:: wikimiki.org ::

Function Composition

Function composition

In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. The functions f: X → Y and g: Y → Z can be composed by first applying f to an argument x and then applying g to the result. Thus one obtains a function g o f: X → Z defined by (g o f)(x) = g(f(x)) for all x in X. The notation g o f is read as "g circle f" or "g composed with f". function As an example, suppose that an airplane's height at time t is given by the function h(t) and that the oxygen concentration at height x is given by the function c(x). Then (c o h)(t) describes the oxygen concentration around the plane at time t. In the mid-20th century, some mathematicians decided that writing "g o f" to mean "first apply f, then apply g" was too confusing and decided to change notations. They wrote "xf" for "f(x)" and "xfg" for "g(f(x))". However, this movement never caught on, and nowadays this notation is found only in old books. The composition of functions is always associative. That is, if f, g, and h are three functions with suitably chosen domains and codomains, then f o (g o h) = (f o g) o h. Since there is no distinction between the choices of placement of parentheses, they may be safely left off. As a result the set of bijective functions f: X → X form a group with respect to the composition operator. The functions g and f commute with each other if g o f = f o g. In general, composition of functions will not be commutative. Only the simplest functions will be commutative under composition. Derivatives of compositions involving differentiable functions can be found using the chain rule. "Higher" derivatives of such functions are given by Faà di Bruno's formula.

Functional powers

If Y⊂X then f may compose with itself; this is sometimes denoted f ². Thus: :(f o f)(x) = f(f(x)) = f ²(x) :(f o f o f)(x) = f(f(f(x))) = f ³(x) Repeated composition of a function with itself is sometimes called function iteration. The functional powers f o f ⁿ = f ⁿ o f = f ⁿ⁺¹ for natural n follow immediately.
- By convention, f ⁰ = id_D(f) (the identity map on the domain of f).
- If f:X→X admits an inverse function, negative functional powers f ^-k (k > 0) are defined as the opposite power of the inverse function, (f ⁻¹)^k. Note: If f takes its values in a ring (in particular for real or complex-valued f ), there is a risk of confusion, as f ⁿ could also stand for the n-fold product of f, e.g. f ²(x) = f(x) · f(x). (For usual numerical functions, usually the latter is meant, at least for positive exponents. For example, in trigonometry, this superscript notation represents standard exponentiation when used with trigonometric functions: sin²(x) = sin(x) · sin(x). However, for negative exponents (especially −1), it nevertheless usually refers to the inverse function, e.g., tan⁻¹ = arctan (≠ 1/tan). In some cases, an expression for f in g(x) = f ^r(x) can be derived from the rule for g given non-integer values of r. This is called fractional iteration. Iterated functions occur naturally in the study of fractals and dynamical systems.

Composition operator

Given a function g, the composition operator

C_g

is defined as that operator which maps functions to functions as :

C_g f = f \circ g

Composition operators are studied in the field of operator theory. :See main article composition operator for more details.

Mathematics

Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions. Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. The remarkable fact that the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics." Today, the natural sciences, engineering, economics, and medicine depend heavily on new mathematical discoveries. The word "mathematics" comes from the Greek μάθημα (máthema) meaning "science, knowledge, or learning" and μαθηματικός (mathematikós) meaning "fond of learning". It is often abbreviated maths in Commonwealth English and math in North American English.

History

:Main article: History of mathematics The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g. addition, subtraction, multiplication and division), naturally followed. Monolithic monuments testify to a knowledge of geometry. Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called khipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse. Historically, the major disciplines within mathematics arose, from the start of recorded history, out of the need to do calculations on taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics, into the studies of quantity, structure, space, and change. Mathematics since has been much extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries have been made throughout history and continue to be made today.

Inspiration, pure and applied mathematics, and aesthetics

Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton invented infinitesimal calculus and Feynman his Feynman path integral using a combination of reasoning and physical insight, and today's string theory also inspires new mathematics. Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between pure mathematics and applied mathematics. Within applied mathematics, two major areas have split off and become disciplines in their own right, statistics and computer science. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty also in a clever proof, such as Euclid's proof that there are infinitely many prime numbers, and in a numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in "A Mathematicians Apology" expressed the belief that these esthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. Main article: Mathematical beauty.

Notation, language, and rigor

Mathematical writing is not easily accessible to the layperson. A Brief History of Time, Stephen Hawking's 1988 bestseller, contained a single mathematical equation. This was the author's compromise with the publisher's advice, that each equation would halve the sales. The reasons for the inaccessibility even of carefully-expressed mathematics can be partially explained. Contemporary mathematicians strive to be as clear as possible in the things they say and especially in the things they write (this they have in common with lawyers). They refer to rigor. To accomplish rigor, mathematicians have extended natural language. There is precisely-defined vocabulary for referring to mathematical objects, and stating certain common relations. There is an accompanying mathematical notation which, like musical notation, has a definite content and also has a strict grammar (under the influence of computer science, more often now called syntax). Some of the terms used in mathematics are also common outside mathematics, such as ring, group and category; but are not such that one can infer the meanings. Some are specific to mathematics, such as homotopy and Hilbert space. It was said that Henri Poincaré was only elected to the Académie Française so that he could tell them how to define automorphe in their dictionary. Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow mechanically from axioms by means of formal axiomatic reasoning. This is to avoid mistaken 'theorems', based on fallible intuitions; of which plenty of examples have occurred in the history of the subject (for example, in mathematical analysis). Axioms in traditional thought were 'self-evident truths', but that conception turns out not to be workable in pushing the mathematical boundaries. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (strong enough) axiom system has undecidable formulas; and so a final axiomatization of mathematics is unavailable. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.

