Question: Georg Ferdinand Ludwig Philipp Cantor ( KAN-tor; German: ['geoRk 'feRdinant 'lu:tvIc 'fIlIp 'kantoR]; March 3 [O.S. February 19] 1845 - January 6, 1918) was a German mathematician. He invented set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are more numerous than the natural numbers.

Cantor's 1874 Crelle paper was the first to invoke the notion of a 1-to-1 correspondence, though he did not use that phrase. He then began looking for a 1-to-1 correspondence between the points of the unit square and the points of a unit line segment. In an 1877 letter to Richard Dedekind, Cantor proved a far stronger result: for any positive integer n, there exists a 1-to-1 correspondence between the points on the unit line segment and all of the points in an n-dimensional space. About this discovery Cantor wrote to Dedekind: "Je le vois, mais je ne le crois pas!" ("I see it, but I don't believe it!") The result that he found so astonishing has implications for geometry and the notion of dimension.  In 1878, Cantor submitted another paper to Crelle's Journal, in which he defined precisely the concept of a 1-to-1 correspondence and introduced the notion of "power" (a term he took from Jakob Steiner) or "equivalence" of sets: two sets are equivalent (have the same power) if there exists a 1-to-1 correspondence between them. Cantor defined countable sets (or denumerable sets) as sets which can be put into a 1-to-1 correspondence with the natural numbers, and proved that the rational numbers are denumerable. He also proved that n-dimensional Euclidean space Rn has the same power as the real numbers R, as does a countably infinite product of copies of R. While he made free use of countability as a concept, he did not write the word "countable" until 1883. Cantor also discussed his thinking about dimension, stressing that his mapping between the unit interval and the unit square was not a continuous one.  This paper displeased Kronecker, and Cantor wanted to withdraw it; however, Dedekind persuaded him not to do so and Weierstrass supported its publication. Nevertheless, Cantor never again submitted anything to Crelle.

Using a quote from the above article, answer the following question: What is one-to-one correspondence?
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Answer: He then began looking for a 1-to-1 correspondence between the points of the unit square and the points of a unit line segment.

Problem: R.E.M. was an American rock band from Athens, Georgia, that was formed in 1980 by drummer Bill Berry, guitarist Peter Buck, bassist/backing vocalist Mike Mills, and lead vocalist Michael Stipe. One of the first alternative rock bands, R.E.M. was noted for Buck's ringing, arpeggiated guitar style, Stipe's distinctive vocal quality and obscure lyrics, Mills' melodic basslines and backing vocals, and Berry's tight, economical style of drumming. R.E.M. released its first single--"Radio Free Europe"--in 1981 on the independent record label Hib-Tone. The single was followed by the Chronic Town EP in 1982, the band's first release on I.R.S. Records.

After the band released two slow-paced albums in a row, R.E.M.'s 1994 album Monster was, as Buck said, "a 'rock' record, with the rock in quotation marks." In contrast to the sound of its predecessors, the music of Monster consisted of distorted guitar tones, minimal overdubs, and touches of 1970s glam rock. Like Out of Time, Monster topped the charts in both the US and UK. The record sold about nine million copies worldwide. The singles "What's the Frequency, Kenneth?" and "Bang and Blame" were the band's last American Top 40 hits, although all the singles from Monster reached the Top 30 on the British charts. Warner Bros. assembled the music videos from the album as well as those from Automatic for the People for release as Parallel in 1995.  In January 1995, R.E.M. set out on its first tour in six years. The tour was a huge commercial success, but the period was difficult for the group. On March 1, Berry collapsed on stage during a performance in Lausanne, Switzerland, having suffered a brain aneurysm. He had surgery immediately and recovered fully within a month. Berry's aneurysm was only the beginning of a series of health problems that plagued the Monster tour. Mills had to undergo abdominal surgery to remove an intestinal adhesion in July; a month later, Stipe had to have an emergency surgery to repair a hernia. Despite all the problems, the group had recorded the bulk of a new album while on the road. The band brought along eight-track recorders to capture its shows, and used the recordings as the base elements for the album. The final three performances of the tour were filmed at the Omni Coliseum in Atlanta, Georgia and released in home video form as Road Movie.  R.E.M. re-signed with Warner Bros. Records in 1996 for a reported $80 million (a figure the band constantly asserted originated with the media), rumored to be the largest recording contract in history at that point. The group's 1996 album New Adventures in Hi-Fi debuted at number two in the US and number one in the UK. The five million copies of the album sold were a reversal of the group's commercial fortunes of the previous five years. Time writer Christopher John Farley argued that the lesser sales of the album were due to the declining commercial power of alternative rock as a whole. That same year, R.E.M. parted ways with manager Jefferson Holt, allegedly due to sexual harassment charges levied against him by a member of the band's home office in Athens. The group's lawyer Bertis Downs assumed managerial duties.

Did they win any awards for the album?

Answer with quotes:
the largest recording contract in history