Hans Hahn (mathematician)
Hans Hahn was an Austrian mathematician who made contributions to functional analysis, set theory, the calculus of variations, real analysis, order theory. Born at Vienna as the son of a higher government official of the k.k. Telegraphen-Korrespondenz-Bureau, in 1898 Hahn became a student at the Universität Wien starting with a study of law. In 1899 he switched over to mathematics and spent some time at the universities of Strasbourg, Munich and Göttingen. In 1902 he took his Ph. D. in Vienna, on the subject "Zur Theorie der zweiten Variation einfacher Integrale". He was a student of Gustav von Escherich, he was appointed to the teaching staff in Vienna in 1905. After 1905/1906 as a stand-in for Otto Stolz at Innsbruck and some further years as a Privatdozent in Vienna, he was nominated in 1909 Professor extraordinarius in Czernowitz, at that time a town within the empire of Austria. After joining the Austrian army in 1915, he was badly wounded in 1916 and became again Professor extraordinarius, now in Bonn.
In 1917 he was nominated a regular Professor there and in 1921 he returned to Vienna with this title, where he stayed until his rather early death in 1934 at the age of 54, following cancer surgery. He had married Eleonore Minor in 1909 and they had a daughter, Nora, he was interested in philosophy, was part of a discussion group concerning Mach's positivism with Otto Neurath and Phillip Frank prior to the First World War. In 1922, he helped arrange Moritz Schlick's entry into the group, which led to the founding of the Vienna Circle, the group, at the center of logical positivist thought in the 1920s, his most famous student was Kurt Gödel, whose Ph. D. thesis was completed in 1929. Within the Vienna Circle, Hahn was known for using his mathematical and philosophical work to study psychic phenomena. Hahn's contributions to mathematics include the Hahn–Banach theorem and the uniform boundedness principle. Other theorems include: the Hahn decomposition theorem. Hahn authored the book: according to Arthur Rosenthal, "... formed a great advance in the Theory of Real functions and had a great influence on the further development of this theory".
He was a co-author of the book Set Functions, published in 1948 by Arthur Rosenthal, fourteen years after his death in Vienna in 1934. In 1921 he received the Richard Lieben Prize. In 1926 he was the president of the German Mathematical Society. In 1928 he was an Invited Speaker at the ICM in Bologna. All his mathematical and philosophical works, except all books and all but one of his book reviews, are published in the three volumes, of his "Collected papers". Hahn, Theorie der reellen Funktionen. Erster Band, Berlin–Heidelberg: Springer-Verlag, pp. VII+600, doi:10.1007/978-3-642-52624-4, ISBN 978-3-642-52570-4, JFM 48.0261.09. Hahn, Reelle Funktionen. Tl. 1. Punktfunktionen, Mathematik und ihre Anwendungen in Monographien und Lehrbüchern, Band 13, Leipzig: Akademische Verlagsgesellschaft, pp. XII+415, JFM 58.0242.05, Zbl 0005.38903 Hahn, Hans. Band 1/Vol. 1, Wien: Springer-Verlag, pp. xii+511, ISBN 978-3-211-82682-9, MR 1361405, Zbl 0859.01030 Hahn, Gesammelte Abhandlungen/Collected works. Band 2/Vol.
2, Wien: Springer-Verlag, pp. xiii+545, ISBN 978-3-211-82750-5, MR 1394443, Zbl 0847.01033. Hahn, Gesammelte Abhandlungen/Collected works. Band 3/Vol. 3, Wien: Springer-Verlag, pp. xiii+581, ISBN 978-3-211-82781-9, MR 1452103, Zbl 0881.01046. Domain Mathematical analysis O'Connor, John J.. "Hans Hahn", MacTutor History of Mathematics archive, University of St Andrews. Josef Lense, "Hahn, Mathematiker", Neue Deutsche Biographie, 7, Berlin: Duncker & Humblot, pp. 506–506
A mathematician is someone who uses an extensive knowledge of mathematics in his or her work to solve mathematical problems. Mathematics is concerned with numbers, quantity, space and change. One of the earliest known mathematicians was Thales of Miletus, he is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. The number of known mathematicians grew when Pythagoras of Samos established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number", it was the Pythagoreans who coined the term "mathematics", with whom the study of mathematics for its own sake begins. The first woman mathematician recorded by history was Hypatia of Alexandria, she succeeded her father as Librarian at the Great Library and wrote many works on applied mathematics. Because of a political dispute, the Christian community in Alexandria punished her, presuming she was involved, by stripping her naked and scraping off her skin with clamshells.
