was a neo-Gothic design by New York architects John Mead Howells and Raymond Hood; the newspaper sponsored a pioneering attempt at Arctic aviation in 1929, an attempted round-trip to Europe across Greenland and Iceland in a Sikorsky amphibious aircraft. But, the aircraft was destroyed by ice on July 15, 1929, near Ungava Bay at the tip of Labrador, Canada.
The crew were rescued by the Canadian science ship CSS Acadia. The Tribune's reputation for innovation extended to radio—it bought an early station, WDAP, in 1924 and renamed it WGN, the station call letters standing for the paper's self-description as the "Worl
Indiana is a U. S. state located in the Midwestern and Great Lakes regions of North America. Indiana is the 17th most populous of the 50 United States, its capital and largest city is Indianapolis. Indiana was admitted to the United States as the 19th U. S. state on December 11, 1816. Indiana borders Lake Michigan to the northwest, Michigan to the north, Ohio to the east, Kentucky to the south and southeast, Illinois to the west. Before becoming a territory, various indigenous peoples and Native Americans inhabited Indiana for thousands of years. Since its founding as a territory, settlement patterns in Indiana have reflected regional cultural segmentation present in the Eastern United States. Indiana has a diverse economy with a gross state product of $359.12 billion in 2017. Indiana has several metropolitan areas with populations greater than 100,000 and a number of smaller industrial cities and towns. Indiana is home to professional sports teams, including the NFL's Indianapolis Colts and the NBA's Indiana Pacers, hosts several notable athletic events, such as the Indianapolis 500 and Brickyard 400 motorsports races.
The state's name means "Land of the Indians", or "Indian Land". It stems from Indiana's territorial history. On May 7, 1800, the United States Congress passed legislation to divide the Northwest Territory into two areas and named the western section the Indiana Territory. In 1816, when Congress passed an Enabling Act to begin the process of establishing statehood for Indiana, a part of this territorial land became the geographic area for the new state. A resident of Indiana is known as a Hoosier; the etymology of this word is disputed, but the leading theory, as advanced by the Indiana Historical Bureau and the Indiana Historical Society, has "Hoosier" originating from Virginia, the Carolinas, Tennessee as a term for a backwoodsman, a rough countryman, or a country bumpkin. The first inhabitants in what is now Indiana were the Paleo-Indians, who arrived about 8000 BC after the melting of the glaciers at the end of the Ice Age. Divided into small groups, the Paleo-Indians were nomads, they created stone tools made out of chert by chipping and flaking.
The Archaic period, which began between 5000 and 4000 BC, covered the next phase of indigenous culture. The people developed new tools as well as techniques to cook food, an important step in civilization; such new tools included different types of spear knives, with various forms of notches. They made ground-stone tools such as woodworking tools and grinding stones. During the latter part of the period, they built earthwork mounds and middens, which showed that settlements were becoming more permanent; the Archaic period ended at about 1500 BC, although some Archaic people lived until 700 BC. The Woodland period commenced around 1500 BC. During this period, the people created ceramics and pottery, extended their cultivation of plants. An early Woodland period group named the Adena people had elegant burial rituals, featuring log tombs beneath earth mounds. In the middle portion of the Woodland period, the Hopewell people began developing long-range trade of goods. Nearing the end of the stage, the people developed productive cultivation and adaptation of agriculture, growing such crops as corn and squash.
The Woodland period ended around 1000 AD. The Mississippian culture emerged, lasting from 1000 AD until the 15th century, shortly before the arrival of Europeans. During this stage, the people created large urban settlements designed according to their cosmology, with large mounds and plazas defining ceremonial and public spaces; the concentrated settlements depended on the agricultural surpluses. One such complex was the Angel Mounds, they had large public areas such as plazas and platform mounds, where leaders lived or conducted rituals. Mississippian civilization collapsed in Indiana during the mid-15th century for reasons that remain unclear; the historic Native American tribes in the area at the time of European encounter spoke different languages of the Algonquian family. They included the Shawnee and Illini, they were joined by refugee tribes from eastern regions including the Delaware who settled in the White and Whitewater River Valleys. In 1679, French explorer René-Robert Cavelier, Sieur de La Salle was the first European to cross into Indiana after reaching present-day South Bend at the Saint Joseph River.
