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In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that a Cartesian product of a collection of non-empty sets is non-empty. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of one object from each bin if the collection is infinite. Formally, it states that for every indexed family i ∈ I of nonempty sets there exists an indexed family i ∈ I of elements such that x i ∈ S i for every i ∈ I; the axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem. In many cases, such a selection can be made without invoking the axiom of choice. An illustrative example is sets picked from the natural numbers. From such sets, one may always select the smallest number, e.g. in the smallest elements are. In this case, "select the smallest number" is a choice function. If infinitely many sets were collected from the natural numbers, it will always be possible to choose the smallest element from each set to produce a set.

That is, the choice function provides the set of chosen elements. However, no choice function is known for the collection of all non-empty subsets of the real numbers. In that case, the axiom of choice must be invoked. Bertrand Russell coined an analogy: for any collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate selection. For an infinite collection of pairs of socks, there is no obvious way to make a function that selects one sock from each pair, without invoking the axiom of choice. Although controversial, the axiom of choice is now used without reservation by most mathematicians, it is included in the standard form of axiomatic set theory, Zermelo–Fraenkel set theory with the axiom of choice. One motivation for this use is that a number of accepted mathematical results, such as Tychonoff's theorem, require the axiom of choice for their proofs. Contemporary set theorists study axioms that are not compatible with the axiom of choice, such as the axiom of determinacy.

The axiom of choice is avoided in some varieties of constructive mathematics, although there are varieties of constructive mathematics in which the axiom of choice is embraced. A choice function is a function f, defined on a collection X of nonempty sets, such that for every set A in X, f is an element of A. With this concept, the axiom can be stated: Formally, this may be expressed as follows: ∀ X. Thus, the negation of the axiom of choice states that there exists a collection of nonempty sets that has no choice function; each choice function on a collection X of nonempty sets is an element of the Cartesian product of the sets in X. This is not the most general situation of a Cartesian product of a family of sets, where a given set can occur more than once as a factor; the axiom of choice asserts the existence of such elements. In this article and other discussions of the Axiom of Choice the following abbreviations are common: AC – the Axiom of Choice. ZF – Zermelo–Fraenkel set theory omitting the Axiom of Choice.

ZFC – Zermelo–Fraenkel set theory, extended to include the Axiom of Choice. There are many other equivalent statements of the axiom of choice; these are equivalent in the sense that, in the presence of other basic axioms of set theory, they imply the axiom of choice and are implied by it. One variation avoids the use of choice functions by, in effect, replacing each choice function with its range. Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains one element in common with each of the sets in X; this guarantees for any partition of a set X the existence of a subset C of X containing one element from each part of the partition. Another equivalent axiom only considers collections X that are powersets of other sets: For any set A, the power set of A has a choice function. Authors who use this formulation speak of the choice function on A, but be advised that this is a different notion of choice function, its domain is the powerset of A, s

The 2018 Copa CONMEBOL Libertadores Femenina was the tenth edition of the CONMEBOL Libertadores Femenina, South America's premier women's club football tournament organized by CONMEBOL. The tournament was held in Manaus, Brazil from 18 November to 2 December 2018. Planned from 4 to 18 November the tournament was pushed back two weeks because of the 2019 FIFA Women's World Cup qualification CONCACAF–CONMEBOL play-off which fell in the time frame. Atlético Huila defeated Santos in the final on penalties to win their first tournament title. Iranduba defeated Colo-Colo to finish third. Audax were the defending champions, having won the title the previous year as a joint team with Corinthians, they were eliminated in the group stage. There were three bids for the 2018 Copa Libertadores Femenina: Manaus, Santa Cruz de la Sierra and São Paulo proposed by Iranduba, Deportivo ITA and Corinthians, respectively. On 11 June 2018, CONMEBOL announced that the tournament would be held in Manaus and Iranduba gained the host association additional berth.

The competition was contested by 12 teams: the champions of all ten CONMEBOL associations were given one entry, additionally the title holders re-entered and the host association qualified one more team. Notes Initially two stadiums would host the tournament. On 14 November 2018, CONMEBOL announced that the Estádio Ismael Benigno would no longer host matches, matches to be played there would be moved to Arena da Amazônia. During the tournament, CONMEBOL decided to move the matches scheduled on 25 and 26 November at Estádio Roberto Simonsen to protect the football field of Arena da Amazônia. Matches were played in Manaus; the stadiums were: Arena da Amazônia Estádio Roberto Simonsen The draw for the tournament was held on 7 November 2018, 19:00 AMT, at the Arena da Amazônia in Manaus. The 12 teams were drawn into three groups of four containing a team from each of the four pots; the defending champions Audax were automatically seeded into Pot 1 and allocated to position A1 in the group stage. For the remaining two teams from hosts Brazil, the first representative was seeded into Pot 2 and the second representative was seeded into Pot 4.

