1.
Nine-point circle
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In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle and these nine points are, The midpoint of each side of the triangle The foot of each altitude The midpoint of the line segment from each vertex of the triangle to the orthocenter. Its center is the nine-point center of the triangle, the diagram above shows the nine significant points of the nine-point circle. Points D, E, and F are the midpoints of the three sides of the triangle, points G, H, and I are the feet of the altitudes of the triangle. Points J, K, and L are the midpoints of the segments between each altitudes vertex intersection and the triangles orthocenter. For an acute triangle, six of the lie on the triangle itself. But soon after Feuerbach, mathematician Olry Terquem himself proved the existence of the circle and he was the first to recognize the added significance of the three midpoints between the triangles vertices and the orthocenter. Thus, Terquem was the first to use the name nine-point circle, the circle which passes through the feet of the altitudes of a triangle is tangent to all four circles which in turn are tangent to the three sides of the triangle. The triangle center at which the incircle and the nine-point circle touch is called the Feuerbach point, the radius of a triangles circumcircle is twice the radius of that triangles nine-point circle. Figure 3 A nine-point circle bisects a line segment going from the triangles orthocenter to any point on its circumcircle. Figure 4 The center N of the nine-point circle bisects a segment from the orthocenter H to the circumcenter O, the nine-point center N is one-fourth of the way along the Euler line from the centroid G to the orthocenter H, HN = 3NG. The nine-point circle of a triangle is the circumcircle of both the reference triangles medial triangle and its orthic triangle. The center of all rectangular hyperbolas that pass through the vertices of a triangle lies on its nine-point circle, examples include the well-known rectangular hyperbolas of Keipert, Jeřábek and Feuerbach. This fact is known as the Feuerbach conic theorem, if an orthocentric system of four points A, B, C and H is given, then the four triangles formed by any combination of three distinct points of that system all share the same nine-point circle. This is a consequence of symmetry, the sides of one triangle adjacent to a vertex that is an orthocenter to another triangle are segments from that second triangle, a third midpoint lies on their common side. Consequently, these four triangles have circumcircles with identical radii, let N represent the common nine-point center and P be an arbitrary point in the plane of the orthocentric system. As P approaches N the locus of P for the corresponding constant K, furthermore the nine-point circle is the locus of P such that PA2+PB2+PC2+PH2 = 4R2. The centers of the incircle and excircles of a triangle form an orthocentric system, the nine-point circle created for that orthocentric system is the circumcircle of the original triangle
2.
Diagram
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A diagram is a symbolic representation of information according to some visualization technique. Diagrams have been used since ancient times, but became prevalent during the Enlightenment. Sometimes, the uses a three-dimensional visualization which is then projected onto a two-dimensional surface. The word graph is used as a synonym for diagram. Specific kind of display, This is the genre that shows qualitative data with shapes that are connected by lines, arrows. In science the term is used in both ways, on the other hand, Lowe defined diagrams as specifically abstract graphic portrayals of the subject matter they represent. Or in Halls words diagrams are simplified figures, caricatures in a way and these simplified figures are often based on a set of rules. The basic shape according to White can be characterized in terms of elegance, clarity, ease, pattern, simplicity, elegance is basically determined by whether or not the diagram is the simplest and most fitting solution to a problem. g. Many of these types of diagrams are generated using diagramming software such as Visio. Chart Diagrammatic reasoning Diagrammatology List of graphical methods Mathematical diagram Plot commons, michael Anderson, Peter Cheng, Volker Haarslev. Theory and Application of Diagrams, First International Conference, Diagrams 2000, edinburgh, Scotland, UK, September 1–3,2000. Garcia, M The Diagrams of Architecture
3.
Set (mathematics)
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In mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right. For example, the numbers 2,4, and 6 are distinct objects when considered separately, Sets are one of the most fundamental concepts in mathematics. Developed at the end of the 19th century, set theory is now a part of mathematics. In mathematics education, elementary topics such as Venn diagrams are taught at a young age, the German word Menge, rendered as set in English, was coined by Bernard Bolzano in his work The Paradoxes of the Infinite. A set is a collection of distinct objects. The objects that make up a set can be anything, numbers, people, letters of the alphabet, other sets, Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same elements. Cantors definition turned out to be inadequate, instead, the notion of a set is taken as a notion in axiomatic set theory. There are two ways of describing, or specifying the members of, a set, one way is by intensional definition, using a rule or semantic description, A is the set whose members are the first four positive integers. B is the set of colors of the French flag, the second way is by extension – that is, listing each member of the set. An extensional definition is denoted by enclosing the list of members in curly brackets, one often has the choice of specifying a set either intensionally or extensionally. In the examples above, for instance, A = C and B = D, there are two important points to note about sets. First, in a definition, a set member can be listed two or more times, for example. However, per extensionality, two definitions of sets which differ only in one of the definitions lists set members multiple times, define, in fact. Hence, the set is identical to the set. The second important point is that the order in which the elements of a set are listed is irrelevant and we can illustrate these two important points with an example, = =. For sets with many elements, the enumeration of members can be abbreviated, for instance, the set of the first thousand positive integers may be specified extensionally as, where the ellipsis indicates that the list continues in the obvious way. Ellipses may also be used where sets have infinitely many members, thus the set of positive even numbers can be written as
4.
