set

set, in mathematics, collection of entities, called elements of the set, that may be real objects or conceptual entities. Set theory not only is involved in many areas of mathematics but has important applications in other fields as well, e.g., computer technology and atomic and nuclear physics.

Definition of Sets

A set must be well defined; i.e., for any given object, it must be unambiguous whether or not the object is an element of the set. For example, if a set contains all the chairs in a designated room, then any chair can be determined either to be in or not in the set. If there were no chairs in the room, the set would be called the empty, or null, set, i.e., one containing no elements. A set is usually designated by a capital letter. If A is the set of even numbers between 1 and 9, then A={2, 4, 6, 8}. The braces, {}, are commonly used to enclose the listed elements of a set. The elements of a set may be described without actually being listed. If B is the set of real numbers that are solutions of the equation x²=9, then the set can be written as B={x:x²=9} or B={x~~pipe~;x²=9}, both of which are read: B is the set of all x such that x²=9; hence B is the set {3,-3}.

Membership in a set is indicated by the symbol ∈ and nonmembership by ∉; thus, x∈A means that element x is a member of the set A (read simply as "x is a member of A") and y∉A means y is not a member of A. The symbols ⊂ and ⊃ are used to indicate that one set A is contained within or contains another set B; A⊂B means that A is contained within, or is a subset of, B; and A⊃B means that A contains, or is a superset of, B.

Operations on Sets

There are three basic set operations: intersection, union, and complementation. The intersection of two sets is the set containing the elements common to the two sets and is denoted by the symbol ⋒. The union of two sets is the set containing all elements belonging to either one of the sets or to both, denoted by the symbol ∪. Thus, if C={1, 2, 3, 4} and D={3, 4, 5}, then C⋒D={3, 4} and C∪D={1, 2, 3, 4, 5}. These two operations each obey the associative law and the commutative law, and together they obey the distributive law.

In any discussion the set of all elements under consideration must be specified, and it is called the universal set. If the universal set is U={1, 2, 3, 4, 5} and A={1, 2, 3}, then the complement of A (written A') is the set of all elements in the universal set that are not in A, or A'={4, 5}. The intersection of a set and its complement is the empty set (denoted by  ), or A⋒A'= ; the union of a set and its complement is the universal set, or A∪A'=U. See also symbolic logic.

Branch of mathematics that deals with the properties of sets. It is most valuable as applied to other areas of mathematics, which borrow from and adapt its terminology and concepts. These include the operations of union (∪), and intersection (∩). The union of two sets is a set containing all the elements of both sets, each listed once. The intersection is the set of all elements common to both original sets. Set theory is useful in analyzing difficult concepts in mathematics and logic. It was placed on a firm theoretical footing by Georg Cantor, who discovered the value of clearly formulated sets in the analysis of problems in symbolic logic and number theory.

Learn more about set theory with a free trial on Britannica.com.

Dance for sets of four couples standing in square formation. The most popular type of U.S. folk dance, it derived from the quadrille and was originally called a square dance to distinguish it from the contra, or longways, dance (for a double line of couples) and the round dance (for a circle of couples). The U.S. square dance progresses through specific patterns called or sung out to the dancers by a caller and accompanied by lively music played on instruments such as fiddle, banjo, accordion, guitar, and piano.

Learn more about square dance with a free trial on Britannica.com.

In measurement, a device consisting of two straightedges set at a right angle. It is used by carpenters and machinists to check the correctness of right angles, as a guide when drawing lines on materials before cutting, or for locating holes. In mechanical drawing or drafting, a T-shaped instrument known as a T square is used to establish a horizontal reference on the drafting board.

Learn more about square with a free trial on Britannica.com.

Geometrical representation of the traditional logical relations of opposition (contradictories, contraries, subalterns) among the four types of propositions (universal affirmative or A-proposition, universal negative or E, particular affirmative or I, particular negative or O) in the Aristotelian doctrine of the syllogism. At the top left is the A; at the top right is the E; below the A is the I; and below the E is the O. The A and the O, like the E and I, are contradictories (diagonal relations); the A and the E are contraries; and the I is subaltern to (hence implied by) the A, the O subaltern to the E. Whereas contradictories have opposite truth-values (one true, the other false), contraries cannot both be true but can both be false.