The remaining teams were seeded based on the results of their association in the 2017 Copa Libertadores Femenina. Teams from the same association could not be drawn into the same group. In the group stage, the teams were ranked according to points. If tied on points, tiebreakers would be applied in the following order: Goal difference; the winners of each group and the best runners-up among all groups advanced to the semi-finals. All times are local, AMT; the semi-final matchups were: Group A winner vs. Group C winner Group B winner vs. Best runner-upThe semi-final winners and losers played in the final and third place match respectively. If tied after full time, extra time would not be played, the penalty shoot-out would be used to determine the winner. CONMEBOL Libertadores Femenina Brasil 2018, CONMEBOL Libertadores Feminina, Confederação Brasileira de Futebol

The Cambridge Companion to Freud is a 1991 collection of articles about Sigmund Freud, the founder of psychoanalysis, edited by the philosopher Jerome Neu. The book received both negative reviews; some of the individual contributions received praise, but commentators criticized the exclusion of particular topics and the failure to include particular authors as contribuors. The Cambridge Companion to Freud includes an introduction by Jerome Neu, essays on various topics related to Freud; these discussions include those by the cultural historian Carl Emil Schorske on Freud's views on "the implications of individual psychodynamics for civilization as a whole", the intellectual historian Gerald N. Izenberg on Freud's seduction theory, the philosopher Clark Glymour on the relation of Freud's views to cognitive psychology, the philosopher James Hopkins on Freud's theory of dreams, the philosopher Sebastian Gardner on the unconscious, the psychoanalyst Bennett Simon and the psychologist Rachel B.

Blass on Oedipus complex, Neu on Freud's views on sexual perversion and sexuality in general, the philosopher Jennifer Church on morality and the superego, the psychoanalyst Nancy Chodorow on Freud's views on women, the philosopher Richard Wollheim on the relevance of Freud's views to art, the anthropologist Robert A. Paul on Freud's "cultural books" Totem and Taboo, Group Psychology and the Analysis of the Ego and Its Discontents, Moses and Monotheism, the philosopher John Deigh on The Future of an Illusion and Civilization and Its Discontents; the book reprints a critical review of the philosopher Adolf Grünbaum's The Foundations of Psychoanalysis by the philosopher David Sachs. Jerome Neu, "Introduction" Carl Emil Schorske, "Freud: The psychoarcheology of civilizations" Gerald N. Izenberg, "Seduced and abandoned: The rise and fall of Freud's seduction theory" Clark Glymour, "Freud's androids" James Hopkins, "The interpretation of dreams" Sebastian Gardner, "The unconscious" Bennett Simon and Rachel B.

Blass, "The development and vicissitudes of Freud's ideas on the Oedipus complex" Jerome Neu, "Freud and perversion" Jennifer Church, "Morality and the internalized other" Nancy Chodorow "Freud on women" Richard Wollheim, "Freud and the understanding of art" Robert A. Paul, "Freud's anthropology: A reading of the'cultural books'" John Deigh, "Freud's theory of civilization: Changes and implications" David Sachs, "In fairness to Freud: A critical notice of The Foundations of Psychoanalysis, by Adolf Grünbaum" The Cambridge Companion to Freud was published by Cambridge University Press in 1991; the Cambridge Companion to Freud received positive reviews from the historian Sander Gilman in Medical History, the philosopher Marcia Cavell in Ethics, Leonard Groopman in the Journal of the American Psychoanalytic Association, a mixed review from the philosopher Andrew Brook in Philosophy in Review. The book received negative reviews from the sociologist Christopher Badcock in Sociology and the British Journal of Sociology, Brayton Polka in History of European Ideas.

Gilman considered the book well-written. He praised the contributions by Schorske, Hopkins, Chodorow and Paul and Deigh, he believed. However, he felt that Gardner's contribution would have benefited from "attention to the older discussion of the pre-history of the unconscious", that the book as a whole would have been improved by the inclusion of a discussion of the young Freud and his interests. Cavell described the book as useful. Though she noted that some important topics were omitted, she added that "most of the salient ones are here and thoughtfully treated." She considered it appropriate that all the authors represented in the book "write from a position somewhere within the Freudian camp", though she noted that some would regard it as a weakness. She wrote that the essays by Paul and Deigh, Izenberg and Blass, Chodorow contained valuable overviews of changes in Freud's views on particular topics, she credited Hopkins with providing the most lucid explanation of "ordinary explanations of actions in terms of motives or beliefs and desires" that she had seen.

She found Paul's discussion of Freud's "myth of the primal horde" interesting, but criticized him for "leaving in place... Freud's arguably grandiose view of the oedipal complex as an inscription in the brain of the human infant" and for failing to "question such Freudian assumptions as that explaining intrapsychic conflict requires two basic instincts." She criticized Deigh and Church for failing to "question Freud's narrow view of self-interest" and his "assumption that only self-interest can ground love for the other."Groopman considered the essays "too diverse in aim and quality to point in any single direction or convey a single message." However, he praised the contributions by Wollheim and Chodorow, concluded that the work as a whole helped to show the diversity of Freud's work. Brook considered most the essays included be of high quality, praised the contributions by Chodorow, Gardner and Blass, Paul. However, he believed that the work as a whole was limited by the fact that it discussed only a few of Freud's interests and theories.