Venn diagram
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A Venn diagram is a diagram that shows all possible logical relations between a finite collection of different sets. These diagrams depict elements as points in the plane, and sets as regions inside closed curves, a Venn diagram consists of multiple overlapping closed curves, usually circles, each representing a set. In Venn diagrams the curves are overlapped in every possible way and they are thus a special case of Euler diagrams, which do not necessarily show all relations. Venn diagrams were conceived around 1880 by John Venn and they are used to teach elementary set theory, as well as illustrate simple set relationships in probability, logic, statistics, linguistics and computer science. A Venn diagram in which in addition the area of each shape is proportional to the number of elements it contains is called an area-proportional or scaled Venn diagram and this example involves two sets, A and B, represented here as coloured circles. The orange circle, set A, represents all living creatures that are two-legged, the blue circle, set B, represents the living creatures that can fly. Each separate type of creature can be imagined as a point somewhere in the diagram, living creatures that both can fly and have two legs—for example, parrots—are then in both sets, so they correspond to points in the region where the blue and orange circles overlap. That region contains all such and only living creatures. Humans and penguins are bipedal, and so are then in the circle, but since they cannot fly they appear in the left part of the orange circle. Mosquitoes have six legs, and fly, so the point for mosquitoes is in the part of the circle that does not overlap with the orange one. Creatures that are not two-legged and cannot fly would all be represented by points outside both circles, the combined region of sets A and B is called the union of A and B, denoted by A ∪ B. The union in this case contains all living creatures that are either two-legged or that can fly, the region in both A and B, where the two sets overlap, is called the intersection of A and B, denoted by A ∩ B. For example, the intersection of the two sets is not empty, because there are points that represent creatures that are in both the orange and blue circles. They are rightly associated with Venn, however, because he comprehensively surveyed and formalized their usage, Venn himself did not use the term Venn diagram and referred to his invention as Eulerian Circles. Of these schemes one only, viz. that commonly called Eulerian circles, has met with any general acceptance, the first to use the term Venn diagram was Clarence Irving Lewis in 1918, in his book A Survey of Symbolic Logic. Venn diagrams are similar to Euler diagrams, which were invented by Leonhard Euler in the 18th century. Baron has noted that Leibniz in the 17th century produced similar diagrams before Euler and she also observes even earlier Euler-like diagrams by Ramon Lull in the 13th Century. In the 20th century, Venn diagrams were further developed, D. W. Henderson showed in 1963 that the existence of an n-Venn diagram with n-fold rotational symmetry implied that n was a prime number
5.
Leonhard Euler
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He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy, Euler was one of the most eminent mathematicians of the 18th century, and is held to be one of the greatest in history. He is also considered to be the most prolific mathematician of all time. His collected works fill 60 to 80 quarto volumes, more than anybody in the field and he spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia. A statement attributed to Pierre-Simon Laplace expresses Eulers influence on mathematics, Read Euler, read Euler, Leonhard Euler was born on 15 April 1707, in Basel, Switzerland to Paul III Euler, a pastor of the Reformed Church, and Marguerite née Brucker, a pastors daughter. He had two sisters, Anna Maria and Maria Magdalena, and a younger brother Johann Heinrich. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, Paul Euler was a friend of the Bernoulli family, Johann Bernoulli was then regarded as Europes foremost mathematician, and would eventually be the most important influence on young Leonhard. Eulers formal education started in Basel, where he was sent to live with his maternal grandmother. In 1720, aged thirteen, he enrolled at the University of Basel, during that time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupils incredible talent for mathematics. In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono, at that time, he was unsuccessfully attempting to obtain a position at the University of Basel. In 1727, he first entered the Paris Academy Prize Problem competition, Pierre Bouguer, who became known as the father of naval architecture, won and Euler took second place. Euler later won this annual prize twelve times, around this time Johann Bernoullis two sons, Daniel and Nicolaus, were working at the Imperial Russian Academy of Sciences in Saint Petersburg. In November 1726 Euler eagerly accepted the offer, but delayed making the trip to Saint Petersburg while he applied for a physics professorship at the University of Basel. Euler arrived in Saint Petersburg on 17 May 1727 and he was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he worked in close collaboration. Euler mastered Russian and settled life in Saint Petersburg. He also took on a job as a medic in the Russian Navy. The Academy at Saint Petersburg, established by Peter the Great, was intended to improve education in Russia, as a result, it was made especially attractive to foreign scholars like Euler
6.
Set theory
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Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics, the language of set theory can be used in the definitions of nearly all mathematical objects. The modern study of set theory was initiated by Georg Cantor, Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals. Mathematical topics typically emerge and evolve through interactions among many researchers, Set theory, however, was founded by a single paper in 1874 by Georg Cantor, On a Property of the Collection of All Real Algebraic Numbers. Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathematicians in the East, especially notable is the work of Bernard Bolzano in the first half of the 19th century. Modern understanding of infinity began in 1867–71, with Cantors work on number theory, an 1872 meeting between Cantor and Richard Dedekind influenced Cantors thinking and culminated in Cantors 1874 paper. Cantors work initially polarized the mathematicians of his day, while Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of mathematical constructivism, did not. This utility of set theory led to the article Mengenlehre contributed in 1898 by Arthur Schoenflies to Kleins encyclopedia, in 1899 Cantor had himself posed the question What is the cardinal number of the set of all sets. Russell used his paradox as a theme in his 1903 review of continental mathematics in his The Principles of Mathematics, in 1906 English readers gained the book Theory of Sets of Points by William Henry Young and his wife Grace Chisholm Young, published by Cambridge University Press. The momentum of set theory was such that debate on the paradoxes did not lead to its abandonment, the work of Zermelo in 1908 and Abraham Fraenkel in 1922 resulted in the set of axioms ZFC, which became the most commonly used set of axioms for set theory. The work of such as Henri Lebesgue demonstrated the great mathematical utility of set theory. Set theory is used as a foundational system, although in some areas category theory is thought to be a preferred foundation. Set theory begins with a binary relation between an object o and a set A. If o is a member of A, the notation o ∈ A is used, since sets are objects, the membership relation can relate sets as well. A derived binary relation between two sets is the relation, also called set inclusion. If all the members of set A are also members of set B, then A is a subset of B, for example, is a subset of, and so is but is not. As insinuated from this definition, a set is a subset of itself, for cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined
7.