Science and mathematics in the Islamic world during the Middle Ages followed various models and modes of funding varied based on scholars. It was extensive patronage and strong intellectual policies implemented by specific rulers that allowed scientific knowledge to develop in many areas. Funding for translation of scientific texts in other languages was ongoing throughout the reign of certain caliphs, it turned out that certain scholars became experts in the works they translated and in turn received further support for continuing to develop certain sciences; as these sciences received wider attention from the elite, more scholars were invited and funded to study particular sciences. An example of a translator and mathematician who benefited from this type of support was al-Khawarizmi. A notable feature of many scholars working under Muslim rule in medieval times is that they were polymaths. Examples include the work on optics and astronomy of Ibn al-Haytham; the Renaissance brought an increased emphasis on science to Europe.
During this period of transition from a feudal and ecclesiastical culture to a predominantly secular one, many notable mathematicians had other occupations: Luca Pacioli. As time passed, many mathematicians gravitated towards universities. An emphasis on free thinking and experimentation had begun in Britain's oldest universities beginning in the seventeenth century at Oxford with the scientists Robert Hooke and Robert Boyle, at Cambridge where Isaac Newton was Lucasian Professor of Mathematics & Physics. Moving into the 19th century, the objective of universities all across Europe evolved from teaching the “regurgitation of knowledge” to “encourag productive thinking.” In 1810, Humboldt convinced the King of Prussia to build a university in Berlin based on Friedrich Schleiermacher’s liberal ideas. Thus and laboratories started to evolve. British universities of this period adopted some approaches familiar to the Italian and German universities, but as they enjoyed substantial freedoms and autonomy the changes there had begun with the Age of Enlightenment, the same influences that inspired Humboldt.
The Universities of Oxford and Cambridge emphasized the importance of research, arguably more authentically implementing Humboldt’s idea of a university than German universities, which were subject to state authority. Overall, science became the focus of universities in the 20th centuries. Students could conduct research in seminars or laboratories and began to produce doctoral theses with more scientific content. According to Humboldt, the mission of the University of Berlin was to pursue scientific knowledge; the German university system fostered professional, bureaucratically regulated scientific research performed in well-equipped laboratories, instead of the kind of research done by private and individual scholars in Great Britain and France. In fact, Rüegg asserts that the German system is responsible for the development of the modern research university because it focused on the idea of “freedom of scientific research and study.” Mathematicians cover a breadth of topics within mathematics in their undergraduate education, proceed to specialize in topics of their own choice at the graduate level.
In some universities, a qualifying exam serves to test both the breadth and depth of a student's understanding of mathematics. Mathematicians involved with solving problems with applications in real life are called applied mathematicians. Applied mathematicians are mathematical scientists who, with their specialized knowledge and professional methodology, approach many of the imposing problems presented in related scientific fields. With professional focus on a wide variety of problems, theoretical systems, localized constructs, applied mathematicians work in the study and formulation of mathematical models. Mathematicians and applied mathematicians are considered to be two of the STEM careers; the discipline of applied mathematics concerns
Austria the Republic of Austria, is a country in Central Europe comprising 9 federated states. Its capital, largest city and one of nine states is Vienna. Austria has an area of 83,879 km2, a population of nearly 9 million people and a nominal GDP of $477 billion, it is bordered by the Czech Republic and Germany to the north and Slovakia to the east and Italy to the south, Switzerland and Liechtenstein to the west. The terrain is mountainous, lying within the Alps; the majority of the population speaks local Bavarian dialects as their native language, German in its standard form is the country's official language. Other regional languages are Hungarian, Burgenland Croatian, Slovene. Austria played a central role in European History from the late 18th to the early 20th century, it emerged as a margraviate around 976 and developed into a duchy and archduchy. In the 16th century, Austria started serving as the heart of the Habsburg Monarchy and the junior branch of the House of Habsburg – one of the most influential royal houses in history.
As archduchy, it was a major component and administrative centre of the Holy Roman Empire. Following the Holy Roman Empire's dissolution, Austria founded its own empire in the 19th century, which became a great power and the leading force of the German Confederation. Subsequent to the Austro-Prussian War and the establishment of a union with Hungary, the Austro-Hungarian Empire was created. Austria was involved in both world wars. Austria is a parliamentary representative democracy with a President as head of state and a Chancellor as head of government. Major urban areas of Austria include Graz, Linz and Innsbruck. Austria is ranked as one of the richest countries in the world by per capita GDP terms; the country has developed a high standard of living and in 2018 was ranked 20th in the world for its Human Development Index. The republic declared its perpetual neutrality in foreign political affairs in 1955. Austria has been a member of the United Nations since 1955 and joined the European Union in 1995.