Learn more about opposition, square of with a free trial on Britannica.com.

One of the largest public squares in the world, originally designed and built in Beijing in 1651 and enlarged in 1958. It is named for the massive stone “Gate of Heavenly Peace” (Tiananmen) at its northern end. It contains and is surrounded by halls, museums, and monuments, including the Mao Zedong Memorial Hall, where Mao's body rests in state. Tiananmen Square is the site of numerous parades and other celebrations. It has also been the rallying point for political demonstrations, including events during the May Fourth Movement in 1919 and during May–June 1989.

Learn more about Tiananmen Square with a free trial on Britannica.com.

Formally organized social group consisting of every male (or female) of comparable age. In societies where the practice traditionally occurs (e.g., the Nuer of the southern Sudan or the Masai of Kenya and Tanzania), a person belongs, either from birth or from a determined age, to a named age set that passes through a series of life stages, or age grades, each of which has a distinctive status or social and political role. Seealso rite of passage; social status.

Learn more about age set with a free trial on Britannica.com.

A set-top box (STB) or set-top unit (STU) is a device that connects to a television and an external source of signal, turning the signal into content which is then displayed on the television screen.

History

Before the All-Channel Receiver Act of 1962 required US television receivers to be able to tune the entire VHF and UHF range (which in North America was NTSC-M channels 2 through 83 on 54 to 890 MHz), a set-top box known as a UHF converter would be installed at the receiver to shift a portion of the UHF-TV spectrum onto low-VHF channels for viewing. As some 1960s-era twelve-channel TV sets remained in use for many years, and Canada and Mexico were slower than the US to require UHF tuners to be factory-installed in new TV's, a market for these converters continued to exist for much of the 1970s.

Cable television represented a possible alternative to deployment of UHF converters as broadcasts could be frequency-shifted to VHF channels at the cable head-end instead of the final viewing location. Unfortunately, cable brought a new problem; most cable systems could not accommodate the full 54-890 MHz VHF/UHF frequency range and the twelve channels of VHF space were quickly exhausted on most systems. Adding any additional channels therefore needed to be done by inserting the extra signals into cable systems on non-standard frequencies, typically either below VHF channel 7 (midband) or directly above VHF channel 13 (superband).

These frequencies corresponded to non-television services (such as two-way radio) over-the-air and were therefore not on standard TV receivers. Before cable-ready TV sets became common in the late 1980s, a set-top box known as a cable converter box was needed to receive the additional analog cable TV channels and convert them to frequencies that could be seen on a regular TV. These boxes often provided a wired or wireless remote control which could be used to shift one selected channel to a low-VHF frequency (most often channels 3 or 2) for viewing. Block conversion of the entire affected frequency band onto UHF, while less common, was used by some models to provide full VCR compatibility and the ability to drive multiple TV sets, albeit with a somewhat non-standard channel numbering scheme.

Newer television receivers greatly reduced the need for external set-top boxes, although cable converter boxes continue to be used to descramble premium cable channels and to receive digital cable channels, along with using interactive services like video on demand, pay per view, and home shopping through television. Satellite and microwave-based services also require specific external receiver hardware, so the use of set-top boxes of various formats never completely disappeared.

Digital television

Special digital set-top boxes are available for receiving digital television broadcasts on TV sets that do not have a built in digital tuner. In the case of direct broadcast satellite (mini-dish) systems such as SES Astra, Dish Network, or DirecTV, the set-top box is an integrated receiver/decoder (or IRD).

In the United Kingdom, digital set-top boxes (often referred to as digiboxes, after Sky Digital's trademark for their unit) are usually for digital terrestrial television through services such as Freeview, a service operated by the Freeview Consortium, or through digital satellite with BSkyB and also with digital cable. They are used to access television as well as audio and interactive services through the "Red Button" promoted by broadcasters such as the BBC with BBCi or Sky with Sky Active. Current Freeview set-top boxes and digital televisions are not capable of decoding the protocol DVB-T2 that terrestrial High-definition will use in 2009, so viewers will need to purchase an HD receiver when the time comes.