He returned the following year to learn about the region. French-Canadian fur traders soon arrived, bringing blankets, tools and weapons to trade for skins with the Native Americans. By 1702, Sieur Juchereau established the first trading post near Vincennes. In 1715, Sieur de Vincennes built Fort Miami at Kekionga, now Fort Wayne. In 1717, another Canadian, Picote de Beletre, built Fort Ouiatenon on the Wabash River, to try to control Native American trade routes from Lake Erie to the Mississippi River. In 1732, Sieur de Vincennes built a second fur trading post at Vincennes. French Canadian settlers, who had left the earlier post because of hostilities, returned in larger numbers. In a period of a few years, British colonists arrived from the East and contended against the Canadians for control of the lucrative fur trade. Fighting between the French and British colonists occurred throughout the 1750s as a result; the Native American tribes of Indiana sided with th
Martin Luther University of Halle-Wittenberg
The Martin Luther University of Halle-Wittenberg referred to as MLU, is a public, research-oriented university in the cities of Halle and Wittenberg in Saxony-Anhalt, Germany. MLU offers German and international courses leading to academic degrees such as BA, BSc, MA, MSc, doctoral degrees and Habilitation; the university was created in 1817 through the merger of the University of Wittenberg and the University of Halle. The university is named after the Protestant reformer Martin Luther, a professor in Wittenberg. Today, the university itself is located in Halle, while the Leucorea Foundation in Wittenberg serves as MLU's convention centre for seminars as well as for academic and political conferences. Both Halle and Wittenberg are about one hour from Berlin via the Berlin–Halle railway, which offers Intercity-Express trains; the University of Wittenberg was founded in 1502 by Frederick the Wise, Elector of Saxony, as the Renaissance was becoming more and more popular. The foundation of the university was criticized when the Ninety-five Theses reached Albert of Brandenburg, the Archbishop of Mainz.
Ecclesiastically speaking, the Electorate of Saxony was subordinate to Albert. He criticized the elector for Luther's theses, viewing the founded university as a breeding ground for heretical ideas. Under the influence of Philipp Melanchthon, building on the works of Martin Luther, the university became a centre of the Protestant Reformation incorporating, at one point in time, Luther's house in Wittenberg, the Lutherhaus, as part of the campus. Notable alumni include George Müller, Georg Joachim Rheticus and – in fiction – William Shakespeare's Prince Hamlet and Horatio and Christopher Marlowe's Doctor Faustus; the University of Halle was founded in 1694 by Frederick III, Elector of Brandenburg, who became Frederick I, King in Prussia, in 1701. In the late 17th century and early 18th century, Halle became a centre for Pietism within Prussia. In the 17th and 18th centuries, the universities were centers of the German Enlightenment. Christian Wolff was an important proponent of rationalism, he influenced many German scholars, such as Immanuel Kant.
Christian Thomasius was at the same time the first philosopher in Germany to hold his lectures not in Latin, but German. He contributed to a rational programme in philosophy but tried to establish a more common-sense point of view, aimed against the unquestioned superiority of aristocracy and theology; the institutionalisation of the local language as the language of instruction, the prioritisation of rationalism over religious orthodoxy, new modes of teaching, the ceding of control over their work to the professors themselves, were among various innovations which characterised the University of Halle, have led to its being referred to as the first "modern" university, whose liberalism was adopted by the University of Göttingen about a generation and subsequently by other German and most North American universities. The University of Wittenberg was closed in 1813 during the Napoleonic Wars; the town of Wittenberg was granted to Prussia in the Congress of Vienna in 1815, the university was merged with the Prussian University of Halle in 1817.
It took its present name on 10 November 1933. More than a dozen professors were expelled. Others were shifted to Halle-Wittenberg from universities regarded as "better" at the time, which led to the university being called an academic Vorkuta – after the largest center of the Gulag camps in European Russia). Following the continental European academic tradition, MLU has 9 faculties, regrouping academic staff and students according to their field of studies: Faculty of Theology Faculty of Law and Economics Faculty of Medicine Faculty of Philosophy I Faculty of Philosophy II Faculty of Philosophy III Faculty of Natural Sciences I Faculty of Natural Sciences II Faculty of Natural Sciences III The Botanical Garden of Martin Luther University of Halle-Wittenberg, founded in 1698. MLU's historical observatory, built in 1788 by Carl Gotthard Langhans. MLU is enclosed by a variety of research institutions, which have either institutional or personal links with the university or cooperate in their respective fields of studies: The German Academy of Sciences Leopoldina The Halle Institute for Economic Research The Fraunhofer Institute for Mechanics of Materials The Leibniz Institute of Agricultural Development in Central and Eastern Europe The Leibniz Institute of Plant Biochemistry The Max Planck Research Unit for Enzymology of Protein Folding The Max Planck Institute for Social Anthropology The Max Planck Institute of Microstructure Physics The Helmholtz Centre for Environmental Research Even though MLU is an academic, research oriented institution, not an academy of music or conservatory, the university has an academic orchestra, founded in 1779, a rather prestigious choir, founded in 1950, which together constitute the so-called Collegium musicum.
