1.
Specials (Unicode block)
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Specials is a short Unicode block allocated at the very end of the Basic Multilingual Plane, at U+FFF0–FFFF. Of these 16 codepoints, five are assigned as of Unicode 9, U+FFFD � REPLACEMENT CHARACTER used to replace an unknown, unrecognized or unrepresentable character U+FFFE <noncharacter-FFFE> not a character. FFFE and FFFF are not unassigned in the sense. They can be used to guess a texts encoding scheme, since any text containing these is by not a correctly encoded Unicode text. The replacement character � is a found in the Unicode standard at codepoint U+FFFD in the Specials table. It is used to indicate problems when a system is unable to render a stream of data to a correct symbol and it is usually seen when the data is invalid and does not match any character, Consider a text file containing the German word für in the ISO-8859-1 encoding. This file is now opened with an editor that assumes the input is UTF-8. The first and last byte are valid UTF-8 encodings of ASCII, therefore, a text editor could replace this byte with the replacement character symbol to produce a valid string of Unicode code points. The whole string now displays like this, f�r, a poorly implemented text editor might save the replacement in UTF-8 form, the text file data will then look like this, 0x66 0xEF 0xBF 0xBD 0x72, which will be displayed in ISO-8859-1 as fï¿½r. Since the replacement is the same for all errors this makes it impossible to recover the original character, a better design is to preserve the original bytes, including the error, and only convert to the replacement when displaying the text. This will allow the text editor to save the original byte sequence and it has become increasingly common for software to interpret invalid UTF-8 by guessing the bytes are in another byte-based encoding such as ISO-8859-1. This allows correct display of both valid and invalid UTF-8 pasted together, Unicode control characters UTF-8 Mojibake Unicodes Specials table Decodeunicodes entry for the replacement character
2.
International standard
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International standards are standards developed by international standards organizations. International standards are available for consideration and use worldwide, the most prominent organization is the International Organization for Standardization. International standards may be used either by application or by a process of modifying an international standard to suit local conditions. Technical barriers arise when different groups together, each with a large user base. Establishing international standards is one way of preventing or overcoming this problem, the implementation of standards in industry and commerce became highly important with the onset of the Industrial Revolution and the need for high-precision machine tools and interchangeable parts. Henry Maudslay developed the first industrially practical screw-cutting lathe in 1800, maudslays work, as well as the contributions of other engineers, accomplished a modest amount of industry standardization, some companies in-house standards spread a bit within their industries. Joseph Whitworths screw thread measurements were adopted as the first national standard by companies around the country in 1841 and it came to be known as the British Standard Whitworth, and was widely adopted in other countries. By the end of the 19th century differences in standards between companies were making trade increasingly difficult and strained, the Engineering Standards Committee was established in London in 1901 as the worlds first national standards body. After the First World War, similar national bodies were established in other countries, by the mid to late 19th century, efforts were being made to standardize electrical measurement. An important figure was R. E. B, Crompton, who became concerned by the large range of different standards and systems used by electrical engineering companies and scientists in the early 20th century. Many companies had entered the market in the 1890s and all chose their own settings for voltage, frequency, current, adjacent buildings would have totally incompatible electrical systems simply because they had been fitted out by different companies. Crompton could see the lack of efficiency in this system and began to consider proposals for a standard for electric engineering. In 1904, Crompton represented Britain at the Louisiana Purchase Exposition in Saint Louis as part of a delegation by the Institute of Electrical Engineers. He presented a paper on standardisation, which was so well received that he was asked to look into the formation of a commission to oversee the process. By 1906 his work was complete and he drew up a permanent constitution for the first international standards organization, the body held its first meeting that year in London, with representatives from 14 countries. In honour of his contribution to electrical standardisation, Lord Kelvin was elected as the bodys first President, the International Federation of the National Standardizing Associations was founded in 1926 with a broader remit to enhance international cooperation for all technical standards and specifications. The body was suspended in 1942 during World War II, after the war, ISA was approached by the recently formed United Nations Standards Coordinating Committee with a proposal to form a new global standards body. List of international common standards List of technical standard organisations Global Frameworks and standards organized along function lines, accessed 2014 ^ Cordova
3.