New math
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New Mathematics or New Math was a brief, dramatic change in the way mathematics was taught in American grade schools, and to a lesser extent in European countries, during the 1960s. The phrase is used now to describe any short-lived fad which quickly became highly discredited. The name is given to a set of teaching practices introduced in the U. S. Topics introduced in the New Math include modular arithmetic, algebraic inequalities, bases other than 10, matrices, symbolic logic, Boolean algebra, All of these topics have been greatly de-emphasized or eliminated in U. S. elementary schools and high schools curricula since the 1960s. Quine wrote that the air of Cantorian set theory was not to be associated with the New Math. According to Quine, the New Math involved merely the Boolean algebra of classes, though the New Math did not succeed in its time, it did reflect on great developments occurring in society. Boolean logic is an ingredient of digital design and binary data are the machine level representation of the data managed in digital machines. The Boolean logic and the rules of sets would later prove to be very valuable with the onset of databases, in particular, the notion of relation as used in relational databases is a realization of a variant of the idea of n-ary relation in set theory. In this and other ways, the New Math proved to be an important link to the computer revolution and this naturally includes all manner of programming. In this sense, the New Math was ahead of its time, many programmers of the 1980s and later hearkened back to their experience with the New Math. The material also put new demands on teachers, many of whom were required to teach material they did not fully understand, parents were concerned that they did not understand what their children were learning and could not help them with their studies. In an effort to learn the material, many parents attended their childrens classes, New Math found some later success in the form of enrichment programs for gifted students from the 1980s onward in Project MEGSSS. I dont think it is worth while teaching such material, in 1973, Morris Kline published his critical book Why Johnny Cant Add, the Failure of the New Math. It explains the desire to be relevant with mathematics representing something more modern than traditional topics, furthermore, noting the trend to abstraction in New Math, Kline says abstraction is not the first stage, but the last stage, in a mathematical development. In West Germany the changes were seen as part of a process of Bildungsreform. Again, the met with a mixed reception, but for different reasons. For example, the end-users of mathematics studies were at that time mostly in the sciences and engineering. Some compromises have since been required, given that discrete mathematics is the language of computing
8.
British Isles
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The British Isles are a group of islands off the north-western coast of continental Europe that consist of the islands of Great Britain, Ireland and over six thousand smaller isles. Situated in the North Atlantic, the islands have an area of approximately 315,159 km2. Two sovereign states are located on the islands, Ireland and the United Kingdom of Great Britain, the oldest rocks in the group are in the north west of Scotland, Ireland and North Wales and are 2,700 million years old. During the Silurian period the north-western regions collided with the south-east, the topography of the islands is modest in scale by global standards. Ben Nevis rises to an elevation of only 1,344 metres, and Lough Neagh, the climate is temperate marine, with mild winters and warm summers. The North Atlantic Drift brings significant moisture and raises temperatures 11 °C above the average for the latitude. This led to a landscape which was dominated by temperate rainforest. The region was re-inhabited after the last glacial period of Quaternary glaciation, Ireland, which became an island by 12,000 BC, was not inhabited until after 8000 BC. Great Britain became an island by 5600 BC, Hiberni, Pictish and Britons tribes, all speaking Insular Celtic, inhabited the islands at the beginning of the 1st millennium AD. Much of Brittonic-controlled Britain was conquered by the Roman Empire from AD43, the first Anglo-Saxons arrived as Roman power waned in the 5th century and eventually dominated the bulk of what is now England. Viking invasions began in the 9th century, followed by permanent settlements. Most of Ireland seceded from the United Kingdom after the Irish War of Independence, the term British Isles is controversial in Ireland, where there are objections to its usage due to the association of the word British with Ireland. The Government of Ireland does not recognise or use the term, as a result, Britain and Ireland is used as an alternative description, and Atlantic Archipelago has had limited use among a minority in academia, while British Isles is still commonly employed. Within them, they are sometimes referred to as these islands. The earliest known references to the islands as a group appeared in the writings of sea-farers from the ancient Greek colony of Massalia. The original records have been lost, however, later writings, e. g. Avienuss Ora maritima, in the 1st century BC, Diodorus Siculus has Prettanikē nēsos, the British Island, and Prettanoi, the Britons. Strabo used Βρεττανική, and Marcian of Heraclea, in his Periplus maris exteri, historians today, though not in absolute agreement, largely agree that these Greek and Latin names were probably drawn from native Celtic-language names for the archipelago. Along these lines, the inhabitants of the islands were called the Πρεττανοί, the shift from the P of Pretannia to the B of Britannia by the Romans occurred during the time of Julius Caesar
9.