In Australia set-top boxes are the principal means of receiving digital terrestrial broadcasts as comparably few television sets have in-built digital tuners. The Foxtel set-top boxes (including the Foxtel iQ unit) are also used to receive subscription television from Foxtel. For HDTV receiving they are using Beyonwiz manufactured media centers which came to market at March 2007.

In the United States, deployment of a very basic coupon-eligible converter box is supported through a $40 federal subsidy to encourage viewers of over-the-air television to adopt ATSC standards ahead of the shutdown of full-power analog broadcasts planned for February 17, 2009. These boxes are not readily available in Canada and Mexico, where broadcasters are not yet required to transition to digital television, although ATSC-capable tuners are appearing in some new TV's and television-related products such as computer video capture cards, satellite receivers and DVD recorders.

Globally, some boxes also have a built-in digital video recorder (or DVR) which often utilises the electronic programme guide scheduling data and records content to an internal hard drive.

Many TV signal sources

The signal source might be an ethernet cable , a satellite dish, a coaxial cable (see cable television), a telephone line (including DSL connections), Broadband over Power Line, or even an ordinary VHF or UHF antenna. Content, in this context, could mean any or all of video, audio, Internet webpages, interactive games, or other possibilities.

IPTV

In IPTV networks, the set-top box is a small computer providing two-way communications on an IP network, and decoding the video streaming media. IP set-top boxes have a built in home network interface which can be Ethernet or one of the existing wire home networking technologies such as HomePNA.

In the US and France, IPTV is being used by telephone companies (often on ADSL or optical fibre networks) as a means to compete with traditional local cable television monopolies.

Most of the set top boxes in France are distributed by the Internet providers and allow the consumer to have access to IPTV, VoIP, Internet and media centre functionalities.

Ambiguities in the definition

With the advent of flat panel televisions set-top boxes are now deeper in profile than the tops of most modern TV sets. Because of this set-top boxes are often placed beneath televisions and the term set-top box has become something of a misnomer.

A set-top box does not necessarily contain a tuner of its own. A box connected to a television (or VCR) set's SCART connector is fed with the baseband television signal from the set's tuner, and can ask the television to display the returned processed signal instead.

This SCART feature had been used for connection to analogue decoding equipment by Pay TV operators in Europe, and in the past was used for connection to teletext equipment before the decoders became built-in. The outgoing signal could be of the same nature as the incoming signal, or RGB component video, or even an "insert" over the original signal, thanks to the "fast switching" feature of SCART.

In case of analogue pay-TV, this approach avoided the need for a second remote control. The use of digital television signals in more modern pay-TV schemes requires that decoding take place before the digital-to-analogue conversion step, rendering the video outputs of an analogue SCART connector no longer suitable for interconnection to decryption hardware. Standards such as DVB's Common Interface and ATSC's CableCARD therefore use a PCMCIA-like card inserted as part of the digital signal path as their alternative to a tuner-equipped set-top box.

The distinction between external tuner or demodulator boxes (traditionally considered to be "set-top boxes") and storage devices (such as VCR, DVD or disc-based PVR units) is also blurred by the increasing deployment of satellite and cable tuner boxes with hard discs, network or USB interfaces built-in.

Devices with computer terminal-like capabilities, such as the WebTV thin-client, also fall into a grey area.

Software quality

As complexity of the set-top box increases, the software quality practices of the industry become obvious and many systems have bugs; this is a particularly troublesome issue with digital television apparatus being rushed to market before the government-mandated shutdown of full-power analogue television broadcasts. See Comparison of CECB units for details on individual units.

However, users of computer-based solutions such as Linux MCE and MythTV have a very flexible list of possible features ranging from basic DVR-like functionality to features such as DVD copying, home automation, and house-wide music/video file playing. They also can fix any software bugs just by joining the development team.

See also

Related Technologies:

• ATSC tuner
• Audio
• Cable modem
• Connectivity: RS-232, USB, Bluetooth and Wi-Fi
• Digital television adapter (DTA)
• DOCSIS
• DVB
• Free-to-air
• Integrated Services Digital Broadcasting
• Interactive television
• IPTV
• QAM tuner
• QPSK
• Satellite dish
• SlingBox
• Symbol rate
• Cable Converter Box
• Media extender, Digital media receiver or HDD media player

FLOSS Technologies:

• The Freedom Box, a STB with Free Software: www.freedombox.tv

References

External links