This article does not cite any references or sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (May 2009)

In mathematics and logic, the phrase "there is one and only one" is used to indicate that exactly one object with a certain property exists. In mathematical logic, this sort of quantification is known as uniqueness quantification or unique existential quantification.

Uniqueness quantification is often denoted with the symbols "∃!" or ∃₌₁". For example, the formal statement

Failed to parse (Missing texvc executable; please see math/README to configure.): \exists! n \in \mathbb{N}\,(n - 2 = 4)

may be read aloud as "there is exactly one natural number n such that n - 2 = 4".

Proving uniqueness[link]

Proving uniqueness turns out to be mostly easier than that of existence or expressibility. The most common technique to proving uniqueness is to assume there exists two quantities (say, a and b) that satisfies the condition given, and then logically deducing their equality, i.e. a = b.

As a simple high school example, to show x + 2 = 5 has only one solution, we assume there are two solutions first, namely, a and b, satisfying x + 2 = 5. Thus

Failed to parse (Missing texvc executable; please see math/README to configure.): a + 2 = 5\text{ and }b + 2 = 5. \,

By transitivity of equality,

Failed to parse (Missing texvc executable; please see math/README to configure.): a + 2 = b + 2. \,

By cancellation,

Failed to parse (Missing texvc executable; please see math/README to configure.): a = b. \,

This simple example shows how a proof of uniqueness is done, the end result being the equality of the two quantities that satisfy the condition. We must say, however, that existence/expressibility must be proven before uniqueness, or else we cannot even assume the existence of those two quantities to begin with.

Reduction to ordinary existential and universal quantification[link]

Uniqueness quantification can be expressed in terms of the existential and universal quantifiers of predicate logic by defining the formula ∃!x P(x) to mean

Failed to parse (Missing texvc executable; please see math/README to configure.): \exists x P(x) \wedge \neg \exists x,y \, ( P(x) \wedge P(y) \wedge (x \neq y))

where an equivalence is:

Failed to parse (Missing texvc executable; please see math/README to configure.): \exists x P(x) \wedge \forall y\,(P(y) \to x = y).

An equivalent definition that has the virtue of separating the notions of existence and uniqueness into two clauses, at the expense of brevity, is

Failed to parse (Missing texvc executable; please see math/README to configure.): \exists x P(x) \wedge \forall y\, \forall z\,((P(y) \wedge P(z)) \to y = z).

Another equivalent definition with the advantage of brevity is

Failed to parse (Missing texvc executable; please see math/README to configure.): \exists x\,\forall y\,(P(y) \leftrightarrow x = y).

Generalizations[link]

One generalization of uniqueness quantification is counting quantification. This includes both quantification of the form "exactly k objects exist such that …" as well as "infinitely many objects exist such that …" and "only finitely many objects exist such that…". The first of these forms is expressible using ordinary quantifiers, but the latter two cannot be expressed in ordinary first-order logic.

See also[link]

• One-hot

This mathematics-related article is a stub. You can help Wikipedia by expanding it. v t e

This article is about human hair. For other uses, see Longhair (disambiguation) or Hair

Boy with shoulder length hair on a beach in Melanesia.

Long hair is a hairstyle. Exactly what constitutes long hair can change from culture to culture, or even within cultures. For example, a woman with chin-length hair in some cultures may be said to have short hair, while a man with the same length of hair in some of the same cultures would be said to have long hair.

Scientists view long hair as playing a large part in natural selection among many species, since long, thick and healthy hair or fur is frequently a sign of fertility and youth.^[1] As hair grows slowly, long hair reveals several years of a person's health status and reproductive fitness. Hair length is significantly correlated with female attractiveness, as rated by men as well as women.^[2] Hair length and quality can act as a cue to especially a woman's youth and health, signifying reproductive potential.^[3] The prevalence of trichophilia (hair partialism or fetischism) is 7% in the population, and very long hair is a common subject of devotion in this group.^[4][5]