Is mathematics a science?

Carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. The mathematician-physicist Leon M. Lederman has quipped: "The physicists defer only to mathematicians, and the mathematicians defer only to God (though you may be hard pressed to find a mathematician that modest)." If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics. [http://info.med.yale.edu/therarad/summers/ziman.htm] In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences.

Overview of fields of mathematics

As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations) and to the empirical mathematics of the various sciences (applied mathematics). The study of quantity starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are characterized in arithmetic. The deeper properties of whole numbers are studied in number theory. The study of structure began with investigations of Pythagorean triples. Neolithic monuments on the British Isles are constructed using Pythagorean triples. Eventually, this led to the invention of more abstract numbers, such as the square root of two. The deeper structural properties of numbers are studied in abstract algebra and the investigation of groups, rings, fields and other abstract number systems. Included is the important concept of vectors, generalized to vector spaces and studied in linear algebra. The study of vectors combines three of the fundamental areas of mathematics, quantity, structure, and space. The study of space originates with geometry, beginning with Euclidean geometry. Trigonometry combines space and number. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles, calculus on manifolds. Within algebraic geometry is the description of geometric objects as solution sets of polynomal equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may be the greatest growth area in 20th century mathematics. Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a most useful tool. The central concept used to describe a changing quantity is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. These have been generalized, with the inclusion of the square root of negative one, to the complex numbers, which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior. Beyond quantity, structure, space, and change are areas of pure mathematics that can be approached only by deductive reasoning. In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic, which divides into recursion theory, model theory, and proof theory, is now closely linked to computer science. When electronic computers were first conceived, several essential theoretical concepts in computer science were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, and information theory. Many of those topics are now investigated in theoretical computer science. Discrete mathematics is the common name for the fields of mathematics most generally useful in computer science. An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a part. It is used in all the sciences. Numerical analysis investigates methods for using computers to efficiently solve a broad range of mathematical problems that are typically beyond human capacity, and taking rounding errors or other sources of error into account to obtain credible answers.

Major themes in mathematics

An alphabetical and subclassified list of mathematical topics is available. The following list of themes and links gives just one possible view. For a fuller treatment, see Areas of mathematics or the list of lists of mathematical topics.

Change

:Ways to express and handle change in mathematical functions, and changes between numbers. : :Arithmetic – Calculus – Vector calculus – Analysis – Differential equations – Dynamical systems – Chaos theory – List of functions

Foundations and methods

:Approaches to understanding the nature of mathematics. :philosophy of mathematics – mathematical intuitionism – mathematical constructivism – foundations of mathematics – set theory – symbolic logic – model theory – category theory – Logic – reverse mathematics – table of mathematical symbols

Discrete mathematics

:Discrete mathematics involves techniques that apply to objects that can only take on specific, separated values. : :Combinatorics – Naive set theory – Theory of computation– Cryptography – Graph theory

Applied mathematics

:Applied mathematics uses the full knowledge of mathematics to solve real-world problems. :Mathematical physics – Mechanics – Fluid mechanics – Numerical analysis – Optimization – Probability – Statistics – Mathematical economics – Financial mathematics – Game theory – Mathematical biology – Cryptography – Information theory

Important theorems

:These theorems have interested mathematicians and non-mathematicians alike. :See list of theorems for more :Pythagorean theorem – Fermat's last theorem – Gödel's incompleteness theorems – Fundamental theorem of arithmetic – Fundamental theorem of algebra – Fundamental theorem of calculus – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's identity – classification theorems of surfaces – Gauss-Bonnet theorem – Quadratic reciprocity – Riemann-Roch theorem.

Important conjectures

See list of conjectures for more :These are some of the major unsolved problems in mathematics. :Goldbach's conjecture – Twin Prime Conjecture – Riemann hypothesis – Poincaré conjecture – Collatz conjecture – P=NP? – open Hilbert problems.

History and the world of mathematicians

Mathematics and other fields

:Mathematics and architecture – Mathematics and education – Mathematics of musical scales

Common misconceptions

Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems. Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudoscience and real science. The misconceptions involved are normally based on:
- misunderstanding of the implications of mathematical rigour;
- attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review, with assumptions of bias;
- lack of familiarity with, and therefore underestimation of, the existing literature. The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. And like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne. Mathematics is not accountancy. Although arithmetic computation is crucial to accountants, their main concern is to verify that computations are correct through a system of doublechecks. Advances in abstract mathematics are mostly irrelevant to the efficiency of concrete bookkeeping, but the use of computers clearly does matter. Mathematics is not numerology. Numerology uses modular arithmetic to reduce names and dates down to numbers, but assigns emotions or traits to these numbers intuitively or on the basis of traditions. Mathematical concepts and theorems need not correspond to anything in the physical world. In the case of geometry, for example, it is not relevant to mathematics to know whether points and lines exist in any physical sense, as geometry starts from axioms and postulates about abstract entities called "points" and "lines" that we feed into the system. While these axioms are derived from our perceptions and experience, they are not dependent on them. And yet, mathematics is extremely useful for solving real-world problems. It is this fact that led Eugene Wigner to write an essay on The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Mathematics is not about unrestricted theorem proving, any more than literature is about the construction of grammatically correct sentences. However, theorems are elements of formal theories, and in some cases computers can generate proofs of these theorems more or less automatically, by means of automated theorem provers. These techniques have proven useful in formal verification of programs and hardware designs. However, they are unlikely to generate (in the near term, at least) mathematics with any widely recognized aesthetic value.

