In mathematics, fuzzy sets are somewhat like sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh and Dieter Klaua in 1965 as an extension of the classical notion of set. At the same time, Salii defined a more general kind of structure called an L-relation, which he studied in an abstract algebraic context. Fuzzy relations, which are used now in different areas, such as linguistics, decision-making, clustering, are special cases of L-relations when L is the unit interval. In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent condition — an element either belongs or does not belong to the set. By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set. Fuzzy sets generalize classical sets, since the indicator functions of classical sets are special cases of the membership functions of fuzzy sets, if the latter only take values 0 or 1. In fuzzy set theory, classical bivalent sets are called crisp sets.

The fuzzy set theory can be used in a wide range of domains in which information is incomplete or imprecise, such as bioinformatics. A fuzzy set is a pair where U is a m: U → a membership function; the reference set U is called universe of discourse, for each x ∈ U, the value m is called the grade of membership of x in. The function m = μ A is called the membership function of the fuzzy set A =. For a finite set U =, the fuzzy set is denoted by. Let x ∈ U. X is called not included in the fuzzy set if m = 0 included if m = 1 included if 0 < m < 1. The set of all fuzzy sets on a universe U is denoted with S F. For any fuzzy set A = and α ∈ the following crisp sets are defined: A ≥ α = A α = is called its α-cut A > α = A α ′ = is called its strong α-cut S = S u p p = A > 0 = is called its support C = C o r e = A = 1 = is called its core. Note that some authors understand'kernel' in a different way, see below. A fuzzy set A = (

Redbelt is a 2008 American martial arts film written and directed by David Mamet and starring Chiwetel Ejiofor, Tim Allen, Alice Braga, Randy Couture, Ricky Jay, Joe Mantegna, Emily Mortimer, David Paymer, Rebecca Pidgeon, Rodrigo Santoro. The film features a number of martial-arts professionals, it opened in wide release in the United States and Canada on May 9, 2008. While closing his Brazilian Jiu-Jitsu studio one evening, martial arts teacher Mike Terry is approached by attorney Laura Black, seeking the owner of the vehicle she accidentally sideswiped. Off-duty police officer Joe Collins, receiving a private lesson from Mike, sees that Laura is distressed and tries to take her coat. Startled, Laura grabs Joe's gun and it goes off, shattering the studio's front window. To avoid having Laura charged with attempted murder and Joe agree to conceal the event. Mike's insurance, will not cover his act of God claim that the window was broken by a strong wind. Mike's wife Sondra, whose fashion business profits are the only thing keeping the struggling studio afloat, requests that Mike ask for a loan from her brother Ricardo, a mixed martial arts champion.

At Ricardo's nightclub, Mike meets with Sondra's other brother and learns that Joe quit as the club's bouncer because Bruno never paid him. Mike is rebuffed. Mike declines Bruno's offer to fight on the undercard of an upcoming match between Ricardo and Japanese legend Morisaki, which could pay out $50,000. Mike believes competitions with money as the incentive weaken the fighter. Meanwhile, aging Hollywood action star Chet Frank enters the nightclub without security and is accosted by a man with a broken bottle. Mike subdues three men in the process; the following day, Mike receives an invitation to dinner from Chet. Mike gives the watch to Joe to pawn in lieu of his unpaid salary at the nightclub. At the dinner party, Chet's wife Zena arranges an informal business deal to buy a large number of dresses from Sondra's company. Chet, impressed by Mike, invites him to the set of his current film; as Mike and Sondra leave the dinner, Mike explains his unique training method to Chet's business associate Jerry Weiss.

Before a sparring match, each fighter must draw one of three marbles, two white and one black. Mike uses his military experience to answer a few technical questions for Chet on the film set and is offered the role of co-producer; that evening, Mike faxes the details of his training methods to Jerry so they can be used in the film. Joe arrives at the studio and informs Mike that he was suspended from duty for pawning the watch, which turned out to be stolen. During their dinner that evening, Mike relays the information to Jerry who excuses himself to handle the matter, but never returns. At home, Mike learns. Sondra is panicky; as he meets with the loan shark to discuss an extension, Mike notices Bruno and Marty Brown on television using Mike's marble-drawing method as a promotional gimmick for the undercard fights of Ricardo's match. Mike hires Laura to sue, but Marty's lawyer threatens that if they do not drop the lawsuit, he will give the police an empty shell casing with Laura's fingerprints, as proof that she attempted to kill an off-duty cop.