Ways of life often viewed as more rigid, such as soldiers and religious cultures, often have explicit rules regarding hair length. For example, Buddhist monks shave their heads as part of their order of worship.^[6] Even outside religious structures, cultures often associate male long hair with ways of life outside of what is culturally accepted. Subservient cultures, for example, are sometimes detected by their rulers through hair length, as was the case with the Gaelic Irish under English rule and the Moors under Spanish rule in Medieval Spain.^[7]

Again, though, there are exceptions to these rules, notably among the long-haired and religiously devoted Nazarites of the Hebrew Bible (Samson being a famous example)^[8] and among the Sikhs.^[9]

East Asian cultures have traditionally seen long, unkempt hair in a woman as a sign of sexual intent or a recent sexual encounter, as usually their hair is tied up in styles such as the ponytail, plait, or any bun.^[10]

Transferred meaning[link]

The traditional connotation of "long hair" in English meant, roughly, someone artistically knowledgeable or wise, an aesthete.^[11] As a descriptive term, it has been applied to Merovingians and classical music enthusiasts, as well as hippies and aesthetes.^[11]

Hair lengths[link]

A Polish woman with hair reaching the mid-back

A Swedish male with waist-length hair

Painting of a woman with hair grown down to her hips, by Franz Xavier Winterhalter

Woman with knee-length braided hair

Hair length is measured (in centimeters or inches) from the line of the scalp on the forehead up over the highest point of the skull to its termination. In cosmetology, hair lengths are usually categorized according to the part of the body where the bulk of the longest hair terminates.^[12] Hair usually takes about two years to reach shoulder length,^[13] and at approximately 80–90 cm (32–36 inches) for most people, waist-length hair usually takes about seven years to grow, including occasional trims.^[12][13]

Maximum hair length[link]

The maximum hair length that is possible to reach is about 15 cm for infants (below the age of 1), about 60 cm for children (below the age of 13), and generally 100 cm for adults. Documentation for decrease of the maximum length with age cannot be found in the literature. Some individuals can reach excessive lengths. Lengths greater than 150 cm are frequently observed in long hair contests.^[14] Xie Qiuping had the longest documented hair in the world, measuring 5.627 m (18 ft 5.54 in) in May 2004.^[15]

The maximum terminal hair length depends on the length of the anagen (period of hair growth) for the individual. Waist-length hair or longer is only possible to reach for people with long anagen. The anagen lasts between 2 and 7 years, for some individuals even longer, and follows by shorter catagen (transition) and telogen (resting) periods. Between 85% and 90% of the hair strains are in anagen.

Psychological significance[link]

Anthropologists speculate that the functional significance of long head hair may be adornment, a by-product of secondary natural selection once other androgenic/somatic hair (body hair) had largely been lost. Another possibility is that long head hair is a result of Fisherian runaway sexual selection, where long lustrous hair is a visible marker for a healthy individual. For some groups or individuals, however, short hair is the selected trait.^[1]

By seven to nine months, infants can tell the sexes apart based on hair length, voice pitch and faces.^[16][17]

Cultural history[link]

Hair is one of the most important ways humans have of presenting themselves, being one of the parts of their body which is easiest to manipulate. Also, males having short, cut hair (or a shaven head) is often viewed as being under society's control, such as while in military service or prison or as punishment for a crime, while males having long hair signifies being outside of the mainstream.^[18]

In Jewish and Christian scriptures[link]

In the Old Testament, the Nazirites would go for long periods of time without cutting their hair to show devotion to God.^[19] Samson is one example; his strength depended upon his refraining from cutting his hair.^[8] The New Testament says, "Doth not even nature itself teach you, that, if a man have long hair, it is a shame unto him? But if a woman has long hair, it is a glory to her: for her hair is given her for a covering,"^[20] something followed by women in some apostolic pentecostal denominations in some countries as an act of holiness.