It is a founding member of the OECD and Interpol. Austria signed the Schengen Agreement in 1995, adopted the euro currency in 1999; the German name for Austria, Österreich, derives from the Old High German Ostarrîchi, which meant "eastern realm" and which first appeared in the "Ostarrîchi document" of 996. This word is a translation of Medieval Latin Marchia orientalis into a local dialect. Another theory says that this name comes from the local name of the mountain whose original Slovenian name is "Ostravica" - because it is steep on both sides. Austria was a prefecture of Bavaria created in 976; the word "Austria" was first recorded in the 12th century. At the time, the Danube basin of Austria was the easternmost extent of Bavaria; the Central European land, now Austria was settled in pre-Roman times by various Celtic tribes. The Celtic kingdom of Noricum was claimed by the Roman Empire and made a province. Present-day Petronell-Carnuntum in eastern Austria was an important army camp turned capital city in what became known as the Upper Pannonia province.
Carnuntum was home for 50,000 people for nearly 400 years. After the fall of the Roman Empire, the area was invaded by Bavarians and Avars. Charlemagne, King of the Franks, conquered the area in AD 788, encouraged colonization, introduced Christianity; as part of Eastern Francia, the core areas that now encompass Austria were bequeathed to the house of Babenberg. The area was known as the marchia Orientalis and was given to Leopold of Babenberg in 976; the first record showing the name Austria is from 996, where it is written as Ostarrîchi, referring to the territory of the Babenberg March. In 1156, the Privilegium Minus elevated Austria to the status of a duchy. In 1192, the Babenbergs acquired the Duchy of Styria. With the death of Frederick II in 1246, the line of the Babenbergs was extinguished; as a result, Ottokar II of Bohemia assumed control of the duchies of Austria and Carinthia. His reign came to an end with his defeat at Dürnkrut at the hands of Rudolph I of Germany in 1278. Thereafter, until World War I, Austria's history was that of its ruling dynasty, the Habsburgs.
In the 14th and 15th centuries, the Habsburgs began to accumulate other provinces in the vicinity of the Duchy of Austria. In 1438, Duke Albert V of Austria was chosen as the successor to his father-in-law, Emperor Sigismund. Although Albert himself only reigned for a year, henceforth every emperor of the Holy Roman Empire was a Habsburg, with only one exception; the Habsburgs began to accumulate territory far from the hereditary lands. In 1477, Archduke Maximilian, only son of Emperor Frederick III, married the heiress Maria of Burgundy, thus acquiring most of the Netherlands for the family. In 1496, his son Philip the Fair married Joanna the Mad, the heiress of Castile and Aragon, thus acquiring Spain and its Italian and New World appendages for the Habsburgs. In 1526, following the Battle of Mohács, Bohemia and the part of Hungary not occupied by the Ottomans came under Austrian rule. Ottoman expansion into Hungary led to frequent conflicts between the two empires evident in the Long War of 1593 to 1606.
The Turks made incursions into Styria nearly 20 times, of which some are c
Michiel Hazewinkel is a Dutch mathematician, Emeritus Professor of Mathematics at the Centre for Mathematics and Computer and the University of Amsterdam known for his 1978 book Formal groups and applications and as editor of the Encyclopedia of Mathematics. Born in Amsterdam to Jan Hazewinkel and Geertrude Hendrika Werner, Hazewinkel studied at the University of Amsterdam, he received his BA in Mathematics and Physics in 1963, his MA in Mathematics with a minor in Philosophy in 1965 and his PhD in 1969 under supervision of Frans Oort and Albert Menalda for the thesis "Maximal Abelian Extensions of Local Fields". After graduation Hazewinkel started his academic career as Assistant Professor at the University of Amsterdam in 1969. In 1970 he became Associate Professor at the Erasmus University Rotterdam, where in 1972 he was appointed Professor of Mathematics at the Econometric Institute. Here he was thesis advisor of Roelof Stroeker, M. van de Vel, Jo Ritzen, Gerard van der Hoek. From 1973 to 1975 he was Professor at the Universitaire Instelling Antwerpen, where Marcel van de Vel was his PhD student.