Logical conjunction
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In logic and mathematics, and is the truth-functional operator of logical conjunction, the and of a set of operands is true if and only if all of its operands are true. The logical connective that represents this operator is written as ∧ or ⋅. A and B is true only if A is true and B is true, an operand of a conjunction is a conjunct. Related concepts in other fields are, In natural language, the coordinating conjunction, in programming languages, the short-circuit and control structure. And is usually denoted by an operator, in mathematics and logic, ∧ or ×, in electronics, ⋅. In Jan Łukasiewiczs prefix notation for logic, the operator is K, logical conjunction is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both of its operands are true. The conjunctive identity is 1, which is to say that AND-ing an expression with 1 will never change the value of the expression. In keeping with the concept of truth, when conjunction is defined as an operator or function of arbitrary arity. The truth table of A ∧ B, As a rule of inference, conjunction introduction is a classically valid, the argument form has two premises, A and B. Intuitively, it permits the inference of their conjunction, therefore, A and B. or in logical operator notation, A, B ⊢ A ∧ B Here is an example of an argument that fits the form conjunction introduction, Bob likes apples. Therefore, Bob likes apples and oranges, Conjunction elimination is another classically valid, simple argument form. Intuitively, it permits the inference from any conjunction of either element of that conjunction, therefore, A. or alternately, A and B. In logical operator notation, A ∧ B ⊢ A. falsehood-preserving, yes When all inputs are false, walsh spectrum, Nonlinearity,1 If using binary values for true and false, then logical conjunction works exactly like normal arithmetic multiplication. Many languages also provide short-circuit control structures corresponding to logical conjunction. Logical conjunction is used for bitwise operations, where 0 corresponds to false and 1 to true,0 AND0 =0,0 AND1 =0,1 AND0 =0,1 AND1 =1. The operation can also be applied to two binary words viewed as bitstrings of length, by taking the bitwise AND of each pair of bits at corresponding positions. For example,11000110 AND10100011 =10000010 and this can be used to select part of a bitstring using a bit mask. For example,10011101 AND00001000 =00001000 extracts the fifth bit of an 8-bit bitstring
4.
Logical disjunction
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In logic and mathematics, or is the truth-functional operator of disjunction, also known as alternation, the or of a set of operands is true if and only if one or more of its operands is true. The logical connective that represents this operator is written as ∨ or +. A or B is true if A is true, or if B is true, or if both A and B are true. In logic, or by means the inclusive or, distinguished from an exclusive or. An operand of a disjunction is called a disjunct, related concepts in other fields are, In natural language, the coordinating conjunction or. In programming languages, the short-circuit or control structure, or is usually expressed with an infix operator, in mathematics and logic, ∨, in electronics, +, and in most programming languages, |, ||, or or. In Jan Łukasiewiczs prefix notation for logic, the operator is A, logical disjunction is an operation on two logical values, typically the values of two propositions, that has a value of false if and only if both of its operands are false. More generally, a disjunction is a formula that can have one or more literals separated only by ors. A single literal is often considered to be a degenerate disjunction, the disjunctive identity is false, which is to say that the or of an expression with false has the same value as the original expression. In keeping with the concept of truth, when disjunction is defined as an operator or function of arbitrary arity. Falsehood-preserving, The interpretation under which all variables are assigned a value of false produces a truth value of false as a result of disjunction. The mathematical symbol for logical disjunction varies in the literature, in addition to the word or, and the formula Apq, the symbol ∨, deriving from the Latin word vel is commonly used for disjunction. For example, A ∨ B is read as A or B, such a disjunction is false if both A and B are false. In all other cases it is true, all of the following are disjunctions, A ∨ B ¬ A ∨ B A ∨ ¬ B ∨ ¬ C ∨ D ∨ ¬ E. The corresponding operation in set theory is the set-theoretic union, operators corresponding to logical disjunction exist in most programming languages. Disjunction is often used for bitwise operations, for example, x = x | 0b00000001 will force the final bit to 1 while leaving other bits unchanged. Logical disjunction is usually short-circuited, that is, if the first operand evaluates to true then the second operand is not evaluated, the logical disjunction operator thus usually constitutes a sequence point. In a parallel language, it is possible to both sides, they are evaluated in parallel, and if one terminates with value true
5.
Negation
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Negation is thus a unary logical connective. It may be applied as an operation on propositions, truth values, in classical logic, negation is normally identified with the truth function that takes truth to falsity and vice versa. In intuitionistic logic, according to the Brouwer–Heyting–Kolmogorov interpretation, the negation of a proposition p is the proposition whose proofs are the refutations of p. Classical negation is an operation on one logical value, typically the value of a proposition, that produces a value of true when its operand is false and a value of false when its operand is true. So, if statement A is true, then ¬A would therefore be false, the truth table of ¬p is as follows, Classical negation can be defined in terms of other logical operations. For example, ¬p can be defined as p → F, conversely, one can define F as p & ¬p for any proposition p, where & is logical conjunction. The idea here is that any contradiction is false, while these ideas work in both classical and intuitionistic logic, they do not work in paraconsistent logic, where contradictions are not necessarily false. But in classical logic, we get an identity, p → q can be defined as ¬p ∨ q. Algebraically, classical negation corresponds to complementation in a Boolean algebra and these algebras provide a semantics for classical and intuitionistic logic respectively. The negation of a proposition p is notated in different ways in various contexts of discussion and fields of application. Among these variants are the following, In set theory \ is also used to indicate not member of, U \ A is the set of all members of U that are not members of A. No matter how it is notated or symbolized, the negation ¬p / −p can be read as it is not the case p, not that p. Within a system of logic, double negation, that is. In intuitionistic logic, a proposition implies its double negation but not conversely and this marks one important difference between classical and intuitionistic negation. Algebraically, classical negation is called an involution of period two and this result is known as Glivenkos theorem. De Morgans laws provide a way of distributing negation over disjunction and conjunction, ¬ ≡, in Boolean algebra, a linear function is one such that, If there exists a0, a1. An ∈ such that f = a0 ⊕ ⊕, another way to express this is that each variable always makes a difference in the truth-value of the operation or it never makes a difference. Negation is a logical operator
6.