Members are gifted students of all faculties, but academic staff and alumni. The university choir performs at the international Handel Festival in George Frideric Handel’s birthplace, Halle. MLU has many international partner universities, including: Argentina: National University of La Plata Australia: University of Queensland Austria: Johannes Kepler University Linz Canada: University of Ottawa Colombi
Germany the Federal Republic of Germany, is a country in Central and Western Europe, lying between the Baltic and North Seas to the north, the Alps to the south. It borders Denmark to the north and the Czech Republic to the east and Switzerland to the south, France to the southwest, Luxembourg and the Netherlands to the west. Germany includes 16 constituent states, covers an area of 357,386 square kilometres, has a temperate seasonal climate. With 83 million inhabitants, it is the second most populous state of Europe after Russia, the most populous state lying in Europe, as well as the most populous member state of the European Union. Germany is a decentralized country, its capital and largest metropolis is Berlin, while Frankfurt serves as its financial capital and has the country's busiest airport. Germany's largest urban area is the Ruhr, with its main centres of Essen; the country's other major cities are Hamburg, Cologne, Stuttgart, Düsseldorf, Dresden, Bremen and Nuremberg. Various Germanic tribes have inhabited the northern parts of modern Germany since classical antiquity.
A region named Germania was documented before 100 AD. During the Migration Period, the Germanic tribes expanded southward. Beginning in the 10th century, German territories formed a central part of the Holy Roman Empire. During the 16th century, northern German regions became the centre of the Protestant Reformation. After the collapse of the Holy Roman Empire, the German Confederation was formed in 1815; the German revolutions of 1848–49 resulted in the Frankfurt Parliament establishing major democratic rights. In 1871, Germany became a nation state when most of the German states unified into the Prussian-dominated German Empire. After World War I and the revolution of 1918–19, the Empire was replaced by the parliamentary Weimar Republic; the Nazi seizure of power in 1933 led to the establishment of a dictatorship, the annexation of Austria, World War II, the Holocaust. After the end of World War II in Europe and a period of Allied occupation, Austria was re-established as an independent country and two new German states were founded: West Germany, formed from the American and French occupation zones, East Germany, formed from the Soviet occupation zone.
Following the Revolutions of 1989 that ended communist rule in Central and Eastern Europe, the country was reunified on 3 October 1990. Today, the sovereign state of Germany is a federal parliamentary republic led by a chancellor, it is a great power with a strong economy. As a global leader in several industrial and technological sectors, it is both the world's third-largest exporter and importer of goods; as a developed country with a high standard of living, it upholds a social security and universal health care system, environmental protection, a tuition-free university education. The Federal Republic of Germany was a founding member of the European Economic Community in 1957 and the European Union in 1993, it is part of the Schengen Area and became a co-founder of the Eurozone in 1999. Germany is a member of the United Nations, NATO, the G7, the G20, the OECD. Known for its rich cultural history, Germany has been continuously the home of influential and successful artists, musicians, film people, entrepreneurs, scientists and inventors.
Germany has a large number of World Heritage sites and is among the top tourism destinations in the world. The English word Germany derives from the Latin Germania, which came into use after Julius Caesar adopted it for the peoples east of the Rhine; the German term Deutschland diutisciu land is derived from deutsch, descended from Old High German diutisc "popular" used to distinguish the language of the common people from Latin and its Romance descendants. This in turn descends from Proto-Germanic *þiudiskaz "popular", derived from *þeudō, descended from Proto-Indo-European *tewtéh₂- "people", from which the word Teutons originates; the discovery of the Mauer 1 mandible shows that ancient humans were present in Germany at least 600,000 years ago. The oldest complete hunting weapons found anywhere in the world were discovered in a coal mine in Schöningen between 1994 and 1998 where eight 380,000-year-old wooden javelins of 1.82 to 2.25 m length were unearthed. The Neander Valley was the location where the first non-modern human fossil was discovered.
