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 siblinghood is symmetric ( if A is a sibling of B, then B is a sibling of A ) and transitive on any 3 distinct people ( if A is a sibling of B and C is a sibling of B, then A is a sibling of C, provided A is not C ( Note that " is a sibling of " is NOT a transitive relation, since A R B, and B R A implies A R A by transitivity )), it is not reflexive ( A cannot be a sibling of A ).

Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic.

Although Jersey is for most day-to-day purposes entirely self-governing in relation to its internal affairs, the Crown retains residual responsibility for the " good government " of the island.

Until the day I die people will keep saying, ‘ Leni is a Nazi ’, and I ’ ll keep saying, ‘ But what did she do ?’” Although she won more than 50 libel cases against people accusing her of collaborating with the Nazis, there are many unanswered questions about her relation to National Socialism in particular and fascism more generally.

Although power-law relations are attractive for many theoretical reasons, demonstrating that data do indeed follow a power-law relation requires more than simply fitting a particular model to the data.

Although Peirce uses words like concordance and correspondence to describe one aspect of the pragmatic sign relation, he is also quite explicit in saying that definitions of truth based on mere correspondence are no more than nominal definitions, which he accords a lower status than real definitions.

Although planets can sometimes be mistaken for stars as one observes the night sky, the planets actually change position from night to night in relation to the stars.

Although it is known today that the planets revolve around the sun, the same terms continue to be used in order to describe the movement of the planets in relation to the stars as they are observed from Earth.

Although this employs analytic expressions which in principle can be solved, calculating the thermal conductivity of a dense fluid or solid using this relation requires the use of molecular dynamics computer simulation.

Although they made small gains in relation to the amount of seats at dissolution, they fell by 2 seats to 49 in comparison to the 51 they received in 2006.

In this relation it has been quoted by the author that " Although eighteenth century writers did not use the term, there was indeed in existence a collection of scientific and engineering knowledge and expertise that can be identified as the state of the art for that time.

Although often depicted in relation to the underworld, Xolotl was not a psychopomp in the Western sense.

Although biology usually determines genetically whether a human being is male or female, ( though intersex people are born ), the state of being neither a man or a woman is sometimes considered in relation to the individual's gender role in society, gender identity, sexual orientation or any other characteristic.

Although not directly a doctrinal belief, the principle of conscientious objection and the obedient relation of the believer to the state became a distinct position of " Socinianism " as it was formalized in the Racovian publications.

Although this day doesn't have much significance in relation to the events of Holy week, it is placed there according to calendar date.

Although it has been reported that Indro Montanelli, the 20th century Italian journalist, is the grandson of Giuseppe Montanelli, there is in fact no relation between the two.

Although the girls are not to be in the show, they are to influence the show's cast and production choices in a story which uses their music but bears no relation to their personal story ; similar to that of ABBA's music in Mamma Mia !.

Although the abbreviations are the same, the pennyweight bears no relation to the actual weight of the penny nail, but to the price for a hundred nails in 15th century England: the larger the nail, the higher the cost per hundred.

Although the GST was promoted as revenue-neutral in relation to the MST, a large proportion of the Canadian population disapproved of the tax.

Although education is normally a matter of exclusive provincial jurisdiction under the Constitution of Canada, there is a special power for the federal government in relation to separate schools.

Although colored quarks are attached to a QCD string and have infinite mass, the spin-statistics relation for quarks can be proved in the short distance limit.

Although originally " microbrewery " was used in relation to the size of breweries, it gradually came to reflect an alternative attitude and approach to brewing flexibility, adaptability, experimentation and customer service.

Although one meets with occasional extremists like Zachrisson or, very recently, Arthur Wade-Evans the majority of scholars have taken a middle position between the extremes of scepticism and gullibility.

Although the false glamour surrounding bourbon or other whisky commercials is possibly no more fatuous than the pseudo-sophistication with which TV soft-drinks are downed or toothpaste applied, there is a sad difference between enticing a viewer into sipping Oopsie-Cola and gulling him into downing bourbon.

Although the tape was run for over 1 hr., a steady state was not reached, and it was concluded that the reason for this was that the back pressure of the manometer was built up from the material fed from between the blocks and this was available at a very slow rate.