Western culture[link]

Ancient Greece and Rome[link]

In ancient Greece, long male hair was a symbol of wealth and power, while a shaven head was appropriate for a slave. The ancient Greeks had several Gods and heroes who wore their hair long, including Zeus, Achilles, Apollo, and Poseidon. Greek soldiers are said to have worn their hair long in battle. Such warriors considered it a sign of aristocracy and are said to have combed it openly in order to show off. Also, in order to keep enemies from getting a hold of it in battle, they were known to cut the front short, but leave it long in the back, where it was more out of reach. A widely held alternative interpretation of the conventional belief that they cut it short in the front in order to keep their enemies from getting a hold of it during battle is that, they did not in fact cut it in the front, but rather tied it back in a style known as a pony tail in order to keep it out of their enemies' reach. The pony tail method allowed warriors, who often traveled to battle with a minimal amount of equipment in order to avoid excessively heavy loads over long marches, to keep their hair manageable with a small piece of string to hold their pony tail in place and a knife to cut the back to length with one simple slice. Around the sixth century, however, the Greek men shifted to shorter hairstyles, with the exception of the Spartans. Women in the culture remained with the longer style, which for them showed freedom, health, and wealth, as well as good behavior.^[21] In men, it was considered a sign of false pride by this time.^[22] Women in Roman times valued long hair, usually with a center part. Men's hair was usually shorter than women's, (but in the early times, the custom was quite the same as in Greece) although other cultures of the time, such as Greeks in the east, considered long hair to be typical of philosophers, who were thought to be too engrossed in learning to bother with hair.^[23] Strictly in the province of Rome, however the shorter hairstyle was especially popular.^[22] When Julius Caesar conquered the Gauls, who favored long hair, he ordered it to be cut short.^[24]

Middle Ages[link]

In the European middle ages, shorter hair often signified servitude and peasantry, while long hair was often attributed to freemen, as was the case with the Germanic Goths and Merovingians.

The Gaelic Irish (both men and women)^[25] took great pride in their long hair—for example, a person could be heavily fined for cutting a man's hair short against his will.^[26] When the Anglo-Normans and the English colonized Ireland, hair length came to signify one's allegiance. Irishmen who cut their hair short were deemed to be forsaking their Irish heritage. Likewise, English colonists who wore their hair long in the back were deemed to be forsaking their role as English subjects and giving in to the Irish life. Thus, hair length was one of the most common ways of judging a true Englishman in this period. Muslims in Christian areas were ordered to keep their hair short and parted, as their longer style was considered rebellious and barbaric.^[7]

Empress Elisabeth of Austria wearing thigh-length hair in 1865.

A long hair fad was widespread among English and French men in the 11th and 12th centuries, though otherwise it was considered, mostly because of endorsement of the Roman Catholic Church, proper for men to have shorter, and women, longer hair. The fad was largely brought about by monarchs who rejected the shorter hairstyle, causing the people to follow. Wulfstan, a religious leader, worried that those with longer hair would fight like women, and be unable to protect England from foreign invasion. This idea can be found in later military leaders as well, such as those of the American Confederacy.^[27] Knights and rulers would also sometimes cut or pull out their hair in order to show penitence and mourning, and a squire's hair was generally shorter than a knight's. Married women who let their hair flow out in public were frowned upon, as this was normally reserved for the unwed, although they were allowed to let it out in mourning, to show their distressed state. Through these centuries it was expected of Eastern Christians to wear long hair as well as long beards, which was especially expected from clergy and monks.

In England, during the English Civil Wars of 1642 to 1651, male hair length was emblematic of the disputes between Cavaliers and Roundheads (Puritans). Cavaliers wore longer hair, and were less religious-minded, thought of by the Roundheads as lecherous. The more devout Roundheads had short hair, although there were exceptions.^[6]

Recent meanings[link]

A man with waist length hair

Before World War One men generally had longer hair and beards. However, short hair on men has often been enforced as a mean of control, in police, military and other forces that require obedience and discipline. Slaves and defeated armies were often required to shave their heads. The trench warfare in 1914 to 1918 exposed men to flea and lice infestations, which prompted the order to cut short hair, establishing a norm that has persisted.^[12]