Subset
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In mathematics, especially in set theory, a set A is a subset of a set B, or equivalently B is a superset of A, if A is contained inside B, that is, all elements of A are also elements of B. The relationship of one set being a subset of another is called inclusion or sometimes containment, the subset relation defines a partial order on sets. The algebra of subsets forms a Boolean algebra in which the relation is called inclusion. For any set S, the inclusion relation ⊆ is an order on the set P of all subsets of S defined by A ≤ B ⟺ A ⊆ B. We may also partially order P by reverse set inclusion by defining A ≤ B ⟺ B ⊆ A, when quantified, A ⊆ B is represented as, ∀x. So for example, for authors, it is true of every set A that A ⊂ A. Other authors prefer to use the symbols ⊂ and ⊃ to indicate proper subset and superset, respectively and this usage makes ⊆ and ⊂ analogous to the inequality symbols ≤ and <. For example, if x ≤ y then x may or may not equal y, but if x < y, then x definitely does not equal y, and is less than y. Similarly, using the convention that ⊂ is proper subset, if A ⊆ B, then A may or may not equal B, the set A = is a proper subset of B =, thus both expressions A ⊆ B and A ⊊ B are true. The set D = is a subset of E =, thus D ⊆ E is true, any set is a subset of itself, but not a proper subset. The empty set, denoted by ∅, is also a subset of any given set X and it is also always a proper subset of any set except itself. These are two examples in both the subset and the whole set are infinite, and the subset has the same cardinality as the whole. The set of numbers is a proper subset of the set of real numbers. In this example, both sets are infinite but the set has a larger cardinality than the former set. Another example in an Euler diagram, Inclusion is the partial order in the sense that every partially ordered set is isomorphic to some collection of sets ordered by inclusion. The ordinal numbers are a simple example—if each ordinal n is identified with the set of all ordinals less than or equal to n, then a ≤ b if and only if ⊆. For the power set P of a set S, the partial order is the Cartesian product of k = |S| copies of the partial order on for which 0 <1. This can be illustrated by enumerating S = and associating with each subset T ⊆ S the k-tuple from k of which the ith coordinate is 1 if and only if si is a member of T
10.
Intersection (set theory)
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In mathematics, the intersection A ∩ B of two sets A and B is the set that contains all elements of A that also belong to B, but no other elements. For explanation of the used in this article, refer to the table of mathematical symbols. The intersection of A and B is written A ∩ B, formally, A ∩ B = that is x ∈ A ∩ B if and only if x ∈ A and x ∈ B. For example, The intersection of the sets and is, the number 9 is not in the intersection of the set of prime numbers and the set of odd numbers. More generally, one can take the intersection of sets at once. The intersection of A, B, C, and D, Intersection is an associative operation, thus, A ∩ = ∩ C. Additionally, intersection is commutative, thus A ∩ B = B ∩ A, inside a universe U one may define the complement Ac of A to be the set of all elements of U not in A. We say that A intersects B if A intersects B at some element, a intersects B if their intersection is inhabited. We say that A and B are disjoint if A does not intersect B, in plain language, they have no elements in common. A and B are disjoint if their intersection is empty, denoted A ∩ B = ∅, for example, the sets and are disjoint, the set of even numbers intersects the set of multiples of 3 at 0,6,12,18 and other numbers. The most general notion is the intersection of a nonempty collection of sets. If M is a nonempty set whose elements are themselves sets, then x is an element of the intersection of M if, the notation for this last concept can vary considerably. Set theorists will sometimes write ⋂M, while others will instead write ⋂A∈M A, the latter notation can be generalized to ⋂i∈I Ai, which refers to the intersection of the collection. Here I is a nonempty set, and Ai is a set for every i in I. In the case that the index set I is the set of numbers, notation analogous to that of an infinite series may be seen. When formatting is difficult, this can also be written A1 ∩ A2 ∩ A3 ∩, even though strictly speaking, A1 ∩ (A2 ∩ (A3 ∩. Finally, let us note that whenever the symbol ∩ is placed before other symbols instead of them, it should be of a larger size. Note that in the section we excluded the case where M was the empty set
11.
Disjoint sets
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In mathematics, two sets are said to be disjoint if they have no element in common. Equivalently, disjoint sets are sets whose intersection is the empty set, for example, and are disjoint sets, while and are not. This definition of disjoint sets can be extended to any family of sets, a family of sets is pairwise disjoint or mutually disjoint if every two different sets in the family are disjoint. For example, the collection of sets is pairwise disjoint, two sets are said to be almost disjoint sets if their intersection is small in some sense. For instance, two sets whose intersection is a finite set may be said to be almost disjoint. In topology, there are notions of separated sets with more strict conditions than disjointness. For instance, two sets may be considered to be separated when they have disjoint closures or disjoint neighborhoods, similarly, in a metric space, positively separated sets are sets separated by a nonzero distance. Disjointness of two sets, or of a family of sets, may be expressed in terms of their intersections, two sets A and B are disjoint if and only if their intersection A ∩ B is the empty set. It follows from definition that every set is disjoint from the empty set. A family F of sets is pairwise disjoint if, for two sets in the family, their intersection is empty. If the family more than one set, this implies that the intersection of the whole family is also empty. However, a family of one set is pairwise disjoint, regardless of whether that set is empty. Additionally, a family of sets may have an empty intersection without being pairwise disjoint, for instance, the three sets have an empty intersection but are not pairwise disjoint. In fact, there are no two disjoint sets in this collection, also the empty family of sets is pairwise disjoint. A Helly family is a system of sets within which the only subfamilies with empty intersections are the ones that are pairwise disjoint. For instance, the intervals of the real numbers form a Helly family, if a family of closed intervals has an empty intersection and is minimal. A partition of a set X is any collection of mutually disjoint non-empty sets whose union is X, every partition can equivalently be described by an equivalence relation, a binary relation that describes whether two elements belong to the same set in the partition. A disjoint union may mean one of two things, most simply, it may mean the union of sets that are disjoint
12.