He threatens Mike as a witness who covered up the crime by bribing the cop with a stolen watch. When told of the situation, Joe kills himself. Mike feels obligated to help Joe's financially struggling wife and, in desperate need of money himself, decides to compete as an undercard fighter in the upcoming competition. At the arena, Mike discovers the fights are being fixed via a magician using sleight of hand to surreptitiously switch the white and black marbles. Disgusted by this revelation, Mike confronts the conspirators: Marty and Bruno who confirm that unknown to the competitors, the fights are handicapped by the fight promoters so as to ensure winning bets, they reveal that Ricardo is intentionally losing the fight to Morisaki so they can make money on the rematch. Jerry tells Mike that Sondra is the one who told them about Laura shooting the window and Bruno justifies her betrayal by explaining that his sister is too smart to stay with someone who cannot provide for her; as Mike is exiting the arena, he meets Laura.

Their conversation is not audible. Mike re-enters the arena, he incapacitates several security guards trying to stop him and is engaged by Ricardo. The audience and camera crews take notice as Ricardo face off in the arena's corridors. Inspired by the Professor, an elderly martial arts master attending the match, Mike manages to slip a difficult choke hold and defeats Ricardo, making it onto the ring to speak to the Professor personally, he is approached by Morisaki, who offers Mike his ivory-studded belt referred to as a Japanese national treasure, as a sign of respect. He is approached by the Professor himself, who proceeds to award the coveted red belt to an incredulous Mike, embraces him, acknowledging his dedication to the art. Chiwetel Ejiofor as Mike Terry Tim Allen as Chet Frank Alice Braga as Sondra Terry Rodrigo Santoro as Bruno Silva Jose Pablo Cantillo as Snowflake Randy Couture as Dylan Flynn Caroline de Souza Correa as Monica Mike Goldberg as Sports Announcer #2 Damon Herriman as

Osmund Kaldheim is a Norwegian businessman, civil servant and politician for the Conservative Party. He grew up in Etne, he worked as an adviser for the Conservative Party parliamentary group from 1987, was a private secretary in the Ministry of Social Affairs from 1989 to 1990. He was the mayor of Fet municipality from 1995 to 1999, he was an partner in Geelmuyden. Kiese from 1990 to 1995 and a manager from 1999 to 2001, he was a State Secretary in the Ministry of Social Affairs from 2001 to 2002 and the Ministry of Labour and Administration from 2002 to 2004, both in Bondevik's Second Cabinet. From 2004 to 2005 he was a director in Telenor Broadcast. In 2005 he was hired as director of the new Directorate of Diversity, he left in March 2010 to become chief administrative officer in Drammen municipality. He was a board chairman of Nedre Romerike Vannverk from 1996 to 1999 and a board member of Canal Digital from 2001 and of Wright Trafikkskole from 2004

Maureen Wheeler, is a Northern Irish-Australian businesswoman, who co-founded Lonely Planet with her husband Tony Wheeler. She was born in Belfast, Northern Ireland and moved to London at the age 20, where she met her future husband, Tony Wheeler, they travelled from London through Europe and Asia on to Australia. That trip resulted in a guidebook Across Asia on the Cheap and laid the foundations of the travel publisher Lonely Planet, they wrote their second book South-East Asia on a Shoestring. Tony Wheeler says: “Although we’ve continually refined the information organisation, in 2004 pushed through a complete reorganisation and redesign, the pattern we established with that first serious book has remained remarkably consistent to the present day.”Australia became their permanent home during the 1970s, but Maureen was convinced that they could never support themselves through Lonely Planet and began study at La Trobe University in February 1976 and completed a Bachelor of Social Work in 1980.