Beat poets during the 1950s wore longer hairstyles, as did many of the urban gay culture, although long hair was far from popular. By 1960, a small "beatnik" community in Newquay, Cornwall, England (including a young Wizz Jones) had attracted the attention and the abhorrence of their neighbors for growing their hair to a length past the shoulders, resulting in a television interview with Alan Whicker on BBC television's Tonight series.^[28] The 1960s also introduced The Beatles, who started a more widespread longer hair fad. The social revolution of the 1960s led to a renaissance of unchecked hair growth,^[12] and long hair, especially on men, was worn as a political or countercultural symbol or protest. This cultural symbol extended to several Western countries in the Americas, Western Europe, South Africa, and Australia.^[29] Specific long hairstyles such as dreadlocks have been part of counterculture movements seeking to define other alternative cultures and lifestyles since this time.^[10] Longer hair in general remained popular among the youth rebellion throughout the liberal decade of the 1960s.^[29] Homosexual men, many of whom had adopted a long hairstyle in the early 1950s, have largely abandoned this trend. Some people saw the long hair fad as a threat to gender identity, cultural, and religious norms as it grew with the spread of the hippie movement in the 1960s.^[29] Notably, some country-and-western performers during this period (and many fans) also sported longer hair.^[30][31]

In the 1970s, the popularity of Jamaica's reggae music and musician Bob Marley prompted interest in dreadlocks internationally. The anti-establishment philosophy of Rastafari, echoed in much of the reggae of the time, resonated with leaning youth of all ethnicities — especially and primarily among African Americans and other Blacks, but among counterculture whites as well.^[32] The Eastern Christians are encouraged to wear long hair with long beard. In the 1980s the view of long hair as a solitary signifier of political or counter-cultural identity was countered and parodied in films such as Rambo and many other militaristic heroes of media which challenged then-contemporary views of what was masculine.^[33] Today, longer hairstyles remain popular among rock enthusiasts.^[34] Long hair may be grown for the purpose of being donated to an organization, such as Locks of Love, for hairpieces to help those who could not have hair otherwise, such as those who are diagnosed with alopecia areata.

Women[link]

Women often have a stronger inclination towards long hair than men do. Younger women tend to have longer hair than older women. Hair length and quality can act as a cue to a woman's youth and health, signifying reproductive potential.^[35] Growing and wearing long hair was almost universal among women in the western world until World War One.^[12]

Sikhism[link]

The Sikhs were commanded by Guru Gobind Singh at the Baisakhi Amrit Sanchar in 1699 to wear long uncut hair called Kesh at all times. This was one of the 'five requisites of faith', collectively called Kakars that form the external visible identifiers to clearly affirm a Sikh's commitment and dedication to the order (Hukam) of the tenth master and made one a member of the Khalsa. The Khalsa is the "Saint-Soldier" of Guru Gobind Singh who stated the following: "He does not recognize anyone else except One Lord, not even the bestowal of charities, performance of merciful acts, austerities and restraint on pilgrim-stations; the perfect light of the Lord illuminates his heart, then consider him as the immaculate Khalsa." (Guru Gobind Singh in the Dasam Granth page 1350)

The Kesh or unshorn long hair is an indispensable part of the human body as created by Vaheguru that is the mainstay of the 'Jivan Jaach' and the Rehni that was prescribed by Guru Gobind Singh Sahib by which a Sikh is clearly and quickly identified, Kesh. The kanga, another requisite of faith is usually tucked behind the "Rishi Knot" and tied under the turban. The uncut long head hair and the beard in the case of men forms the main kakar for the Sikhs.

KESADHARI, a term defining a Sikh as one who carries on his head the full growth of his kes (hair) which he never trims or cuts for any reason. Anyone, Sikh or non-Sikh, may keep the hair unshorn, but for the Sikh kes, unshorn hair, is a requisite of faith and an inviolable vow. The Sikh Rahit Maryada published by the Shiromani Gurdwara Prabandhak Committee, statutory body for the control and management of Sikh shrines and by extension for laying down rules about Sikh beliefs and practices, issued in 1945, after long and minute deliberations among Sikh scholars and theologians, defines a Sikh thus: Every Sikh who has been admitted to the rites of amrit, i.e. who has been initiated as a Sikh, must allow his hair to grow to its full length. This also applies to those born of Sikh families but [who] have not yet received the rites of amrit of the tenth master, Guru Gobind Singh. ".