The Neanderthal 1 fossils are known to be 40,000 years old. Evidence of modern humans dated, has been found in caves in the Swabian Jura near Ulm; the finds included 42,000-year-old bird bone and mammoth ivory flutes which are the oldest musical instruments found, the 40,000-year-old Ice Age Lion Man, the oldest uncontested figurative art discovered, the 35,000-year-old Venus of Hohle Fels, the oldest uncontested human figurative art discovered. The Nebra sky disk is a bronze artefact created during the European Bronze Age attributed to a site near Nebra, Saxony-Anhalt, it is part of UNESCO's Memory of the World Programme. The Germanic tribes are thought to date from the Pre-Roman Iron Age. From southern Scandinavia and north Germany, they expanded south and west from the 1st century BC, coming into contact with the Celtic tribes of Gaul as well
University of California, Los Angeles
The University of California, Los Angeles is a public research university in Los Angeles. It became the Southern Branch of the University of California in 1919, making it the third-oldest undergraduate campus of the 10-campus University of California system, it offers 337 graduate degree programs in a wide range of disciplines. UCLA enrolls about 31,000 undergraduate and 13,000 graduate students and had 119,000 applicants for Fall 2016, including transfer applicants, making the school the most applied-to of any American university; the university is organized into six undergraduate colleges, seven professional schools, four professional health science schools. The undergraduate colleges are the College of Science; as of 2017, 24 Nobel laureates, three Fields Medalists, five Turing Award winners, two Chief Scientists of the U. S. Air Force have been affiliated with UCLA as researchers, or alumni. Among the current faculty members, 55 have been elected to the National Academy of Sciences, 28 to the National Academy of Engineering, 39 to the Institute of Medicine, 124 to the American Academy of Arts and Sciences.
The university was elected to the Association of American Universities in 1974. UCLA is considered one of the country's Public Ivies, meaning that it is a public university thought to provide a quality of education comparable with that of the Ivy League. In 2018, US News & World Report named UCLA the best public university in the United States. UCLA student-athletes compete as the Bruins in the Pac-12 Conference; the Bruins have won 126 national championships, including 116 NCAA team championships, more than any other university except Stanford, who has won 117. UCLA student-athletes and staff won 251 Olympic medals: 126 gold, 65 silver, 60 bronze. UCLA student-athletes competed in every Olympics since 1920 with one exception and won a gold medal in every Olympics the U. S. participated in since 1932. In March 1881, the California State Legislature authorized the creation of a southern branch of the California State Normal School in downtown Los Angeles to train teachers for the growing population of Southern California.
The Los Angeles branch of the California State Normal School opened on August 29, 1882, on what is now the site of the Central Library of the Los Angeles Public Library system. The facility included an elementary school where teachers-in-training could practice their technique with children; that elementary school is related to the present day UCLA Lab School. In 1887, the branch campus became independent and changed its name to Los Angeles State Normal School. In 1914, the school moved to a new campus on Vermont Avenue in East Hollywood. In 1917, UC Regent Edward Augustus Dickson, the only regent representing the Southland at the time, Ernest Carroll Moore, Director of the Normal School, began to lobby the State Legislature to enable the school to become the second University of California campus, after UC Berkeley, they met resistance from UC Berkeley alumni, Northern California members of the state legislature, Benjamin Ide Wheeler, President of the University of California from 1899 to 1919, who were all vigorously opposed to the idea of a southern campus.
However, David Prescott Barrows, the new President of the University of California, did not share Wheeler's objections. On May 23, 1919, the Southern Californians' efforts were rewarded when Governor William D. Stephens signed Assembly Bill 626 into law, which transformed the Los Angeles Normal School into the Southern Branch of the University of California; the same legislation added the College of Letters and Science. The Southern Branch campus opened on September 15 of that year, offering two-year undergraduate programs to 250 Letters and Science students and 1,250 students in the Teachers College, under Moore's continued direction. Under University of California President William Wallace Campbell, enrollment at the Southern Branch expanded so that by the mid-1920s the institution was outgrowing the 25 acre Vermont Avenue location; the Regents searched for a new location and announced their selection of the so-called "Beverly Site"—just west of Beverly Hills—on March 21, 1925 edging out the panoramic hills of the still-empty Palos Verdes Peninsula.