Element (mathematics)
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In mathematics, an element, or member, of a set is any one of the distinct objects that make up that set. Writing A = means that the elements of the set A are the numbers 1,2,3 and 4, sets of elements of A, for example, are subsets of A. For example, consider the set B =, the elements of B are not 1,2,3, and 4. Rather, there are three elements of B, namely the numbers 1 and 2, and the set. The elements of a set can be anything, for example, C =, is the set whose elements are the colors red, green and blue. The relation is an element of, also called set membership, is denoted by the symbol ∈, writing x ∈ A means that x is an element of A. Equivalent expressions are x is a member of A, x belongs to A, x is in A and x lies in A, another possible notation for the same relation is A ∋ x, meaning A contains x, though it is used less often. The negation of set membership is denoted by the symbol ∉, writing x ∉ A means that x is not an element of A. The symbol ϵ was first used by Giuseppe Peano 1889 in his work Arithmetices principia nova methodo exposita, here he wrote on page X, Signum ϵ significat est. Ita a ϵ b legitur a est quoddam b. which means The symbol ϵ means is, so a ϵ b is read as a is a b. The symbol itself is a stylized lowercase Greek letter epsilon, the first letter of the word ἐστί, the Unicode characters for these symbols are U+2208, U+220B and U+2209. The equivalent LaTeX commands are \in, \ni and \notin, mathematica has commands \ and \. The number of elements in a set is a property known as cardinality, informally. In the above examples the cardinality of the set A is 4, an infinite set is a set with an infinite number of elements, while a finite set is a set with a finite number of elements. The above examples are examples of finite sets, an example of an infinite set is the set of positive integers =. Using the sets defined above, namely A =, B = and C =,2 ∈ A ∈ B3,4 ∉ B is a member of B Yellow ∉ C The cardinality of D = is finite, the cardinality of P = is infinite. Halmos, Paul R. Naive Set Theory, Undergraduate Texts in Mathematics, NY, Springer-Verlag, ISBN 0-387-90092-6 - Naive means that it is not fully axiomatized, not that it is silly or easy. Jech, Thomas, Set Theory, Stanford Encyclopedia of Philosophy Suppes, Patrick, Axiomatic Set Theory, NY, Dover Publications, Inc
13.
Empty set
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In mathematics, and more specifically set theory, the empty set is the unique set having no elements, its size or cardinality is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, in other theories, many possible properties of sets are vacuously true for the empty set. Null set was once a synonym for empty set, but is now a technical term in measure theory. The empty set may also be called the void set, common notations for the empty set include, ∅, and ∅. The latter two symbols were introduced by the Bourbaki group in 1939, inspired by the letter Ø in the Norwegian, although now considered an improper use of notation, in the past,0 was occasionally used as a symbol for the empty set. The empty-set symbol ∅ is found at Unicode point U+2205, in LaTeX, it is coded as \emptyset for ∅ or \varnothing for ∅. In standard axiomatic set theory, by the principle of extensionality, hence there is but one empty set, and we speak of the empty set rather than an empty set. The mathematical symbols employed below are explained here, in this context, zero is modelled by the empty set. For any property, For every element of ∅ the property holds, There is no element of ∅ for which the property holds. Conversely, if for some property and some set V, the two statements hold, For every element of V the property holds, There is no element of V for which the property holds. By the definition of subset, the empty set is a subset of any set A. That is, every element x of ∅ belongs to A. Indeed, since there are no elements of ∅ at all, there is no element of ∅ that is not in A. Any statement that begins for every element of ∅ is not making any substantive claim and this is often paraphrased as everything is true of the elements of the empty set. When speaking of the sum of the elements of a finite set, the reason for this is that zero is the identity element for addition. Similarly, the product of the elements of the empty set should be considered to be one, a disarrangement of a set is a permutation of the set that leaves no element in the same position. The empty set is a disarrangment of itself as no element can be found that retains its original position. Since the empty set has no members, when it is considered as a subset of any ordered set, then member of that set will be an upper bound. For example, when considered as a subset of the numbers, with its usual ordering, represented by the real number line
14.
Syllogism
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A syllogism is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two or more propositions that are asserted or assumed to be true. In its earliest form, defined by Aristotle, from the combination of a statement and a specific statement. For example, knowing that all men are mortal and that Socrates is a man, Syllogistic arguments are usually represented in a three-line form, All men are mortal. In antiquity, two theories of the syllogism existed, Aristotelian syllogistic and Stoic syllogistic. Aristotle defines the syllogism as. a discourse in which certain things having been supposed, despite this very general definition, in Aristotles work Prior Analytics, he limits himself to categorical syllogisms that consist of three categorical propositions. From the Middle Ages onwards, categorical syllogism and syllogism were usually used interchangeably and this article is concerned only with this traditional use. The use of syllogisms as a tool for understanding can be dated back to the logical reasoning discussions of Aristotle, the onset of a New Logic, or logica nova, arose alongside the reappearance of Prior Analytics, the work in which Aristotle develops his theory of the syllogism. Prior Analytics, upon re-discovery, was regarded by logicians as a closed and complete body of doctrine, leaving very little for thinkers of the day to debate. Aristotles theories on the syllogism for assertoric sentences was considered especially remarkable, Aristotles Prior Analytics did not, however, incorporate such a comprehensive theory on the modal syllogism—a syllogism that has at least one modalized premise. Aristotles terminology in this aspect of his theory was deemed vague and in many cases unclear and his original assertions on this specific component of the theory were left up to a considerable amount of conversation, resulting in a wide array of solutions put forth by commentators of the day. The system for modal syllogisms laid forth by Aristotle would ultimately be deemed unfit for practical use, boethius contributed an effort to make the ancient Aristotelian logic more accessible. While his Latin translation of Prior Analytics went primarily unused before the twelfth century and his perspective on syllogisms can be found in other works as well, such as Logica Ingredientibus. With the help of Abelards distinction between de dicto modal sentences and de re modal sentences, medieval logicians began to shape a coherent concept of Aristotles modal syllogism model. For two hundred years after Buridans discussions, little was said about syllogistic logic, the Aristotelian syllogism dominated Western philosophical thought for many centuries. In the 17th century, Sir Francis Bacon rejected the idea of syllogism as being the best way to draw conclusions in nature. Instead, Bacon proposed a more inductive approach to the observation of nature, in the 19th century, modifications to syllogism were incorporated to deal with disjunctive and conditional statements. Kant famously claimed, in Logic, that logic was the one completed science, though there were alternative systems of logic such as Avicennian logic or Indian logic elsewhere, Kants opinion stood unchallenged in the West until 1879 when Frege published his Begriffsschrift. This introduced a calculus, a method of representing categorical statements by the use of quantifiers, in the last 20 years, Bolzanos work has resurfaced and become subject of both translation and contemporary study
15.