Islam[link]

Transjordanian Bedouin with shoulder length hair, during World War II

In the Muslim world, it seems the trend of hair styles is now favouring short over long in men. In the past, Bedouin Muslims often wore their hair in long braids, but western influences brought on the view that such styles were feminine in nature. Now, Bedouins are much less likely to have long hair.^[36] Islamic countries in North Africa such as Egypt view long hair in men as homosexual and in one case the Egyptian police viewed it as satanic and a sign of an infidel^[37].

Muslims regard Prophet Muhammad as the best example to live by, and try to emulate him whenever possible. The Islamic Prophet Muhammad reportedly in Sahih Muslim had hair that "hung over his shoulders and earlobes".^[38] Sahih Bukhari, regarded the most authentic of hadith, also supports this using a prime example of the prophet Isa.^[39] The Prophet Muhammad has also described Jesus as "having long hair reaching his ear lobes."^[40] Malik's Muwatta 51.2.6 reported, Yahya related to me from Malik from Yahya ibn Said that Abu Qatada al-Ansari said to the Messenger of Allah, may Allah bless him and grant him peace, "I have a lot of hair which comes down to my shoulders, shall I comb it? The Messenger of Allah. may Allah bless him and grant him peace, said, "Yes, and honour it." Sometimes Abu Qatada oiled it twice in one day because the Messenger of Allah, may Allah bless him and grant him peace, said to him. "Honour it."

With regards to women, neither Qur'an nor Sunnah explicitly state that women cannot cut their hair. Hadith does mention that women should not imitate men, and vice versa, and hence many scholars on this assumption, decree that women should let their hair grow longer than the hair of the Prophet, reaching beyond their shoulders, as hadith mentions that the Prophet had his hair between his shoulder and his earlobes. (He described Jesus's hair, which hung to his earlobes, as long.^[41])

However culturally, some Muslims are opposed to men having long hair as it is also important in Islam to have clear differences (in appearance) between sexes. And generally these cultures encourage women to have long hair and men to have short hair.^[42] The Taliban in Afghanistan viewed long hair for men as a western influence, and punished it by arrest and forced haircuts,^[43] albeit this would be a direct contradiction of the sunnah of the Prophet. Similar measures have been taken by Islamists in Iraq.^[44] In spite of this, several Taleban affiliated members of the Mehsud clan are recognisable by their long hair. The Saudi Islamist fighter Amir Khattab was also notable for his long hair. Dervishes of some Sufi orders, such as the Kasnazani, often have long hair and whirl it around during rituals.^[45]

Native Americans[link]

Yakima boy with mid-back level hair. (Photo by Edward S. Curtis, 1868–1952.)

Many Native American men wore long hair before the arrival of western influences on their culture or rather eastern influences for them. (In Cherokee legends, for example, males said to be handsome were often described as having "long hair almost to the ground" or similar formulas.^[46]). Both men and women of these cultures have frequently struggled to maintain their tradition, but have faced heavy opposition. Many consider it a sign of giving in to western influences to have their hair cut.^[47][48] Early American settlers saw long-haired, native men as rebelling against their civilized society. Mountain men and trappers who adopted the customs were also considered amoral, and often identified by their long hair.^[49] Since the cultural movements of the sixties and seventies, however, Native Americans have felt less pressure to have short hair, as different movements have defended their cultural rights.^[50] For example, several states have loosened prison regulations, allowing Native Americans to wear long hair during incarceration, along with other cultural allowances.^[51] There has been resistance to these changes, however, as long hair is sometimes used to hide drugs, as well as to identify with a gang.^[52]

Sub-Saharan Africans[link]

Ngbandi girls with waist-length hair, central Africa, 1905

In West African cultures, women with long hair were highly valued. Long, thick hair was seen as a sign of health, strength, and capability to bear many children. In keeping with this general theme, women who were too young for marriage would shave a portion of their heads to signal so. This tradition, however, did not extend to every African tribe, as several valued shorter hair.^[53]

African Americans[link]

When African slaves were freed in the Americas, they struggled with comments from whites calling their hair ugly and nappy. Many African-American women began straightening their hair. However, there was and always will be a variety of hairstyles, such as dreadlocks, twists, short hair, long hair, curly hair, straight hair, etc. Social pressures were heavily influencing African-American females to have long straight hair. Although African Americans have short or long hair, they can do a lot with their hair. Therefore, they have hair shows to show the large variety of hairstyles. African Americans, as well as other thick curly-haired people, are often pressured to not wear their natural hair in today's society. When they wear their hair natural, it's soft and easy to manage. However, when it comes to not wanting to wear natural hair, it becomes more difficult to manage. Therefore, African Americans are wearing natural hair more, so it's finally becoming more accepted.