Although Miller noted in 1907 that a difference in the pleural blood supply existed between animals, nowhere in his published works is it found that he did a comparative study of the intrapulmonary features of various mammalian lungs other than in the dog and cat ( Miller, '13 ; ;

Although the standard deviation values on which spread of the lines is based are relatively larger for those centers which begin to ossify early ( Table 1 ), there are considerable differences in this value between centers having the closely timed Onsets.

Although I have emphasized the barriers which an aroused nationalism has raised against relations between Christians and non-Christians in Asia, the fact is that this development has also widened the gulf between certain Afro-Asian religions themselves.

Although they " were expecting to see activity in the brain's reward centers ", based on the idea that " people perform altruistic acts because they feel good about it ", what they found was that " another part of the brain was also involved, and it was quite sensitive to the difference between doing something for personal gain and doing it for someone else's gain ".

Although single genes have little effect on complex traits and interact heavily both between themselves and with the external factors, research is underway to unravel possible molecular mechanisms underlying anxiety and comorbid conditions.

Although the range and precise application of defenses varies between jurisdictions, the following represents a list of the defenses that may apply to all levels of assault:

Although the increased contact brought by trade between the Japanese and the Ainu contributed to increased mutual understanding, sometimes it led to conflict, occasionally intensifying into violent Ainu revolts, of which the most important was Shakushain's Revolt ( 1669 – 1672 ).

Although there have been attempts to show that the Ainu language and the Japanese language are related, modern scholars have rejected that the relationship goes beyond contact, such as the mutual borrowing of words between Japanese and Ainu.

Although he described his method as translating " sometimes word for word, sometimes sense for sense ," Alfred's translation actually keeps very close to his original, although through his choice of language he blurred throughout the distinction between spiritual and secular authority.

Although there were low levels of preexisting antibiotic-resistant bacteria before the widespread use of antibiotics, evolutionary pressure from their use has played a role in the development of muiltidrug resistance varieties and the spread of resistance between bacterial species.

Although clubs from other states were at times invited, the final was almost always between the premiers from the two strongest state competitions of the time — South Australia and Victoria — and the majority of matches were played in Adelaide at the request of the SAFA / SAFL.

Although he used the usual French terms " avant J .- C ." ( before Jesus Christ ) and " après J .- C ." ( after Jesus Christ ) to label years elsewhere in his book, the Byzantine historian Venance Grumel used negative years ( identified by a minus sign, −) to label BC years and unsigned positive years to label AD years in a table, possibly to save space, without a year 0 between them.

Although some plants do prefer slightly basic soil ( including vegetables like cabbage and fodder like buffalograss ), most plants prefer a mildly acidic soil ( pH between 6. 0 and 6. 8 ), and alkaline soils can cause problems.

Although no military coups occurred during the 67 years of the Brazilian Empire, the Republican period experienced 4 military coup d ' états in the 75 years between 1889 and 1964.

Although a Conservative, Disraeli was sympathetic to some of the demands of the Chartists and argued for an alliance between the landed aristocracy and the working class against the increasing power of the merchants and new industrialists in the middle class, helping to found the Young England group in 1842 to promote the view that the landed interests should use their power to protect the poor from exploitation by middle-class businessmen.

Although the Nebel stream lay between Fugger's and Marsin's squadrons, the French were forced to change front to meet this new threat, thus forestalling the chance for Marsin to strike at Marlborough's infantry.

Although the book has been historically classified as " prophetic ", the style of writing is apocalyptic which was popular between 200 BCE and 100 CE.

Although Emlyn Hooson held on to the seat of Montgomeryshire, upon Clement Davies death in 1962, the party lost its other Welsh five Welsh seats between 1950 and 1966.

Although historically inaccurate, the legend of the Greek messenger Pheidippides running to Athens with news of the victory became the inspiration for this athletic event, introduced at the 1896 Athens Olympics, and originally run between Marathon and Athens.

Although its Academy Award was presented to Taiwan, Crouching Tiger, Hidden Dragon was in fact an international co-production between companies in four regions: the Chinese company China Film Co-Production Corporation ; the American companies Columbia Pictures Film Production Asia, Sony Pictures Classics and Good Machine ; the Hong Kong company EDKO Film ; and the Taiwanese Zoom Hunt International Productions Company, Ltd ; as well as the unspecified United China Vision, and Asia Union Film & Entertainment Ltd., created solely for this film.

Although contacts between Cubans and foreign visitors were made legal in 1997, extensive censorship has isolated it from the rest of the world.