From 1982 to 1985 he was appointed part-time Professor Extraordinarius in Mathematics at the Erasmus Universiteit Rotterdam, part-time Head of the Department of Pure Mathematics at the Centre for Mathematics and Computer in Amsterdam. In 1985 he was appointed Professor Extraordinarius in Mathematics at the University of Utrecht, where he supervised the promotion of Frank Kouwenhoven, Huib-Jan Imbens, J. Scholma and F. Wainschtein. At the Centre for Mathematics and Computer CWI in Amsterdam in 1988 he became Professor of Mathematics and head of the Department of Algebra and Geometry until his retirement in 2008. Hazewinkel has been managing editor for journals as Nieuw Archief voor Wiskunde since 1977, he was managing editor for the book series Mathematics and Its Applications for Kluwer Academic Publishers in 1977. Hazewinkel was member of 15 professional societies in the field of Mathematics, participated in numerous administrative tasks in institutes, Program Committee, Steering Committee, Consortiums and Boards.
In 1994 Hazewinkel was elected member of the International Academy of Computer Sciences and Systems. Hazewinkel has authored and edited several books, numerous articles. Books, selection: 1970. Géométrie algébrique-généralités-groupes commutatifs. With Michel Demazure and Pierre Gabriel. Masson & Cie. 1976. On invariants, canonical forms and moduli for linear, finite dimensional, dynamical systems. With Rudolf E. Kalman. Springer Berlin Heidelberg. 1978. Formal groups and applications. Vol. 78. Elsevier. 1993. Encyclopaedia of Mathematics. Ed. Vol. 9. Springer. Articles, a selection: Hazewinkel, Michiel. "Moduli and canonical forms for linear dynamical systems II: The topological case". Mathematical Systems Theory. 10: 363–385. Doi:10.1007/BF01683285. Archived from the original on 12 December 2013. Hazewinkel, Michiel. "On Lie algebras and finite dimensional filtering". Stochastics. 7: 29–62. Doi:10.1080/17442508208833212. Archived from the original on 12 December 2013. Hazewinkel, M.. J.. "Nonexistence of finite-dimensional filters for conditional statistics of the cubic sensor problem".
Systems & Control Letters. 3: 331–340. Doi:10.1016/0167-691190074-9. Hazewinkel, Michiel. "The algebra of quasi-symmetric functions is free over the integers". Advances in Mathematics. 164: 283–300. Doi:10.1006/aima.2001.2017. Homepage
Q. E. D. is an initialism of the Latin phrase "quod erat demonstrandum" meaning "what was to be shown" or "thus it has been demonstrated." Traditionally, the abbreviation is placed at the end of a mathematical proof or philosophical argument to indicate that the proof or argument is complete. The phrase, quod erat demonstrandum, is a translation into Latin from the Greek ὅπερ ἔδει δεῖξαι. Translating from the Latin into English yields, "what was to be demonstrated", translating the Greek phrase ὅπερ ἔδει δεῖξαι produces a different meaning. Since the verb "δείκνυμι" means to show or to prove, a better translation from the Greek would read, "The thing it was required to have shown."The Greek phrase was used by many early Greek mathematicians, including Euclid and Archimedes. During the European Renaissance, scholars wrote in Latin, phrases such as Q. E. D. were used to conclude proofs. The most famous use of Q. E. D. in a philosophical argument is found in the Ethics of Baruch Spinoza, published posthumously in 1677.
Written in Latin, it is considered by many to be Spinoza's magnum opus. The style and system of the book are, as Spinoza says, "demonstrated in geometrical order", with axioms and definitions followed by propositions. For Spinoza, this is a considerable improvement over René Descartes's writing style in the Meditations, which follows the form of a diary. There is another Latin phrase with a different meaning shortened but being less common in use. Quod erat faciendum, originating from the Greek geometers' closing ὅπερ ἔδει ποιῆσαι, meaning "which had to be done"; because of the difference in meaning, the two phrases should not be confused. Euclid used the phrase, Quod Erat Faciendum, to close propositions that were not proofs of theorems, but constructions. For example, Euclid's first proposition showing how to construct an equilateral triangle, given one side, is concluded this way. Q. E. D. has acquired many translations in various languages, including: There is no common formal English equivalent, although the end of a proof may be announced with a simple statement such as "this completes the proof", "as required", "hence proved", "ergo", or by using a similar locution.
