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Although and set
Although originally controversial, the axiom of choice is now used without reservation by most mathematicians, and it is included in ZFC, the standard form of axiomatic set theory.
Although " Amazing Grace " set to " New Britain " was popular, other versions existed regionally.
Although there are no set minimum admissions criteria for Barrett College, the average GPA of incoming freshmen was 3. 84, with average SAT scores of 1314 / 1600 and ACT scores of 29.
Although Rodin is generally considered the progenitor of modern sculpture, he did not set out to rebel against the past.
Although he had for many years " harbored an ambition to create work for a public square ", he " had never set foot in New York, and knew nothing about life in a rapidly evolving metropolis.
Although the Umayyads did not have a historical presence in the region ( no member of the Umayyad family was known to have ever set foot in al-Andalus before ) and there were grave concerns about young Abd al-Rahman's inexperience, several of the lower-ranking Yemenite commanders felt they had little to lose and much to gain, and agreed to support the prince.
Although Zermelo's fix allows a class to describe arbitrary ( possibly " large ") entities, these predicates of the meta-language may have no formal existence ( i. e., as a set ) within the theory.
Although Benedictines do not take a vow of silence, hours of strict silence are set, and at other time silence is maintained as much as is practically possible.
Although each series is set in a different era, all follow the " misfortunes " of Edmund Blackadder ( played by Atkinson ), who in each is a member of a British family dynasty present at many significant periods and places in British history.
Although passengers rarely have cause to refer to either document, all travel is subject to the National Rail Conditions of Carriage and all tickets are valid subject to the rules set out in a number of so-called technical manuals, which are centrally produced for the network.
Although counting the number of elements in a set is a rather broad mathematical problem, many of the problems that arise in applications have a relatively simple combinatorial description.
Although the individual philosophical schools differ considerably, they nevertheless share a common vocabulary and set of concerns.
Although a " bijection " seems a more advanced concept than a number, the usual development of mathematics in terms of set theory defines functions before numbers, as they are based on much simpler sets.
Although Cantor himself defined the set in a general, abstract way, the most common modern construction is the Cantor ternary set, built by removing the middle thirds of a line segment.
Although Cesare Borgia's father gave him the power to set up, Cesare ruled the Romagna with skill and tact for the most part.
Although Celera was originally set to sequence a composite of DNA samples, partway through the sequencing, Venter switched the samples for his own DNA.
Although these guidelines have been widely implemented, opponents argue that even when a diagnostic criteria set is accepted across different cultures, it does not necessarily indicate that the underlying constructs have any validity within those cultures ; even reliable application can only demonstrate consistency, not legitimacy.
Although rumors of drug use and other " problems on the set " swirled around her dismissal, the producers were adamant that the pregnancy was the only reason her character was written out.
Although various notations are used throughout the literature to denote that two elements a and b of a set are equivalent with respect to an equivalence relation R, the most common are " a ~ b " and " a ≡ b ", which are used when R is the obvious relation being referenced, and variations of " a ~< sub > R </ sub > b ", " a ≡< sub > R </ sub > b ", or " aRb ".
Although he doled out offices, money, and titles mainly to Christian highlanders in the hope of co-opting would-be Eritrean opponents in early 1967, the imperial secret police of Ethiopia also set up a wide network of informants in Eritrea and conducted disappearances, intimidations and assassinations among the same populace driving several prominent political figures into exile.
Although definitions of ecosystem management abound, there is a common set of principles which underlie these definitions.
Although unsure as to why these saints should choose such a " sinner " as their patron, Einhard nonetheless set about ensuring they continued to receive a resting place fitting of their honour.
Although formal certification of his freedom from Burghley's control was deferred until May 1572, Oxford was finally granted the income of £ 666 which his father had intended him to have earlier, but properties set aside to pay his father's debts would not come his way for another decade.