Sir William Hamilton, 9th Baronet
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Sir William Hamilton, 9th Baronet FRSE DD FSAS was a Scottish metaphysician. He is often referred to as William Stirling Hamilton of Preston, in reference to his mother and he was born in rooms at the University of Glasgow He was from an academic family, his younger brother being Robert Hamilton, the economist. William Hamilton and a brother, Thomas Hamilton, were brought up by their mother. He obtained a first class in lit ens humanioribus and took his B. A. in 1811. He had been intended for the profession, but soon after leaving Oxford he gave up this idea. His life continued to be that of a student, and the years that followed were filled by researches of all kinds, while at the same time he was gradually forming his philosophic system. Two visits to Germany in 1817 and 1820 led to Williams taking up the study of German and later on that of contemporary German philosophy, which was almost entirely neglected in British universities. Soon afterwards he was appointed professor of history, and as such delivered several courses of lectures on the history of modern Europe. The salary was £100 a year, derived from a beer tax. No pupils were compelled to attend, the class dwindled, in January 1827 his mother, to whom he had been devoted, died. In March 1828 he married his cousin, Janet Marshall, around this time he moved to live in a recently built townhouse at 11 Manor Place, in Edinburghs west end. Much about the time he began the preparation of an annotated edition of Thomas Reids works. Before, however, this design had been carried out, he was struck with paralysis of the right side, the edition of Reid appeared in 1846, but with only seven of the intended dissertations, one unfinished. At his death he had not completed the work, notes on the subjects to be discussed were found among his manuscripts. But the elaboration of the scheme in its details and applications continued during the few years to occupy much of his leisure. Out of this arose a controversy with Augustus de Morgan. The essay did not appear, but the results of the labour gone through are contained in the appendices to his Lectures on Logic, Hamilton also prepared extensive materials for a publication which he designed on the personal history, influence and opinions of Martin Luther. Here he advanced so far as to have planned and partly carried out the arrangement of the work, but it did not go further, and still remains in manuscript
16.
Leonhard Paul Euler
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He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy, Euler was one of the most eminent mathematicians of the 18th century, and is held to be one of the greatest in history. He is also considered to be the most prolific mathematician of all time. His collected works fill 60 to 80 quarto volumes, more than anybody in the field and he spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia. A statement attributed to Pierre-Simon Laplace expresses Eulers influence on mathematics, Read Euler, read Euler, Leonhard Euler was born on 15 April 1707, in Basel, Switzerland to Paul III Euler, a pastor of the Reformed Church, and Marguerite née Brucker, a pastors daughter. He had two sisters, Anna Maria and Maria Magdalena, and a younger brother Johann Heinrich. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, Paul Euler was a friend of the Bernoulli family, Johann Bernoulli was then regarded as Europes foremost mathematician, and would eventually be the most important influence on young Leonhard. Eulers formal education started in Basel, where he was sent to live with his maternal grandmother. In 1720, aged thirteen, he enrolled at the University of Basel, during that time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupils incredible talent for mathematics. In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono, at that time, he was unsuccessfully attempting to obtain a position at the University of Basel. In 1727, he first entered the Paris Academy Prize Problem competition, Pierre Bouguer, who became known as the father of naval architecture, won and Euler took second place. Euler later won this annual prize twelve times, around this time Johann Bernoullis two sons, Daniel and Nicolaus, were working at the Imperial Russian Academy of Sciences in Saint Petersburg. In November 1726 Euler eagerly accepted the offer, but delayed making the trip to Saint Petersburg while he applied for a physics professorship at the University of Basel. Euler arrived in Saint Petersburg on 17 May 1727 and he was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he worked in close collaboration. Euler mastered Russian and settled life in Saint Petersburg. He also took on a job as a medic in the Russian Navy. The Academy at Saint Petersburg, established by Peter the Great, was intended to improve education in Russia, as a result, it was made especially attractive to foreign scholars like Euler
17.