Asia[link]

Japanese woman combing her long hair. (Colour woodblock print from 1920 by Hashiguchi Goyō.)

Illustration of The Tale of Genji, painted by Tosa Mitsuoki (1617-1691).

Historically, East Asian cultures viewed long hair as a sign of youth and aesthetic beauty. Long hair is associated with private life and sexuality. East Asian cultures see long, unkempt hair in a woman as a sign of sexual intent or a recent sexual encounter, as usually their hair is tied up.^[10] Lay Buddhists have long hair, while Buddhist monks have shaved heads.^[6]

For Sikhs, Kesh is the practice of allowing one's hair to grow naturally as a symbol of devotion to God and lack of worldliness.^[9]

In Jewish and other cultures, shortening hair signifies mourning and sadness.^[18]

Around the seventeenth century, the Manchu people forced all men in China to adopt a hairstyle called a queue, which was basically a long braid down the back with the hair on the front part of the head shaved. This style lasted well into the nineteenth century, when the Chinese began immigrating to America. Americans at first judged them to be poor workers because their long hair brought an association with women.^[54] Both Islamic and Christian missionaries to the Chinese were strong advocates of shorter hair for their converts, but this was a small group.^[55] Around the Destruction of Four Olds period in 1964, almost anything seen as part of Traditional Chinese culture would lead to problems with the Communist Red Guards. Items that attracted dangerous attention if caught in the public included jewelry and long hair.^[56] These things were regarded as symbols of bourgeois lifestyle, that represented wealth. People had to avoid them or suffer serious consequences such as tortures and beatings by the guards.^[56] More recently, long hair was ridiculed in China from October 1983 to February 1984, as part of the Anti-Spiritual Pollution Campaign.^[57] Li Yang, an unorthodox Chinese English teacher who brands the popular Crazy English, claims the following on his website:

"What [America, England and Japan] want most is for China’s youth to have long hair, wear bizarre clothes, drink soda, listen to Western music, have no fighting spirit, love pleasure and comfort!"^[58]

In Southeast Asia and Indonesia, male long hair was valued in until the seventeenth century, when the area adopted outside influences including Islam and Christianity. Invading cultures enforced shorter hairstyles on men as a sign of servitude, as well. They were also confused at the short hairstyles among women in certain areas, such as Thailand, and struggled to explain why women in the area had such short hair. They came up with several mythical stories, one of which involved a king who found a long hair in his rice and, in a rage, demanded that all women keep their hair short.^[55]

In rural areas in certain Asian countries, for example India, girls still usually let their hair grow long, and knee-length hair is not unusual.^[59]

In the medieval Japan, heian gentleman were not very interested in a woman's physical beauty and rarely had an opportunity to see it. The only physical attribute of interest was a woman's hair, which had to be thick and longer than she was tall. The fascination with long hair was one reason why a woman's becoming a nun was regarded with such seriousness - it could never again grow to its full length. This explains why Genji refuses to let Murasaki (his de facto wife in the classical Japanese novel the Tale of Genji) take the tonsure when she is ill.^[60]

See also[link]

• Grooming
• Haircut
• Hair extensions
• Hairstick
• Lady Godiva
• Long hair fetishism, a form of Trichophilia
• Rapunzel

Images[link]

• Female long hair in art (Gallery at Wikimedia commons)
• Female long hair photos (Gallery at Wikimedia commons)
• Male long hair in art (Gallery at Wikimedia commons)
• Male long hair photos (Gallery at Wikimedia commons)

References[link]