Afterward, she committed herself full-time to developing the business. Maureen says in 1979: "We moved into an office rather than working from our house, we took on a partner, we took on the India book which resulted in the biggest book on India, seen. Up until there were three of us – all the books were stored in this little tin shed out the back and under the beds and everywhere else, it was a amateur, home grown business." In 1981, with a staff of ten, Lonely Planet India was published. After giving birth to her two children, the numerous questions Maureen received by parents wondering if travel had to be postponed until the children were older, prompted Maureen to write a guidebook, her years of experience on the road with her children allowed her to write Travel With Children to give advice on how to make travel as stress-free as possible. Over the next few decades Lonely Planet became a major publishing house, with offices in Melbourne and Oakland, over 500 staff members and 300 authors.

The company sells 90 per cent overseas. Lonely Planet has printed more than 54 million copies of its 600 guides in 17 languages and has $85 million annual turnover. Maureen organised two Lonely Planet travel summits in 1994 and 1997. Maureen has been the driving force behind Lonely Planet's corporate contributions program established to provide financial assistance for humanitarian projects in developing countries; the next step of her philanthropy is in creating the Planet Wheeler Foundation by funding it with money from the sale of Lonely Planet to the BBC. 1999 – Australian Business Women's Network award for the most Inspiring Business Woman for the year. 2001 – Honorary Doctorate awarded by University of Ulster, Northern Ireland 2002 – Lloyd O'Neil Award for Services to the Australian Book Industry awarded jointly to Tony Wheeler and Maureen Wheeler. 2005 – American University's School of Communication and the Society of American Travel Writers Foundation inaugural Eric A. Friedheim Travel Journalism Lifetime Achievement Award.

2014 – Awarded an Order of Australia with her husband in 2014. 2000 to 2006 – Director on the Board of Tourism Tasmania, serving two terms. 2005 – Adjunct Professor in the Department of Sport and Hospitality Management in the Faculty of Law and Management at La Trobe University. 2006 – Director on the Board of NT Tourism. 2008 – Advisory Board of Australian Aid International. 2014 – Director on the Board of Melbourne Theatre Company. Irish Australian Wheeler Centre

The men's épée competition of the fencing events at the 2019 Pan American Games was held on August 5 at the Lima Convention Centre. The épée competition consisted of a qualification round followed by a single-elimination bracket with a bronze medal match between the two semifinal losers. Fencing was done to 15 touches or to the completion of three three-minute rounds if neither fencer reached 15 touches by then. At the end of time, the higher-scoring fencer was the winner; this sudden-death period was further modified by the selection of a draw-winner beforehand. The following are the results of the event. All 18 fencers were put into three groups of six athletes, were each fencer would have five individual matches; the top 14 athletes overall would qualify for next round

Kei Ito is a Japanese photographer and installation artist based in the United States. He is most known for his Afterimage Requiem series. Ito's work addresses issues of generational connection and deep loss as he explores the materiality of photography as a medium, his current work deals with the tragedy and legacy passed on from his grandfather who survived the atomic bombing of Hiroshima, yet died from cancer and the threat of nuclear disaster, still present today. His recent works express a strong connection between the sun and his family history regarding the idea of invisibility in light and shadow, how the unique characteristic of photography as a media has allowed him to express this matter. On August 6th, 1945, at 8:15 AM, Kei Ito's grandfather, Takeshi Ito, witnessed a great tragedy that destroyed nearly everything in Hiroshima, he survived the bombing. As an activist and author, Ito's grandfather fought against the use of nuclear weaponry throughout his life, until he too passed away from cancer when Kei Ito was ten years old.