Although individual buses are generally owned by individuals or small companies, the timetables, fares, and levels of service are set by Strandfaraskip Landsins and the government.

Although and same
Although federal and city narcotic agents sometimes worked together, Sokol continued, rivalries developed when they were `` aiming at the same criminals ''.
Although it had at first been somewhat established in many colonies, in 1861 it was ruled that, except where specifically established, the Church of England had just the same legal position as any other church.
Although Acorn were able to shrink substantially the same functionality as the BBC into just one chip, manufacturing problems meant that very few machines were available for the Christmas period — to the extent that some shops reported eight presales for every delivered machine.
Although this produced a mode 7 that barely impacted upon CPU performance and gave the same visual quality as the BBC Micro, it remained compatible only with software that used the ROM routines for outputting text and graphics and still used 10 kB of memory for the display.
Although a number of different broadcast television systems were in use worldwide, the same principles of operation apply.
Although a committed Christian, she has characterised the issue as one of life and death on which her view had been the same when she was agnostic.
Although both polysaccharide classes share the same galactose-based backbone, agaropectin is heavily modified with acidic side-groups, such as sulfate and pyruvate.
Although the hard-part bearing organisms make up as little as 14 % of the community, these same organisms are found in similar proportions in other Cambrian localities.
Although the same basic components are present in all vertebrate brains, some branches of vertebrate evolution have led to substantial distortions of brain geometry, especially in the forebrain area.
Although he was aware that the Bastarnae were hardly likely to achieve the same success as Hannibal some 40 years earlier, and would most likely end up cut to pieces by the Romans, Philip hoped that the Romans would be distracted long enough to allow him to reoccupy his former possessions in Greece.
Although Douglas was always second billed under Lancaster in these films, their roles were usually more or less the same size with the exceptions of I Walk Alone, in which Douglas played a villain, and in Seven Days in May, where Douglas ' part was larger than Lancaster's but not as dramatic.
Although many assume that the biblical Mark the Cousin of Barnabas ( Colossians 4: 10 ) is the same as John Mark ( Acts 12: 12, 25 ; 13: 5, 13 ; 15: 37 ) and Mark the Evangelist, the traditionally believed author of the Gospel of Mark, according to Hippolytus of Rome, the three " Mark " s are distinct persons.
Although α and − α produce the same differential equation, it is conventional to define different Bessel functions for these two orders ( e. g., so that the Bessel functions are mostly smooth functions of α ).
Although all the nuclei of all atoms belonging to one element will have the same number of protons, they may not necessarily have the same number of neutrons ; such atoms are termed isotopes.
Although high-end computers may be bought retail in the same manner as the common computer, they are frequently built by their owners.
Although the Three Kingdoms were reunified by the Jin Dynasty in 280, this structure was essentially the same until the Wu Hu uprising.
Although choosing two options in the same division is discouraged, it is still possible.
Although there are various paths and catalysts involved in the CNO cycles, simply speaking all these cycles have the same net result:
Although not closely related, the coyote evolved separately to fill roughly the same ecological niche in the Americas that is filled in Eurasia and Africa by the similarly sized jackals.
Although it responds to the same DOS commands, most disk utilities written prior to 1987, most notably fast loaders, are so 1541-specific that they do not work with the 1581.
Although rarely used today, it has the same status in scoring as a place kick.
Although it did not achieve the same levels of commercial success as the band's later releases ( it reached # 73 on the Billboard 200 ), it has been hailed by many fans and critics as Dream Theater's masterpiece and the band's defining album.
Although these letters are not formally published in the Doctrine and Covenants, they are still deemed to be inspired, and are dealt with in the same manner that revelations are ( that is, they must be deliberated and approved by the voting members of a World Conference ).
Although the Cantino planisphere ( 1504 ) and the Ruysch map ( 1507 ) clearly delineate the Maldive Islands, giving them the same names, they do not show any islands to the south which can be identified as the Chagos archipelago.

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