Bibliography

- Benson, Donald C., The Moment Of Proof: Mathematical Epiphanies (1999).
- Courant, R. and H. Robbins, What Is Mathematics? (1941);
- Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Birkhäuser, Boston, Mass., 1980. A gentle introduction to the world of mathematics.
- Boyer, Carl B., History of Mathematics, Wiley, 2nd edition 1998 available, 1st edition 1968 . A concise history of mathematics from the Concept of Number to contemporary Mathematics.
- Gullberg, Jan, Mathematics--From the Birth of Numbers. W.W. Norton, 1996. An encyclopedic overview of mathematics presented in clear, simple language.
- Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer Academic Publishers 2000. A translated and expanded version of a Soviet math encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM.
- Kline, M., Mathematical Thought from Ancient to Modern Times (1973).
- Pappas, Theoni, The Joy Of Mathematics (1989).

External links

- [http://www.cut-the-knot.org/ Interactive Mathematics Miscellany and Puzzles] — A collection of articles on various math topics, with interactive Java illustrations at cut-the-knot
- Rusin, Dave: [http://www.math-atlas.org/ The Mathematical Atlas]. A guided tour through the various branches of modern mathematics.
- Stefanov, Alexandre: [http://us.geocities.com/alex_stef/mylist.html Textbooks in Mathematics]. A list of free online textbooks and lecture notes in mathematics.
- Weisstein, Eric et al.: [http://www.mathworld.com/ MathWorld: World of Mathematics]. An online encyclopedia of mathematics.
- Polyanin, Andrei: [http://eqworld.ipmnet.ru/ EqWorld: The World of Mathematical Equations]. An online resource focusing on algebraic, ordinary differential, partial differential (mathematical physics), integral, and other mathematical equations.
- A mathematical thesaurus maintained by the [http://nrich.maths.org/ NRICH] project at the University of Cambridge (UK), [http://thesaurus.maths.org/ Connecting Mathematics]
- [http://planetmath.org/ Planet Math]. An online math encyclopedia under construction, focusing on modern mathematics. Uses the GFDL, allowing article exchange with Wikipedia. Uses TeX markup.
- [http://www.mathforge.net/ Mathforge]. A news-blog with topics ranging from popular mathematics to popular physics to computer science and education.
- [http://www.youngmath.net/concerns Young Mathematicians Network (YMN)]. A math-blog "Serving the Community of Young Mathematicians". Topics include: Math News, Grad and Undergrad Life, Job Search, Career, Work & Family, Teaching, Research, Misc...
- [http://metamath.org/ Metamath]. A site and a language, that formalize math from its foundations.
- [http://world.std.com/~reinhold/dir/mathmovies.html Math in the Movies]. A guide to major motion pictures with scenes of real mathematics
- [http://math.cofc.edu/faculty/kasman/MATHFICT/default.html Mathematics in fiction]. Links to works of fiction that refer to mathematics or mathematicians.
- [http://www.mathhelpforum.com/math-help Math Help Forum]. A forum, for math help, math discussion and debate.
- [http://www.sosmath.com/CBB S.O.S. Mathematics Cyberboard] a math help forum which incorporates a LaTeX extension, making it easier for members to write and display math formulae.
- [http://www-history.mcs.st-and.ac.uk/~history/ Mathematician Bibliography]. Extensive history and quotes from all famous mathematicians.
- [http://www.physicsmathforums.com/ Physics Math Forums]
- Category:School subjects fiu-vro:Matõmaatiga zh-min-nan:Sò·-ha̍k ko:수학 ms:Matematik ja:数学 simple:Mathematics th:คณิตศาสตร์

20th century

The 20th century lasted from 1901 to 2000 in the Gregorian calendar. Common usage sometimes regards it as lasting from 1900 to 1999, but this is incorrect since counting of calendar years begins with the year 1. The 20th century is also sometimes known as the nineteen hundreds (1900s). Decades are almost always considered as starting with the "0" year and named accordingly ("1960s", etc.). However, a number of arguments have been used to justify the common usage. One was advanced, erroneously, by Stephen Jay Gould. He claimed that the first decade had only nine years, thus contradicting the definition of decade equaled 10 years. Another argument is that the astronomical year numbering system for years does have a year zero, the year normally known as 1 BC. In 2000 the International Organization for Standardization clarified ISO 8601 to use the astronomical year numbering system, which could be interpreted as retrospectively endorsing all the people who had celebrated the new century a few months earlier. The term is also used to describe various periods that overlap with the calendar definition, most notably the Short twentieth century, which claims that the 20th Century spanned from 1914 to 1989, rendering the pre-WWI 1900s into the 19th Century and putting the 1990s at the beginning of the 21st Century. Indeed, the part of the 20th Century before World War I is quite identical to the late 1800s culturally and technologically and the 1990s decade pointed in many ways (such as the rise of the Internet) to the 21st Century and is seen by some as not being truly a part of the 20th Century.