Ito remembers his grandfather saying that day in Hiroshima was like hundreds of suns lighting up the sky. In order to express the connection between the sun and his family's history, Ito has created 108 letter size prints and a 200-foot long scroll, which were made by exposing Type-C photographic paper to sunlight; the pattern on the scroll corresponds to his breath. In a darkened room, Ito pulled the paper in front of a small aperture to expose it to the direct sunlight while inhaling and paused when exhaling, he repeated this action until he breathed eight times. One hundred eight is a number with ritual significance in Japanese Buddhism and culture, called Joya no Kane. It's an event where all of the Buddhist temples in the country strike human size or larger bells one hundred and eight times from New Year's Eve to New Year's Day; the number corresponds to the number of evil desires that we suffer from on earth, believed to cause war. This purifies us for the upcoming year; the ritualistic image making of this repeating act, which corresponds to his breath, has become the act of remembrance.

Ito has shown in over 50 exhibitions worldwide. Among those are solo exhibitions at prominent spaces such as Southeastern Center for Contemporary Art Museum, Winston-Salem, NC. Group exhibitions of note include: the Norton Museum of Art, West Palm Beach, FL. 2019 Archives Aflame, Southeastern Center for Contemporary Art Museum, NC Out of the Box: Camera-less Photography, The Norton Museum of Art, FL Ghostly Traces: Memory and Mortality in Contemporary Photography, Vicki Myhren Gallery at University of Denver, CO2018 InLight Richmond, Virginia Museum of Fine Arts, 1708 Gallery, VA Fotofocus Biennial 2018: Nuclear Fallout, Antioch College, OH The Noorderlicht Summer: Opera Spanga Aida, Noorderlicht: House of Photography, Netherland What Keeps You Up at Night, Mendocino College Art Gallery, Ukiah, CA Afterimage Requiem, Baltimore War Memorial, MD Only What We Can Carry, IA&A at Hillyer, DC Infertile American Dream, 14x48 Art Billboard, 215 Woodpoint Rd, Brooklyn Atomic Traces, Online sponsored by 14x48.org Art Billboard2017 Re:Collection, Museum of Contemporary Photography at Colombia College, Chicago, IL 3rd International Exhibition on Conceptual Art, CICA Museum, South Korea 54th Annual Group Exhibition, Masur Museum of Art, LA Wave Pool 44th Group Exhibition, Field Projects Chelsea Gallery, NY 2019 Artist in Residence at Center for Photography at Woodstock Semifinalist for the 14th Janet & Walter Sondheim Prize, Baltimore, MD Artist in Residence at Creative Alliance, Baltimore, MD2018 Individual Artist Award: Photography, Maryland Arts State Council, MD Artist in Residence at Creative Alliance, Baltimore, MD2017 Working Artist Photography Award/Grant, Working Artist Org, WA Honorable Mention for INFOCUS Sidney Zuber Photography Award, Phoenix Art Museum, AZ Maryland representative Artist for The States Project, Lenscratch2016 7th Manifest One Award, Manifest Gallery, Cincinnati, OH Rubys Artist Project Grants: Full grant recipient, GBCA Honorable Mention for IPA: International Photography Awards 4th Annual New York Times Portfolio Review recipient Shortlisted for Royal Photography Society Annual print exhibition, UK Snider Prize honorable mention Awards for Innovations in Imaging awarded by Society of Photographic Education Shortlisted for Royal Photographic Society Annual print exhibition, UK Shortlisted for Tokyo International Foto Awards Honorable Mention Award at 53rd Annual Juried Exhibition at Masur Museum, Louisiana Washington City Paper, The Best Photo Exhibits of 2018 by Louis Jacobson Exposure Magazine/Medium, Art of the Atomic Legacy: the Work of Kei Ito BBC World News and BBC Culture/Art interviewed about Afterimage Requiem exhibition.

Washington Post Magazine published an article about the project Afterimage Requiem The Baltimore Sun Newspaper featured the project Afterimage Requiem on the front article Washington Post reviewed the solo exhibition at Hillyer Art Space in DC Strange Fire Collective "Q&A: KEI ITO" Velocity Magazine "Making Meaning of the Atomic Bomb" Art Maze Magazine International Issue 3: Summer, 2 page Spread, ART MAZE MAGAZINE LTD, UK Featured on the Chicago Magazine “The 5 Biggest Buys by Chicago Art Museum in 2016” Norton Museum of Art Museum of Contemporary Photography in Chicago Center for Photography at Woods