Christian Weise
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He produced a large number of dramatic works, noted for their social criticism and idiomatic style. In the 1670s he started a fashion for German political novels and he has also been credited with the invention of the mathematical Euler diagram, though this is uncertain. Christian Weise was born in Zittau, the son of Elias Weise, Weise studied theology at the University of Leipzig, gaining a Magisters degree in 1663. His studies expanded into rhetoric, politics, history and poetry, however, in 1668 he secured a post at the court in Halle, as the secretary of Simon Philipp von Leiningen-Westerburg, the minister of Augustus, Duke of Saxe-Weissenfels. In 1670 Weise became Hofmeister for Gustav Adolf von der Schulenburg, however, later that same year he moved again, to teach at a school in Weißenfels, the Gymnasium Illustre Augusteum. During the 1670s Weise produced a number of political and satirical novels. These created a fashion for the German political novel which lasted into the next decade, as Weise had done earlier, Beer worked at the court of Duke Augustus, he followed the court from Halle to Weißenfels when it moved in 1680. In 1678, however, Weise left Weißenfels to become rector of the Gymnasium in Zittau, during his time at the Gymnasium he wrote up to 60 dramatic works, which were staged by his pupils. He intended the moral and political lessons contained in the works to be of educational benefit. As well as continuing the German school drama tradition, he continued the tradition of Protestant biblical drama, with such as Jephtha. His dramas often contained interspersed music, there are complete scores for the two biblical works, by Moritz Edelmann, who also provided music for the comedy Der baürischer Machiavellus. Weise also wrote tragedies, including one of his best known works and he satirized the social and political ills of the time, criticizing the higher levels of society from the viewpoint of those lower down. Unusually for a writer of the Baroque era, he employed an exceptionally sober and his comedies, written in the dialects of Upper Lusatia and North Bohemia, gave a sympathetic portrayal of common people. By the end of the century he introduced German as the language of instruction. In 1708 he gave up his position as Rektor due to failing eyesight, the modern successor to his school is called the Christian-Weise-Gymnasium. During his time as librarian at Zittaus Ratsbibliothek, he made additions to the librarys collections. These are held by Zittaus modern library, which was named the Christian-Weise-Bibliothek in 1954, in the 1950s a revival of his tragedy Masaniello was also staged in Zittau. In the field of logic, Weise has been credited with the first use of circles in diagrams showing logical relationships between mathematical sets, the technique, now known as the Euler diagram, was the precursor of the Venn diagram
18.
John Venn
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John Venn, FRS, FSA, was an English logician and philosopher noted for introducing the Venn diagram, used in the fields of set theory, probability, logic, statistics, and computer science. John Venn was born on 4 August 1834 in Kingston upon Hull, Yorkshire to Martha Sykes and Rev. Henry Venn and his mother died when he was three years old. Venn was descended from a line of church evangelicals, including his grandfather John Venn. Venn was brought up in a very strict atmosphere at home. His father Henry had played a significant part in the Evangelical movement and he was also the secretary of the ‘Society for Missions to Africa and his grandfather was pastor to William Wilberforce of the abolitionist movement, in Clapham. He began his education in London joining Sir Roger Cholmeleys School, now known as Highgate School and he moved on to Islington proprietary school and in October 1853 he went to Gonville and Caius College, Cambridge. In 1857, he obtained his degree in mathematics and became a fellow, in 1903 he was elected President of the College, a post he held until his death. He would follow his vocation and become an Anglican priest, ordained in 1859, serving first at the church in Cheshunt, Hertfordshire. In 1862, he returned to Cambridge as a lecturer in science, studying and teaching logic and probability theory. These duties led to his developing the diagram which would bear his name. I began at once somewhat more work on the subjects. I now first hit upon the device of representing propositions by inclusive and exclusive circles. In 1868, he married Susanna Carnegie Edmonstone with whom he had one son, in 1883, he resigned from the clergy, having concluded that Anglicanism was incompatible with his philosophical beliefs. In that same year, Venn was elected a Fellow of the Royal Society and was awarded a Sc. D. by Cambridge. Venn is commemorated at the University of Hull by the Venn Building, built in 1928 A stained glass window in the hall of Gonville and Caius College, Cambridge. Venn then further developed George Booles theories in the 1881 work Symbolic Logic, Venn compiled Alumni Cantabrigienses, a biographical register of former members of the University of Cambridge. This work is still being updated online, see #External links and his other works include, A Cambridge Alumni Database The Venn archives clarify the confusing timeline of the various Venns. Obituary of John Venn Portrait of Venn by Charles Brock, and a link to a site about Venn Another view of the Venn stained glass window John Venn at Find a Grave
19.
Algorithm
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In mathematics and computer science, an algorithm is a self-contained sequence of actions to be performed. Algorithms can perform calculation, data processing and automated reasoning tasks, an algorithm is an effective method that can be expressed within a finite amount of space and time and in a well-defined formal language for calculating a function. The transition from one state to the next is not necessarily deterministic, some algorithms, known as randomized algorithms, giving a formal definition of algorithms, corresponding to the intuitive notion, remains a challenging problem. In English, it was first used in about 1230 and then by Chaucer in 1391, English adopted the French term, but it wasnt until the late 19th century that algorithm took on the meaning that it has in modern English. Another early use of the word is from 1240, in a manual titled Carmen de Algorismo composed by Alexandre de Villedieu and it begins thus, Haec algorismus ars praesens dicitur, in qua / Talibus Indorum fruimur bis quinque figuris. Which translates as, Algorism is the art by which at present we use those Indian figures, the poem is a few hundred lines long and summarizes the art of calculating with the new style of Indian dice, or Talibus Indorum, or Hindu numerals. An informal definition could be a set of rules that precisely defines a sequence of operations, which would include all computer programs, including programs that do not perform numeric calculations. Generally, a program is only an algorithm if it stops eventually, but humans can do something equally useful, in the case of certain enumerably infinite sets, They can give explicit instructions for determining the nth member of the set, for arbitrary finite n. An enumerably infinite set is one whose elements can be put into one-to-one correspondence with the integers, the concept of algorithm is also used to define the notion of decidability. That notion is central for explaining how formal systems come into being starting from a set of axioms. In logic, the time that an algorithm requires to complete cannot be measured, from such uncertainties, that characterize ongoing work, stems the unavailability of a definition of algorithm that suits both concrete and abstract usage of the term. Algorithms are essential to the way computers process data, thus, an algorithm can be considered to be any sequence of operations that can be simulated by a Turing-complete system. Although this may seem extreme, the arguments, in its favor are hard to refute. Gurevich. Turings informal argument in favor of his thesis justifies a stronger thesis, according to Savage, an algorithm is a computational process defined by a Turing machine. Typically, when an algorithm is associated with processing information, data can be read from a source, written to an output device. Stored data are regarded as part of the state of the entity performing the algorithm. In practice, the state is stored in one or more data structures, for some such computational process, the algorithm must be rigorously defined, specified in the way it applies in all possible circumstances that could arise. That is, any conditional steps must be dealt with, case-by-case
20.