Overview

The twentieth century saw a remarkable shift in the way that vast numbers of people lived, as a result of technological, medical, social, ideological, and political innovations. Terms like ideology, world war, genocide, and nuclear war entered common usage and became an influence on the lives of everyday people. War reached an unprecedented scale and level of sophistication; in the Second World War (1939-1945) alone, approximately 57 million people died, mainly due to massive improvements in weaponry. The trends of mechanization of goods and services and networks of global communication, which were begun in the 19th century, continued at an ever-increasing pace in the 20th. In spite of the terror and chaos, the 20th century saw many attempts at world peace. As the 35th President of the United States John F. Kennedy said: :What kind of peace do we seek? I am talking about a genuine peace, the kind of peace that makes life on earth worth living. Not merely peace in our time, but peace in all time. Our problems are man-made, therefore they can be solved by man. For in the final analysis, our most basic common link is that we all inhabit this small planet, we all breathe the same air, we all cherish our children's future, and we are all mortal. Virtually every aspect of life in virtually every human society changed in some fundamental way or another during the twentieth century and for the first time, any individual could influence the course of history no matter their background. Arguably, the 20th century re-shaped the face of the planet in more ways than any previous century.
- Death rates
- Infant mortality
- Infectious disease
- Life expectancy
- Maternal death rates
- Battles Scientific discoveries such as relativity and quantum physics radically changed the worldview of scientists, causing them to realize that the universe was much more complex than they had previously believed, and dashing the hopes at the end of the preceding century that the last few details of knowledge were about to be filled in. For a more coherent overview of the historical events of the century, see The 20th century in review. The 20th century has sometimes been called, both within and outside the United States, the American Century, though this is a controversial term.

Important developments, events and achievements

Science and technology

- The assembly line and mass production of motor vehicles and other goods allowed manufacturers to produce more and cheaper products. This allowed the automobile to become the most important means of transportation.
- The invention of heavier-than-air flying machines and the jet engine allowed for the world to become "smaller". Space flight increased knowledge of the rest of the universe and allowed for global real-time communications via geosynchronous satellites.
- Mass media technologies such as film, radio, and television allow the communication of political messages and entertainment with unprecedented impact
- Mass availability of the telephone and later, the computer, especially through the Internet, provides people with new opportunities for near-instantaneous communication
- Applied electronics, notably in its miniaturized form as integrated circuits, made possible the above mentioned rise of mass media, telecommunications, ubiquitous computing, and all kinds of "intelligent" appliances; as well as many advances in natural sciences such as physics, by the use of exponentially growing calculation power (see supercomputer).
- The development of Nitrogen fertilizer, pesticides and herbicides resulted in significantly higher agricultural yield.
- Advances in fundamental physics through the theory of relativity and quantum mechanics led to the development of nuclear weapons (known informally as "the Bomb" and dropped on the industrial town of Hiroshima and the historic one of Nagasaki), the nuclear reactor, and the laser. Fusion power was studied extensively but remained an experimental technology at the end of the century.
- Inventions such as the washing machine and air conditioning led to an increase in both the quantity and quality of leisure time for the middle class in Western societies.
- Most influential inventions in the 20th century: antibiotics, oral contraceptives, new plastics, transistors, Internet
- More...

Wars and politics

- Democratic nations began to extend voting privileges to all adults.
- Rising nationalism and increasing national awareness were among the causes of World War I, the first of two wars to involve all the major world powers including Germany, France, Italy, Japan, the United States and the British Commonwealth. World War I led to the creation of many new countries, especially in Eastern Europe. Ironically, it was said by many to be the 'War to end all Wars'.
- The economic and political aftermath of World War I led to the rise of Fascism and Nazism in Europe, and shortly to World War II. This war also involved Asia and the Pacific, in the form of Japanese aggression against China and the United States. While the First World War mainly cost lives among soldiers, civilians suffered greatly in the Second -- from the bombing of cities on both sides, and in the unprecedented German genocide of the Jews and others, known as the Holocaust.
- During World War I, in Russia the Bolshevik putsch led to the Russian Revolution of 1917. After the Soviet Union's involvement in World War II, Communism became a major force in global politics, spreading all over the world: notably, to Eastern Europe, China, Indochina and Cuba. This led to the Cold War and proxy wars with the western world, including wars in Korea (1950-53) and Vietnam (1957 - 75).
- The "fall of Communism" in the late 1980s freed Eastern and Central Europe from Soviet supremacy. It also led to the dissolution of the Soviet Union and Yugoslavia into successor states, many rife with ethnic nationalism, and left the United States as the world's superpower.
- Through the League of Nations and, after World War II, the United Nations, international cooperation increased. Other efforts included the formation of the European Union, leading to a common currency in much of Western Europe, the euro around the turn of the millennium.
- The end of colonialism led to the independence of many African and Asian countries. During the Cold War, many of these aligned with the USA, the USSR, or China for defense.
- The creation of Israel, a Jewish state in a mostly Arab region of the world, fueled many conflicts in the region, which were also influenced by the vast oil fields in many of the Arab countries.
- The term Southeast Asia coined.