WWWWW or W5 – an abbreviation of "Which Was What Was Wanted" – has been used similarly. This is considered to be more tongue-in-cheek than the usual Halmos symbol or Q. E. D. Due to the paramount importance of proofs in mathematics, mathematicians since the time of Euclid have developed conventions to demarcate the beginning and end of proofs. In printed English language texts, the formal statements of theorems and propositions are set in italics by tradition; the beginning of a proof follows thereafter, is indicated by the word "proof" in boldface or italics. On the other hand, several symbolic conventions exist to indicate the end of a proof. While some authors still use the classical abbreviation, Q. E. D, it is uncommon in modern mathematical texts. Paul Halmos pioneered the use of a solid black square at the end of a proof as a Q. E. D symbol, a practice which has become standard, although not universal. Halmos adopted this use of a symbol from magazine typography customs in which simple geometric shapes had been used to indicate the end of an article.
This symbol was called the tombstone or Halmos symbol or a halmos by mathematicians. The Halmos symbol is drawn on chalkboard to signal the end of a proof during a lecture, although this practice is not so common as its use in printed text; the tombstone symbol appears in TeX as the character ◼ and sometimes, as a ◻. In the AMS Theorem Environment for LaTeX, the hollow square is the default end-of-proof symbol. Unicode explicitly provides the "End of proof" character, U+220E; some authors use other Unicode symbols to note the end of a proof, including, ▮, ‣. Other authors have adopted four forward slashes. In other cases, authors have elected to segregate proofs typographically by displaying them as indented blocks. In Joseph Heller's book Catch-22, the Chaplain, having been told to examine a forged letter signed by him, verified that his name was in fact there, his investigator replied, "Then you wrote it. Q. E. D." The chaplain said he didn't write it and that it wasn't his handwriting, to which the investigator replied, "Then you signed your name in somebody else's handwriting again."In the 1978 science-fiction radio comedy, in the television and novel adaptations of The Hitchhiker's Guide to the Galaxy, "Q.
E. D." is referred to in the Guide's entry for the babel fish, when it is claimed that the babel fish – which serves the "mind-bogglingly" useful purpose of being able to translate any spoken language when inserted into a person's ear – is used as evidence for existence and non-existence of God. The exchange from the novel is as follows: "'I refuse to prove I exist,' says God,'for proof denies faith, without faith I am nothing."But,' says Man,'The babel fish is a dead giveaway, isn't it? It could not have evolved by chance, it proves you exist, so therefore, by your own arguments, you don't. QED."Oh dear,' says God,'I hadn't thought of that,' and promptly vanishes in a puff of logic."In Neal Stephenson's 1999 novel Cryptonomicon, Q. E. D. is used as a punchline to several humorous anecdotes in which characters go to great lengths to prove something non-mathematical. Singer-songwriter Thoma
Encyclopedia of Mathematics
The Encyclopedia of Mathematics is a large reference work in mathematics. It is available in book form and on CD-ROM; the 2002 version contains more than 8,000 entries covering most areas of mathematics at a graduate level, the presentation is technical in nature. The encyclopedia is edited by Michiel Hazewinkel and was published by Kluwer Academic Publishers until 2003, when Kluwer became part of Springer; the CD-ROM contains three-dimensional objects. The encyclopedia has been translated from the Soviet Matematicheskaya entsiklopediya edited by Ivan Matveevich Vinogradov and extended with comments and three supplements adding several thousand articles; until November 29, 2011, a static version of the encyclopedia could be browsed online free of charge online. This URL now redirects to the new wiki incarnation of the EOM. A new dynamic version of the encyclopedia is now available as a public wiki online; this new wiki is a collaboration between the European Mathematical Society. This new version of the encyclopedia includes the entire contents of the previous online version, but all entries can now be publicly updated to include the newest advancements in mathematics.
All entries will be monitored for content accuracy by members of an editorial board selected by the European Mathematical Society. Vinogradov, I. M. Matematicheskaya entsiklopediya, Sov. Entsiklopediya, 1977. Hazewinkel, M. Encyclopaedia of Mathematics, Kluwer, 1994. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 1, Kluwer, 1987. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 2, Kluwer, 1988. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 3, Kluwer, 1989. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 4, Kluwer, 1989. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 5, Kluwer, 1990. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 6, Kluwer, 1990. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 7, Kluwer, 1991. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 8, Kluwer, 1992. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 9, Kluwer, 1993. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 10, Kluwer, 1994. Hazewinkel, M. Encyclopaedia of Mathematics, Supplement I, Kluwer, 1997. Hazewinkel, M. Encyclopaedia of Mathematics, Supplement II, Kluwer, 2000.