He criticized the over-representation of philosophers and Americans, under-representation of women and practicing psychoanalysts, among its contributors, the failure to include contributions from Grünbaum and the psychoanalyst Marshall Edelson. Badcock maintained in Sociology that the book incorrectly treats Freud as a philosopher rather than a "pioneering scientific empiricist", and, in the British Journal of Sociology, that it "perpetuates many errors and

Fujiwara Sadaie, better-known as Fujiwara no Teika, was a Japanese poet, calligrapher, anthologist and scholar of the late Heian and early Kamakura periods. His influence was enormous, he is counted as among the greatest of Japanese poets, the greatest master of the waka form – an ancient poetic form consisting of five lines with a total of 31 syllables. Teika's critical ideas on composing poetry were influential and studied until as late as the Meiji era. A member of a poetic clan, Teika was born to the noted poet Fujiwara no Shunzei. After coming to the attention of the Retired Emperor Go-Toba, Teika began his long and distinguished career, spanning multiple areas of aesthetic endeavor, his relationship with Go-Toba was at first cordial and led to commissions to compile anthologies, but resulted in his banishment from the retired emperor's court. His descendants and ideas would dominate classical Japanese poetry for centuries afterwards. Teika was born to a minor and distant branch of the aristocratic and courtly clan, the Fujiwara, in 1162, sometime after the Fujiwara regents had lost their political pre-eminence in the Imperial court during the Hōgen Rebellion.

His branch of the clan sought prestige and power in the court by aligning itself with the Mikohidari family, by specializing in artistic endeavors, principally poetry. Such specialization was not unusual. Teika's grandfather was the venerable poet Fujiwara no Toshitada, his father was Fujiwara no Shunzei, a well known and respected poet, who had compiled the seventh Imperial anthology of waka. His niece would become a well-respected poet of waka and renga, known as Kengozen or Shunzei's Daughter, whom he would seek out for poetic advice, his elder brother, Fujiwara no Nariee, would be somewhat successful in court, but not nearly as much as his niece. Teika's foster-brother, the priest Jakuren or "Sadanaga" c. 1139–1202 would be successful as a poet although his career was cut tragically short. Teika's goals as the senior male of his branch were to inherit and cement his father's position in poetry, to advance his own reputation. While his life would be marked by repeated illness and wildly shifting fortunes – only moderated by his father's long-lasting influence in court, the young and poetically inclined Retired Emperor Go-Toba's patronage would prove to lead to some of Teika's greatest successes.

The Retired Emperor Go-Toba announced, in the second year of his abdication that he would be conducting a poetry contest. Retired Emperors became more influential after their retirement from the office of Emperor rather than as the actual Emperor, since they were free from the restricting ceremonial requirements and politics of the court. Go-Toba was 20. Go-Toba regarded all these pursuits as hobbies, dropping another. One of these was his support of poetry the waka. After his abdication, he had announced that he would hold two poetry contests, each requiring a number of preeminent poets to compose some 100 waka in a particular thematic progression, known as the hyakushu genre of poem sequences; the first contest was considered a crucial political nexus. Teika's diary records, he was 38, had reached middle age. While he was recognized as a talented poet, his career was stagnant, he was "Lesser Commander of the Palace Guards of the Left" with little prospect of further advancement. He had wider political problems: The influence of his patrons, the Kujōs, over the Emperors had declined drastically.

Minamoto no Michichika had insinuated himself into Imperial circles through Go-Toba's former nursemaid. As Ninshi was the daughter of the Kujō's leader Kujō Kanezan

The following list includes notable people who were born or have lived in Hallowell, Maine. Gorham Dummer Abbott, clergyman and author Jacob Abbott, children's writer Martha Ballard, colonial midwife. Swain, philanthropist Benjamin Vaughan, political economist and doctor Abbott Abbott Vaughn Meader, political humorist Amos Stoddard, US Army officer Supply Belcher, composer and compiler of tune books Hiram Belcher, US congressman Joseph R. Bodwell, 40th governor of Maine Scott Cowger, state legislator George Evans, US congressman and senator John Hubbard, 22nd governor of Maine Dale McCormick, first gay member of the Maine state legislature Patrick K. McGowan, state legislator, candidate for governor Amos Nourse, doctor and US senator John Otis, US congressman James L. Reid, state representative and Maine Superior Court Justice Peleg Sprague, US Federal Judge Sharon Treat, state representative for Maine's 79th District Samuel Wells, 25th governor of Maine Reuel Williams, US senator Charlie Waitt, first baseman for the St. Louis Brown Stockings, Chicago White Stockings, Baltimore Orioles, Philadelphia Quakers