Culture and entertainment

- Movies, music and the media had a major influence on fashion and trends in all aspects of life. As many movies and music originate from the United States, American culture spread rapidly over the world.
- After gaining political rights in the United States and much of Europe in the first part of the century, and with the advent of new birth control techniques women became more independent throughout the century.
- Rock and Roll and Jazz styles of music are developed in the United States, and quickly become the dominant forms of popular music in America, and later, the world. The Beatles, a 1960s British Rock and Roll band, becomes one of the most successful acts of all time, and is credited, in their experimental later albums, with permanently changing what was thought possible in popular music.
- Modern art developed new styles such as expressionism, cubism, and surrealism.
- The automobile provided vastly increased transportation capabilities for the average member of Western societies in the early to mid-century, spreading even further later on. City design throughout most of the West became focused on transport via car. The car became a leading symbol of modern society, with styles of car suited to and symbolic of particular lifestyles.
- Sports became an important part of society, becoming an activity not only for the privileged. Watching sports, later also on television, became a popular activity.

Disease and medicine

- Although the availability and quality of medicine continued to improve, epidemic diseases continued to spread, aided by modern transportation. An influenza pandemic, the Spanish Flu, killed 25 million between 1918 and 1919, while AIDS is yet uncured and treatments remain too expensive for wide use in developing countries.
- Advances in medicine, such as the invention of antibiotics, decreased the number of people dying from diseases. Contraceptive drugs and organ transplantation were developed. The discovery of DNA molecules and the advent of molecular biology allowed for cloning and genetic engineering.

Natural resources and the environment

- The widespread use of petroleum in industry -- both as a chemical precursor to plastics and as a fuel for the automobile and airplane -- led to the vital geopolitical importance of petroleum resources. The Middle East, home to many of the world's oil deposits, became a center of geopolitical and military tension throughout the latter half of the century. (For example, oil was a factor in Japan's decision to go to war against the United States in 1941, and the oil cartel, OPEC, used an oil embargo of sorts in the wake of the Yom Kippur War in the 1970s).
- A vast increase in fossil fuel consumption leads to depletion of natural resources, while air pollution has led to the develoment of an ozone hole and, many believe, global warming and both local and global climate change. The problem is increased by world-wide deforestation, also causing a loss of biodiversity. The problem of a depletion of natural resources is decreased by advances in drilling technology which led to a net increase in the amount of fossil fuel that is readily obtainable at the end of the century, as compared with the amount considered obtainable at the beginning of the century.

Significant people

World leaders

- Africa
- Gnassingbe Eyadema, Togo
- Félix Houphouët-Boigny, Côte d'Ivoire
- Kenneth Kaunda, Zambia
- Jomo Kenyatta, Kenya
- Idi Amin, Uganda
- Nelson Mandela, South Africa
- Robert Mugabe, Zimbabwe
- Gamal Abdal Nasser, Egypt
- Kwame Nkrumah, Ghana
- Julius Nyerere, Tanzania
- Habib Bourguiba, Tunisia
- Muammar al-Qaddafi, Libya
- Haile Selassie, Ethiopia
- Léopold Sédar Senghor, Senegal
- Ahmed Sékou Touré, Guinea
- Americas
- Juan Perón, Argentina
- Eva Perón, Argentina
- Getúlio Vargas, Brazil
- Luis Carlos Prestes, Brazil
- Juscelino Kubitschek, Brazil
- Wilfrid Laurier, Canada
- William Lyon Mackenzie King, Canada
- Pierre Trudeau, Canada
- Salvador Allende, Chile
- Augusto Pinochet, Chile
- Fidel Castro, Cuba
- Ernesto 'Che' Guevara, Argentina/Cuba
- Emiliano Zápata, Mexico
- Pancho Villa, Mexico
- Lázaro Cárdenas del Río, Mexico
- Augusto César Sandino, Nicaragua
- Fernando Belaúnde Terry, Peru
- Alberto Kenya Fujimori, Peru
- Theodore Roosevelt, USA
- Woodrow Wilson,USA
- Franklin D. Roosevelt, USA
- Harry S Truman, USA
- Dwight Eisenhower, USA
- John F. Kennedy, USA
- Lyndon B. Johnson, USA
- Richard Nixon, USA
- Ronald Reagan, USA
- Bill Clinton, USA
- George H. W. Bush, USA
- José Batlle y Ordóñez, Uruguay
- Romulo Betancourt, Venezuela
- Asia
- Mahatma Gandhi, India
- Lee Kuan Yew, Singapore
- Ferdinand Marcos, the Philippines
- Corazon Aquino, the Philippines
- Mao Zedong, People's Republic of China
- Deng Xiaoping, People's Republic of China
- Pol Pot, Cambodia
- Muhammad Ali Jinnah, Pakistan
- Indira Gandhi, India
- Mahathir Mohamad, Malaysia
- Jawaharlal Nehru, India
- Emperor Hirohito, Japan
- Ho Chi Minh, Vietnam
- Sun Yat-sen, Republic of China
- Chiang Kai-shek, Republic of China
- Achmad Sukarno, Indonesia
- Suharto, Indonesia
- Australia and Oceania
- Edmund Barton, Australia
- Sir Robert Menzies, Australia
- Peter Fraser, New Zealand
- Michael Joseph Savage, New Zealand
- David Lange, New Zealand
- Europe
- Franz Joseph of Austria, Austria-Hungary
- Václav Havel, Czech Republic
- Franjo Tuđman, Croatia
- Archbishop Makarios III, Cyprus
- Urho Kekkonen, Finland
- Philippe Pétain, France
- Charles de Gaulle, France
- Valéry Giscard d'Estaing, France
- François Mitterrand, France
- Kaiser Wilhelm II, Germany
- Friedrich Ebert, Germany
- Adolf Hitler, Germany
- Konrad Adenauer, West Germany
- Walter Ulbricht, East Germany
- Erich Honecker, East Germany
- Willy Brandt, West Germany
- Helmut Kohl, Germany
- Gerhard Schröder, Germany
- Eleftherios Venizelos, Greece
- Ioannis Metaxas, Greece
- Konstantinos Karamanlis, Greece
- Andreas Papandreou, Greece
- Miklós Horthy, Hungary
- Imre Nagy, Hungary
- Benito Mussolini, Italy
- Aldo Moro, Italy
- Eamon de Valera, Ireland
- Einar Gerhardsen, Norway
- Józef Piłsudski, Poland
- Lech Wałęsa, Poland
- António de Oliveira Salazar, Portugal
- Mário Soares, Portugal
- Nicolae Ceauşescu, Romania
- Milan Kučan, Slovenia
- Francisco Franco, Spain
- Felipe González, Spain
- Adolfo Suárez, Spain
- Olof Palme, Sweden
- Mustafa Kemal Atatürk, Turkey
- Neville Chamberlain, United Kingdom
- Winston Churchill, United Kingdom
- Margaret Thatcher, United Kingdom
- Tony Blair, United Kingdom
- Josip Broz Tito,Yugoslavia
- Slobodan Milošević, Yugoslavia
- Russia and Soviet Union
- Czar Nicholas II
- Vladimir Lenin
- Joseph Stalin
- Leon Trotsky
- Nikita Khrushchev
- Leonid Brezhnev
- Mikhail Gorbachev
- Boris Yeltsin
- Middle East
- Reza Shah Pahlavi, Iran
- Mohammad Reza Pahlavi, Iran
- Mohammad Mosaddeq, Iran
- Ayatollah Khomeini, Iran
- Ayatollah Khamenei, Iran
- Mohammad Khatami, Iran
- Abdul Nasser, Egypt or United Arab Republic
- Anwar Sadat, Egypt or United Arab Republic
- David Ben-Gurion, Israel
- Golda Meir, Israel
- Menachem Begin, Israel
- Yitzhak Rabin, Israel
- Hafez el Assad, Syria
- Saddam Hussein, Iraq
- King Hussein, Jordan
- Yassar Arafat, Palestine