Complement (set theory)
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In set theory, the complement of a set A refers to elements not in A. The relative complement of A with respect to a set B, also termed the difference of sets A and B, written B ∖ A, is the set of elements in B but not in A. When all sets under consideration are considered to be subsets of a given set U, the absolute complement of A is the set of elements in U but not in A. If A and B are sets, then the complement of A in B, also termed the set-theoretic difference of B and A, is the set of elements in B. The relative complement of A in B is denoted B ∖ A according to the ISO 31-11 standard, if R is the set of real numbers and Q is the set of rational numbers, then R ∖ Q is the set of irrational numbers. Let A, B, and C be three sets, the following identities capture notable properties of relative complements, C ∖ = ∪. C ∖ = ∪, with the important special case C ∖ = demonstrating that intersection can be expressed using only the relative complement operation. If A is a set, then the complement of A is the set of elements not in A. Formally. The absolute complement of A is usually denoted by A ∁, other notations include A c, A ¯, A ′, ∁ U A, and ∁ A. Assume that the universe is the set of integers, if A is the set of odd numbers, then the complement of A is the set of even numbers. If B is the set of multiples of 3, then the complement of B is the set of numbers congruent to 1 or 2 modulo 3, assume that the universe is the standard 52-card deck. If the set A is the suit of spades, then the complement of A is the union of the suits of clubs, diamonds, and hearts. If the set B is the union of the suits of clubs and diamonds, then the complement of B is the union of the suits of hearts, let A and B be two sets in a universe U. The following identities capture important properties of complements, De Morgans laws. Complement laws, A ∪ A ∁ = U, if A ⊂ B, then B ∁ ⊂ A ∁. Involution or double complement law, ∁ = A, relationships between relative and absolute complements, A ∖ B = A ∩ B ∁. Relationship with set difference, A ∁ ∖ B ∁ = B ∖ A, the first two complement laws above show that if A is a non-empty, proper subset of U, then is a partition of U. In the LaTeX typesetting language, the command \setminus is usually used for rendering a set difference symbol, when rendered, the \setminus command looks identical to \backslash except that it has a little more space in front and behind the slash, akin to the LaTeX sequence \mathbin
7.
Natural number
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In mathematics, the natural numbers are those used for counting and ordering. In common language, words used for counting are cardinal numbers, texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers. These chains of extensions make the natural numbers canonically embedded in the number systems. Properties of the numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics, the most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a mark, the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers, the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1,10, and all the powers of 10 up to over 1 million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds,7 tens, and 6 ones, and similarly for the number 4,622. A much later advance was the development of the idea that 0 can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, the Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica. The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628, the first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, independent studies also occurred at around the same time in India, China, and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the nature of the natural numbers. A school of Naturalism stated that the numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized God made the integers, in opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not really natural, later, two classes of such formal definitions were constructed, later, they were shown to be equivalent in most practical applications. The second class of definitions was introduced by Giuseppe Peano and is now called Peano arithmetic and it is based on an axiomatization of the properties of ordinal numbers, each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several systems of set theory
8.
0 (number)
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0 is both a number and the numerical digit used to represent that number in numerals. The number 0 fulfills a role in mathematics as the additive identity of the integers, real numbers. As a digit,0 is used as a placeholder in place value systems, names for the number 0 in English include zero, nought or naught, nil, or—in contexts where at least one adjacent digit distinguishes it from the letter O—oh or o. Informal or slang terms for zero include zilch and zip, ought and aught, as well as cipher, have also been used historically. The word zero came into the English language via French zéro from Italian zero, in pre-Islamic time the word ṣifr had the meaning empty. Sifr evolved to mean zero when it was used to translate śūnya from India, the first known English use of zero was in 1598. The Italian mathematician Fibonacci, who grew up in North Africa and is credited with introducing the system to Europe. This became zefiro in Italian, and was contracted to zero in Venetian. The Italian word zefiro was already in existence and may have influenced the spelling when transcribing Arabic ṣifr, modern usage There are different words used for the number or concept of zero depending on the context. For the simple notion of lacking, the words nothing and none are often used, sometimes the words nought, naught and aught are used. Several sports have specific words for zero, such as nil in football, love in tennis and it is often called oh in the context of telephone numbers. Slang words for zero include zip, zilch, nada, duck egg and goose egg are also slang for zero. Ancient Egyptian numerals were base 10 and they used hieroglyphs for the digits and were not positional. By 1740 BC, the Egyptians had a symbol for zero in accounting texts. The symbol nfr, meaning beautiful, was used to indicate the base level in drawings of tombs and pyramids. By the middle of the 2nd millennium BC, the Babylonian mathematics had a sophisticated sexagesimal positional numeral system, the lack of a positional value was indicated by a space between sexagesimal numerals. By 300 BC, a symbol was co-opted as a placeholder in the same Babylonian system. In a tablet unearthed at Kish, the scribe Bêl-bân-aplu wrote his zeros with three hooks, rather than two slanted wedges, the Babylonian placeholder was not a true zero because it was not used alone