Hazewinkel, M. Encyclopaedia of Mathematics, Supplement III, Kluwer, 2002. Hazewinkel, M. Encyclopaedia of Mathematics on CD-ROM, Kluwer, 1998. Encyclopedia of Mathematics, public wiki monitored by an editorial board under the management of the European Mathematical Society. List of online encyclopedias Official website Publications by M. Hazewinkel, at ResearchGate
Infimum and supremum
In mathematics, the infimum of a subset S of a ordered set T is the greatest element in T, less than or equal to all elements of S, if such an element exists. The term greatest lower bound is commonly used; the supremum of a subset S of a ordered set T is the least element in T, greater than or equal to all elements of S, if such an element exists. The supremum is referred to as the least upper bound; the infimum is in a precise sense dual to the concept of a supremum. Infima and suprema of real numbers are common special cases that are important in analysis, in Lebesgue integration. However, the general definitions remain valid in the more abstract setting of order theory where arbitrary ordered sets are considered; the concepts of infimum and supremum are similar to minimum and maximum, but are more useful in analysis because they better characterize special sets which may have no minimum or maximum. For instance, the positive real numbers ℝ+ does not have a minimum, because any given element of ℝ+ could be divided in half resulting in a smaller number, still in ℝ+.
There is, however one infimum of the positive real numbers: 0, smaller than all the positive real numbers and greater than any other real number which could be used as a lower bound. A lower bound of a subset S of a ordered set is an element a of P such that a ≤ x for all x in S. A lower bound a of S is called an infimum of S. An upper bound of a subset S of a ordered set is an element b of P such that b ≥ x for all x in S. An upper bound b of S is called a supremum of S if for all upper bounds z of S in P, z ≥ b. Infima and suprema do not exist. Existence of an infimum of a subset S of P can fail if S has no lower bound at all, or if the set of lower bounds does not contain a greatest element. However, if an infimum or supremum does exist, it is unique. Ordered sets for which certain infima are known to exist become interesting. For instance, a lattice is a ordered set in which all nonempty finite subsets have both a supremum and an infimum, a complete lattice is a ordered set in which all subsets have both a supremum and an infimum.
More information on the various classes of ordered sets that arise from such considerations are found in the article on completeness properties. If the supremum of a subset S exists, it is unique. If S contains a greatest element that element is the supremum. If S contains a least element that element is the infimum; the infimum of a subset S of a ordered set P, assuming it exists, does not belong to S. If it does, it is a minimum or least element of S. Similarly, if the supremum of S belongs to S, it is a maximum or greatest element of S. For example, consider the set of negative real numbers; this set has no greatest element, since for every element of the set, there is another, element. For instance, for any negative real number x, there is another negative real number x 2, greater. On the other hand, every real number greater than or equal to zero is an upper bound on this set. Hence, 0 is the least upper bound of the negative reals, so the supremum is 0; this set has a supremum but no greatest element.
However, the definition of maximal and minimal elements is more general. In particular, a set can have many maximal and minimal elements, whereas infima and suprema are unique. Whereas maxima and minima must be members of the subset, under consideration, the infimum and supremum of a subset need not be members of that subset themselves. A ordered set may have many minimal upper bounds without having a least upper bound. Minimal upper bounds are those upper bounds for which there is no smaller element, an upper bound; this does not say that each minimal upper bound is smaller than all other upper bounds, it is not greater. The distinction between "minimal" and "least" is only possible when the given order is not a total one. In a ordered set, like the real numbers, the concepts are the same; as an example, let S be the set of all finite subsets of natural numbers and consider the ordered set obtained by taking all sets from S together with the set of integers ℤ and the set of positive real numbers ℝ+, ordered by subset inclusion as above.
Both ℤ and ℝ+ are greater than all finite sets of natural numbers. Yet, neither is ℝ+ smaller than ℤ nor is the converse true: both sets are minimal upper bounds but none is a supremum; the least-upper-bound property is an example of the aforementioned completeness properties, typical for the set of real numbers. This property is sometimes called Dedekind completeness. If an ordered set S has the property that every nonempty subset of S having an upper bound has a least upper bound S is said to have the least-upper-bound property; as noted above, the set ℝ of all real numbers has the least-upper-bound property. The set ℤ of integers has the least-upper-bound property.