Scientists

; Biology and Anthropology
- Norman Borlaug
- Francis Crick
- Theodosius Dobzhansky
- Paul Ehrlich
- Jane Goodall
- Stephen Jay Gould
- Hans Adolf Krebs
- Ernst Mayr
- John Maynard Smith
- Albert Szent-Györgyi
- James Watson ; Chemistry
- Elias Corey
- Maria Skłodowska-Curie
- Pierre Curie
- Fritz Haber
- Stanley Miller
- Linus Pauling
- Ernest Rutherford
- J.J. Thomson
- Harold Urey ; Computer Science
- John Backus
- Edsger Dijkstra
- Richard Matthew Stallman
- Linus Torvalds
- Grace Murray Hopper
- John von Neumann
- Claude Shannon
- Alan Turing
- William Gates III ; Mathematics
- Paul Erdős
- Kurt Gödel
- David Hilbert
- Andrey Nikolaevich Kolmogorov
- Benoit Mandelbrot
- John Nash
- John von Neumann ; Medicine and Pharmacy
- Carl Djerassi
- Alexander Fleming
- Howard Walter Florey
- Ma Haide (George Hatem)
- Jonas Salk ; Physics and Astronomy
- Abdus Salam
- Niels Bohr
- Paul Dirac
- Freeman Dyson
- Albert Einstein
- Enrico Fermi
- Richard Feynman
- Stephen Hawking
- Werner Karl Heisenberg
- Edwin Hubble
- Wolfgang Pauli
- Max Planck
- Carl Sagan
- Erwin Schrödinger ; Psychology
- Aaron T. Beck
- Mary Whiton Calkins
- Albert Ellis
- Sigmund Freud
- Carl Jung
- Alfred Kinsey
- Stanley Milgram
- Ivan Pavlov
- Jean Piaget
- B.F. Skinner
- John B. Watson

Humanities

- Art and Literary Theory
- Rudolf Arnheim
- Clive Bell
- Fredric Jameson
- Pauline Kael
- Siegfried Kracauer
- Raymond Williams
- Civil Rights
- Martin Luther King Jr.
- Economics
- John Maynard Keynes
- John Kenneth Galbraith
- Milton Friedman
- Ludwig von Mises
- History
- Stephen Ambrose
- Charles A. Beard
- Marc Bloch
- Fernand Braudel
- Lucien Febvre
- Jacques Le Goff
- Philosophy
- Theodor Adorno
- Louis Althusser
- Hannah Arendt
- Gaston Bachelard
- Walter Benjamin
- Henri Bergson
- Gilles Deleuze
- Michel Foucault
- Jürgen Habermas
- Martin Heidegger
- W. V. Quine
- John Rawls
- Bertrand Russell
- Jean-Paul Sartre
- Alfred North Whitehead
- Ludwig Wittgenstein
- Political Science
- Robert A. Dahl
- Maurice Duverger
- Francis Fukuyama
- Arend Lijphart
- C. Wright Mills

Business

- Paul Allen
- Warren Buffett
- Walt Disney
- Henry Ford
- Bill Gates
- Howard Hughes
- Steve Jobs
- Linus Torvalds
- Donald Trump
- Sam Walton
- Thomas J. Watson

