Another way, called the set-builder notation, is by a statement of what is true of the members of the set, like this:
In spoken English, this reads: "the set of all "x" such that "x" is a natural number and "x" is less than four". The symbol [ipe "|" means "such that" or "so that".
The empty set is written in a special way: formula_75, formula_76 or formula_77.
When object "a" is the member of set formula_26 it is written as:
In spoken English, this reads: ""a" is a member of formula_26".
To illustrate operations on sets mathematicians use Venn diagrams. Venn diagrams use circles to show individual sets. The universe is depicted with a rectangle. Results of operations are shown as colored areas. In the illustration of the intersection operation the left circle shows set formula_26 and the right circle shows set formula_27.
Some sets are very important to mathematics. They are used very often. One of these is the "empty set".
Many of these special sets are written using blackboard bold typeface, and these include:
Each of these sets of numbers has an infinite number of elements, and formula_98.
The mathematician Bertrand Russell found that there are problems with the informal definition of sets. He stated this in a paradox called "Russell's paradox". An easier to understand version, closer to real life, is called the "Barber paradox".
There is a small town somewhere. In that town, there is a barber. All the men in the town do not like beards, so they either shave themselves, or they go to the barber shop to be shaved by the barber.
We can therefore make a statement about the barber himself: "The barber shaves all men that do not shave themselves. He only shaves those men (since the others shave themselves and do not need a barber to give them a shave)."
This of course raises the question: What does the barber do each morning to look clean-shaven? This is the paradox.
If the barber shaves himself, he cannot be a barber, since a barber does not shave himself.
If he does not shave himself, he falls in the category of those who do not shave themselves, and so, cannot be a barber.
The following books explore sets in more detail:
Maldives
The Maldives are a group of islands in the Indian Ocean. They are southwest of India. They are made up of the Maldive and Suadive archipelagos, with over 1,200 islands. These islands are a part of 26 atolls. And they are divided to 20 administrative atolls. People live on about 200 of the islands. It has more than 1126 coral reefs. It is the flattest country in the world. The highest point in the Maldives is only high above the level of the sea.
Malé is the capital, with a population of 150,000 people. It’s official religion is Islam mandated by law.
The Maldives has a tropical monsoon climate (Am) under the Köppen climate classification, which is affected by the large landmass of South Asia to the north. Because the Maldives has the lowest elevation of any country in the world, the temperature is constantly hot and often humid. The presence of this landmass causes differential heating of land and water. These factors set off a rush of moisture-rich air from the Indian Ocean over South Asia, resulting in the southwest monsoon. Two seasons dominate Maldives' weather: the dry season associated with the winter northeastern monsoon and the rainy season which brings strong winds and storms.
The shift from the dry northeast monsoon to the moist southwest monsoon occurs during April and May. During this period, the southwest winds contribute to the formation of the southwest monsoon, which reaches Maldives in the beginning of June and lasts until the end of August. However, the weather patterns of Maldives do not always conform to the monsoon patterns of South Asia. The annual rainfall averages in the north and in the south.
The monsoonal influence is greater in the north of the Maldives than in the south, more influenced by the equatorial currents.
The highest temperature ever recorded in Malé was 39.6 °C (103.3 °F) on 1 January 2000 and the lowest was 20.4 °C (68.7 °F) on 20 March 1989 and 21 January 2017.
Prime Minister of Australia
The Prime Minister of Australia is the head of the executive branch of the government of Australia. The prime leads the cabinet of Australia and is the highest ranking Minister of the Crown in the federal parliament. The prime minister is appointed by the monarch, who is advised by the governor general.
The current Prime Minister of Australia, since August 2018, is Scott Morrison.
As of , there are six living former Australian Prime Ministers.
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Structure
In science, structure is a fundamental concept. It means the arrangement of parts. Examples are the organs in a body, or crystals in a substance like granite.
A structure may refer to buildings, machines and similar things made by people as well as to a rock or a mineral, atoms or sentences. It always reflects the hierarchy of its parts.
Hardcore punk
Hardcore punk (or hardcore) is a style of punk music that usually sounds louder, faster and more angry than earlier punk rock music from the 1970s. Hardcore began in the late 1970s. Some of the important bands of the genre are Bad Brains, Black Flag, D.O.A, Minor Threat and Cro-Mags. Hardcore songs often have more political lyrics (words) than songs from 1970s punk.
No one is quite sure where the term "hardcore punk" came from. The band D.O.A. may have helped to make the term more popular with the title of their album "Hardcore '81" in 1981.
The first hardcore punk song was "Out of Vogue" by the Santa Ana band The Middle Class. This band started to play a loud type of punk with shouted words.
People in hardcore punk bands and people who listened to hardcore punk wore clothes that were of two types. Some people wore clothes and hairstyles that were strange and a bit scary, like Mohawk hairstyles and leather jackets with metal studs. Other people wore regular clothes like black pants and T-shirts.
At hardcore punk concerts, people did a type of dancing called moshing. This type of dance involves people pushing each other and jumping off the stage.
In the early 1980s, hardcore bands from England began to get noticed, such as Discharge, The Exploited, and The Varukers. Discharge played with a sound that included heavy metal guitar styles.
Hardcore led to the creation of the straight edge movement. People who are straight edge do not use alcohol, tobacco, and other drugs. Straight edge led to two new groups, called hardline and youth crew. Hardcore musicians also created independent record labels in the 1980s.
Many bands had a "Do it Yourself" (DIY) attitude. Bands would record their own albums using simple four-track recorders, and the sell the records themselves at shows or through the mail. Hardcore influenced the creation of new styles of music, including alternative rock, grunge, alternative metal, metalcore, thrash metal and post-hardcore.
Many early hardcore punk bands had songs about liberal politics such as anarchism and socialism. They spoke against political leaders, such as American president Ronald Reagan and British prime minister Margaret Thatcher. While George W. Bush was President of the United States, a number of hardcore bands criticized him.
A small number of hardcore musicians have expressed right wing views, including the band Antiseen, whose guitarist Joe Young ran for office as a Libertarian in North Carolina. Former Misfits singer Michale Graves supported George W. Bush.
Hardcore band singers often sing by shouting the words of the song. Hardcore songs often sound like the singer is angry.
Hardcore bands use electric guitars, electric bass guitar, drums, and a singer. The electric guitars in hardcore bands are distorted, noisy, and fuzzy because the guitarists are using electronic boxes called distortion pedals. Some hardcore guitarists have changed the tuning of their guitar strings so they play notes that are lower than those a normal guitar can play.