After the athletic teams entered the Pacific Coast conference in 1926, the Southern Branch student council adopted the nickname "Bruins", a name offered by the student council at UC Berkeley. In 1927, the Regents renamed the Southern Branch the University of California at Los Angeles. In the same year, the state broke ground in Westwood on land sold for $1 million, less than one-third its value, by real estate developers Edwin and Harold Janss, for whom the Janss Steps are named; the campus in Westwood opened to students in 1929. The original four buildings were the College Library, Royce Hall, the Physics-Biology Building, the Chemistry Building, arrayed around a quadrangular courtyard on the 400 acre campus; the first undergraduate classes on the new campus were held in 1929 with 5,500 students. After lobbying by alumni, faculty and community leaders, UCLA was permitted to award the master's degree in 1933, the doctorate in 1936, against continued resistance from UC Berkeley. A timeline of the history can be found on its website, as well
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most to objects that are relevant to mathematics; the language of set theory can be used to define nearly all mathematical objects. The modern study of set theory was initiated by Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, such as Russell's paradox, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with or without the axiom of choice, are the best-known. Set theory is employed as a foundational system for mathematics in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.
Mathematical topics emerge and evolve through interactions among many researchers. Set theory, was founded by a single paper in 1874 by Georg Cantor: "On a Property of the Collection of All Real Algebraic Numbers". Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathematicians in the East, mathematicians had struggled with the concept of infinity. Notable is the work of Bernard Bolzano in the first half of the 19th century. Modern understanding of infinity began in 1870–1874 and was motivated by Cantor's work in real analysis. An 1872 meeting between Cantor and Richard Dedekind influenced Cantor's thinking and culminated in Cantor's 1874 paper. Cantor's work polarized the mathematicians of his day. While Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of mathematical constructivism, did not. Cantorian set theory became widespread, due to the utility of Cantorian concepts, such as one-to-one correspondence among sets, his proof that there are more real numbers than integers, the "infinity of infinities" resulting from the power set operation.
This utility of set theory led to the article "Mengenlehre" contributed in 1898 by Arthur Schoenflies to Klein's encyclopedia. The next wave of excitement in set theory came around 1900, when it was discovered that some interpretations of Cantorian set theory gave rise to several contradictions, called antinomies or paradoxes. Bertrand Russell and Ernst Zermelo independently found the simplest and best known paradox, now called Russell's paradox: consider "the set of all sets that are not members of themselves", which leads to a contradiction since it must be a member of itself and not a member of itself. In 1899 Cantor had himself posed the question "What is the cardinal number of the set of all sets?", obtained a related paradox. Russell used his paradox as a theme in his 1903 review of continental mathematics in his The Principles of Mathematics. In 1906 English readers gained the book Theory of Sets of Points by husband and wife William Henry Young and Grace Chisholm Young, published by Cambridge University Press.
The momentum of set theory was such. The work of Zermelo in 1908 and the work of Abraham Fraenkel and Thoralf Skolem in 1922 resulted in the set of axioms ZFC, which became the most used set of axioms for set theory; the work of analysts such as Henri Lebesgue demonstrated the great mathematical utility of set theory, which has since become woven into the fabric of modern mathematics. Set theory is used as a foundational system, although in some areas—such as algebraic geometry and algebraic topology—category theory is thought to be a preferred foundation. Set theory begins with a fundamental binary relation between an object o and a set A. If o is a member of A, the notation o. Since sets are objects, the membership relation can relate sets as well. A derived binary relation between two sets is the subset relation called set inclusion. If all the members of set A are members of set B A is a subset of B, denoted A ⊆ B. For example, is a subset of, so is but is not; as insinuated from this definition, a set is a subset of itself.
For cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined. A is called a proper subset of B if and only if A is a subset of B, but A is not equal to B. Note that 1, 2, 3 are members of the set but are not subsets of it. Just as arithmetic features binary operations on numbers, set theory features binary operations on sets. The: Union of the sets A and B, denoted A ∪ B, is the set of all objects that are a member of A, or B, or both; the union of and is the set. Intersection of the sets A and B, denoted A ∩ B, is the set of all objects that are members of both A and B; the intersection of and is the set. Set difference of U and A, denoted U \ A, is the set of all members of U that are not members of A; the set difference \ is, conversely, the set difference \ is. When A is a subset of U, the set difference U \ A is called the complement of A in U. In this case, if the choice of U is clear from the context, the notation Ac is sometimes used instead of U \ A if U is a universal set as in the study of Venn diagrams.
Symmetric difference of sets A and B, denoted A △ B or A ⊖ B, is