Aerospace pioneers

- Alberto Santos-Dumont
- Robert Goddard
- Wernher von Braun
- Neil Armstrong
- Louis Bleriot
- Yuri Gagarin
- Vladimir Mikhailovich Komarov
- Freddie Laker
- Charles Lindbergh
- Ron McNair
- Ellison Onizuka
- Herman Potočnik Noordung
- Alan Shepard
- Valentina Tereshkova
- Wright Brothers
- Chuck Yeager

Military leaders

- Moshe Dayan
- Dwight Eisenhower
- Sir Bernard Freyberg
- Charles de Gaulle
- Vo Nguyen Giap
- Che Guevara
- Douglas Haig
- Paul von Hindenburg
- Erich Ludendorff
- Douglas MacArthur
- Rudolf Maister
- Bernard Montgomery
- Chester Nimitz
- George Patton
- Colin Powell
- Erwin Rommel
- Franc Rozman Stane
- Leon Trotsky
- Mao Zedong
- Georgy Zhukov

Spiritual figures

- Pope Pius X
- Pope Pius XII
- Pope John XXIII
- Pope John Paul II
- Sayyid Abul A'la Maududi
- Mother Teresa of Calcutta
- The 13th Dalai Lama of Tibet, Thubten Gyatso
- The 14th Dalai Lama of Tibet, Tenzin Gyatso
- The Rev. Martin Luther King Jr.
- The Rev. Billy Graham
- Mahatma Gandhi
- Aurobindo Ghosh
- Ramana Maharshi
- Maharishi Mahesh Yogi
- Ayatollah Khomeini
- Ayatollah Khamenei
- Rasputin
- Rabbi Menachem Mendel Schneerson
- Rev. Dr. Sun Myung Moon

Artists

- Josef Albers
- Ernst Barlach
- Balthus
- Max Beckmann
- Hans Bellmer
- Joseph Beuys
- Louise Bourgeois
- Constantin Brancusi
- George Braque
- John Cage
- Marc Chagall
- Giorgio de Chirico
- Chuck Close
- Enzo Cucchi
- Salvador Dalí
- Otto Dix
- Marcel Duchamp
- Jacob Epstein
- Max Ernst
- Lyonel Feininger
- Helen Frankenthaler
- Alberto Giacometti
- Juan Gris
- Walter Gropius
- Erich Heckel
- Barbara Hepworth
- Eva Hesse
- Donald Judd
- Frida Kahlo
- Wassily Kandinsky
- Anselm Kiefer
- Ernst Ludwig Kirchner
- Paul Klee
- Yves Klein
- Gustav Klimt
- Oskar Kokoschka
- Käthe Kollwitz
- Willem de Kooning
- Jannis Kounellis
- Le Corbusier
- Sol LeWitt
- Roy Lichtenstein
- El Lissitzky
- René Magritte
- Marino Marini
- Henri Matisse
- Joan Miró
- Amedeo Modigliani
- László Moholy-Nagy
- Piet Mondrian
- Henry Moore
- Robert Motherwell
- Edvard Munch
- Bruce Nauman
- Emil Nolde
- Eduardo Paolozzi
- Pino Pascali
- Max Pechstein
- Pablo Picasso
- Jackson Pollock
- Diego Rivera
- Alexander Rodchenko
- Auguste Rodin
- James Rosenquist
- Mark Rothko
- Henri Rousseau
- Egon Schiele
- Karl Schmidt-Rottluff
- Kurt Schwitters
- Richard Serra
- Robert Smithson
- Andy Warhol
- Frank Lloyd Wright

Music

- ABBA
- King Sunny Ade
- Nusrat Fateh Ali Khan
- Louis Armstrong
- Béla Bartók
- Alban Berg
- Luciano Berio
- Chuck Berry
- Pierre Boulez
- David Bowie
- John Cage
- Ray Charles
- John Coltrane
- Aaron Copland
- Dalida
- Gary Davis
- Miles Davis
- Claude Debussy
- Bob Dylan
- Carlos Gardel
- Marvin Gaye
- George Gershwin
- Philip Glass
- Amy Grant
- Nazia Hassan
- Jimi Hendrix
- Gustav Holst
- Michael Jackson
- Janis Joplin
- Scott Joplin
- Aram Khachaturian
- Kraftwerk
- Fela Kuti
- Led Zeppelin
- Bob Marley
- Olivier Messiaen
- Nirvana
-

Bijective

In mathematics, injections, surjections and bijections are classes of functions distinguished by the manner in which arguments (input expressions from the domain) and images (output expressions from the codomain) are related or mapped to each other.
- A function

f: \; A \to B

is injective (one-to-one) if

f(x)=f(y) \; \to \; x=y

or, equivalently, if

x \ne y \; \to \; f(x) \ne f(y)

. One could also say that every element of the codomain (sometimes called range) is mapped to by at most one element (argument) of the domain; not every element of the codomain, however, need have an argument mapped to it. An injective function is an injection.
- A function is surjective (onto) if every element of the codomain is mapped to by some element (argument) of the domain; some images may be mapped to by more than one argument. (Equivalently, a function where the range is equal to the codomain.) A surjective function is a surjection.
- A function is bijective (one-to-one and onto) if and only if (iff) it is both injective and surjective. (Equivalently, every element of the codomain is mapped to by exactly one element of the domain.) A bijective function is a bijection (one-to-one correspondence). (Note: a one-to-one function is injective, but may fail to be surjective, while a one-to-one correspondence is both injective and surjective.) An injective function need not be surjective (not all elements of the codomain may be associated with arguments), and a surjective function need not be injective (some images may be associated with more than one argument). The four possible combinations of injective and surjective features are illustrated in the following diagrams.

Injection

if and only if A function is injective (one-to-one) if every possible element of the codomain is mapped to by at most one argument. Equivalently, a function is injective if it maps distinct arguments to distinct images. An injective function is an injection. The formal definition is the following. :The function

f: A \to B

is injective iff for all

a,b \in A

, we have

f(a) = f(b) \Rarr a = b.

- A function f : A → B is injective if and only if A is empty or f is left-invertible, that is, there is a function g: B → A such that g o f = identity function on A.
- Since every function is surjective when its codomain is restricted to its range, every injection induces a bijection onto its range. More precisely, every injection f : A → B can be factored as a bijection followed by an inclusion as follows. Let f_R : A → f(A) be f with codomain restricted to its image, and let i : f(A) → B be the inclusion map from f(A) into B. Then f = i o f_R. A dual factorisation is given for surjections below.
- The composition of two injections is again an injection, but if g o f is injective, then it can only be concluded that f is injective. See the figure at right.
- Every embedding is injective.

Surjection

embedding A function is surjective (onto) if every possible image is mapped to by at least one argument. In other words, every element in the codomain has non-empty preimage. Equivalently, a function is surjective if its range is equal to its codomain. A surjective function is a surjection. The formal definition is the following. :The function

f: A \to B

is surjective iff for all

b \in B

, there is

a \in A

such that

f(a) = b.

- A function f : A → B is surjective if and only if it is right-invertible, that is, if and only if there is a function g: B → A such that f o g = identity function on B. (This statement is equivalent to the axiom of choice.)
- By collapsing all arguments mapping to a given fixed image, every surjection induces a bijection defined on a quotient of its domain. More precisely, every surjection f : A → B can be factored as a projection followed by a bijection as follows. Let A/~ be the equivalence classes of A under the following equivalence relation: x ~ y if and only if f(x) = f(y). Equivalently, A/~ is the set of all preimages under f. Let P(~) : A → A/~ be the projection map which sends each x in A to its equivalence class [x]_~, and let f_P : A/~ → B be the well-defined function given by f_P([x]_~) = f(x). Then f = f_P o P(~). A dual factorisation is given for injections above.
- The composition of two surjections is again a surjection, but if g o f is surjective, then it can only be concluded that g is surjective. See the figure at right
- .

Bijection

axiom of choice A function is bijective if it is both injective and surjective. A bijective function is a bijection (one-to-one correspondence). A function is bijective if and only if every possible image is mapped to by exactly one argument. This equivalent condition is formally expressed as follows. :The function

f: A \to B

is bijective iff for all

b \in B

, there is a unique

a \in A

such that

f(a) = b.

- A function f : A → B is bijective if and only if it is invertible, that is, there is a function g: B → A such that g o f = identity function on A and f o g = identity function on B. This function maps each image to its unique preimage.
- The composition of two bijections is again a bijection, but if g o f is a bijection, then it can only be concluded that f is injective and g is surjective. (See the figure at right and the remarks above regarding injections and surjections.)
- The bijections from a set to itself form a group under composition, called the symmetric group.

Cardinality

Suppose you want to define what it means for two sets to "have the same number of elements". One way to do this is to say that two sets "have the same number of elements" if and only if all the elements of one set can be paired with the elements of the other, in such a way that each element is paired with exactly one element. Accordingly, we can define two sets to "have the same number of elements" if there is a bijection between them. We say that the two sets have the same cardinality.

Examples

It is important to specify the domain and codomain of each function since by changing these, functions which we think of as the same may have different jectivity.

Injective and surjective (bijective)

- For every set A the identity function id_A and thus specifically

\mathbf \to \mathbf : x \mapsto x

.
-

\mathbf^+ \to \mathbf^+ : x \mapsto x^2

and thus also its inverse

\mathbf^+ \to \mathbf^+ : x \mapsto \sqrt

.
- The exponential function

\exp : \mathbf \to \mathbf^+ : x \mapsto \mathrm^x

and thus also its inverse the natural logarithm

\ln : \mathbf^+ \to \mathbf : x \mapsto \ln

Injective and non-surjective

- The exponential function

\exp : \mathbf \to \mathbf : x \mapsto \mathrm^x

Non-injective and surjective

\mathbf \to \mathbf : x \mapsto (x-1)x(x+1) = x^3 - x

- The sine function f(

Function composition

Functional powers

Composition operator

See also

Mathematics

History

Inspiration, pure and applied mathematics, and aesthetics

Notation, language, and rigor

Is mathematics a science?

Overview of fields of mathematics

Major themes in mathematics

Quantity

Structure

Space

Change

Foundations and methods

Discrete mathematics

Applied mathematics

Important theorems

Important conjectures

History and the world of mathematicians

Mathematics and other fields

Common misconceptions

See also

Bibliography

External links

20th century

Overview

Important developments, events and achievements

Science and technology

Wars and politics

Culture and entertainment

Disease and medicine

Natural resources and the environment

Significant people

World leaders

Scientists

Humanities

Business

Aerospace pioneers

Military leaders

Spiritual figures

Artists

Music

Bijective

Injection

Surjection

Bijection

Cardinality

Examples

Injective and surjective (bijective)

Injective and non-surjective

Non-injective and surjective