Venn diagrams
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A Venn diagram is a diagram that shows all possible logical relations between a finite collection of different sets. These diagrams depict elements as points in the plane, and sets as regions inside closed curves, a Venn diagram consists of multiple overlapping closed curves, usually circles, each representing a set. In Venn diagrams the curves are overlapped in every possible way and they are thus a special case of Euler diagrams, which do not necessarily show all relations. Venn diagrams were conceived around 1880 by John Venn and they are used to teach elementary set theory, as well as illustrate simple set relationships in probability, logic, statistics, linguistics and computer science. A Venn diagram in which in addition the area of each shape is proportional to the number of elements it contains is called an area-proportional or scaled Venn diagram and this example involves two sets, A and B, represented here as coloured circles. The orange circle, set A, represents all living creatures that are two-legged, the blue circle, set B, represents the living creatures that can fly. Each separate type of creature can be imagined as a point somewhere in the diagram, living creatures that both can fly and have two legs—for example, parrots—are then in both sets, so they correspond to points in the region where the blue and orange circles overlap. That region contains all such and only living creatures. Humans and penguins are bipedal, and so are then in the circle, but since they cannot fly they appear in the left part of the orange circle. Mosquitoes have six legs, and fly, so the point for mosquitoes is in the part of the circle that does not overlap with the orange one. Creatures that are not two-legged and cannot fly would all be represented by points outside both circles, the combined region of sets A and B is called the union of A and B, denoted by A ∪ B. The union in this case contains all living creatures that are either two-legged or that can fly, the region in both A and B, where the two sets overlap, is called the intersection of A and B, denoted by A ∩ B. For example, the intersection of the two sets is not empty, because there are points that represent creatures that are in both the orange and blue circles. They are rightly associated with Venn, however, because he comprehensively surveyed and formalized their usage, Venn himself did not use the term Venn diagram and referred to his invention as Eulerian Circles. Of these schemes one only, viz. that commonly called Eulerian circles, has met with any general acceptance, the first to use the term Venn diagram was Clarence Irving Lewis in 1918, in his book A Survey of Symbolic Logic. Venn diagrams are similar to Euler diagrams, which were invented by Leonhard Euler in the 18th century. Baron has noted that Leibniz in the 17th century produced similar diagrams before Euler and she also observes even earlier Euler-like diagrams by Ramon Lull in the 13th Century. In the 20th century, Venn diagrams were further developed, D. W. Henderson showed in 1963 that the existence of an n-Venn diagram with n-fold rotational symmetry implied that n was a prime number
21.
Louis Couturat
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Louis Couturat was a French logician, mathematician, philosopher, and linguist. Born in Ris-Orangis, Essonne, France, he was educated in philosophy and he held professorships, first at the University of Toulouse, then subsequently at the Collège de France. Like Russell and Whitehead, Couturat saw symbolic logic as a tool to both mathematics and the philosophy thereof. In this, he was opposed by Henri Poincaré, who took exception to Couturats efforts to interest the French in symbolic logic. With the benefit of hindsight, we can see that Couturat was in agreement with the logicism of Russell and Whitehead. His first major publication was Couturat, in 1901, he published La Logique de Leibniz, a detailed study of Leibniz the logician, based on his examination of the huge Leibniz Nachlass in Hannover. Even though Leibniz had died in 1716, his Nachlass was cataloged only in 1895, only then was it possible to determine the extent of Leibnizs unpublished work on logic. In 1903, Couturat published much of that work in large volume, his Opuscules et Fragments Inedits de Leibniz. Couturat was thus the first to appreciate that Leibniz was the greatest logician during the more than 2000 years that separate Aristotle from George Boole, a significant part of the 20th century Leibniz revival is grounded in Couturats editorial and exegetical efforts. This work on Leibniz attracted Russell, also the author of a 1900 book on Leibniz, in 1905, Couturat published a work on logic and the foundations of mathematics which was originally conceived as a translation of Russells Principles of Mathematics. In the same year, he published LAlgèbre de la logique, an introduction to the algebraic logic of George Boole, C. S. Peirce. In 1907, Couturat helped found the artificial language Ido, an offshoot of Esperanto, by pushing Ido, Couturat walked in Leibnizs footsteps, Leibniz called for the creation a universal symbolic and conceptual language he named the characteristica universalis. Couturat, a confirmed pacifist, was killed when his car was hit by a car carrying orders for the mobilization of the French Army and he appears as a character in Joseph Skibells 2010 novel, A Curable Romantic. Leibniz Ernst Schröder Ido Boolean algebra Logicism Primary literature,1896 De Platonicis mythis Thesim Facultati Litterarum Parisiensi proponebat Ludovicus Couturat, donald Rutherfords English translation in progress. Opuscules et Fragments Inédits de Leibniz, les Principes des Mathématiques, avec un appendice sur la philosophie des mathématiques de Kant. Étude sur la dérivation dans la langue internationales, the Search for Mathematical Roots 1870-1940. Bibliography contains 27 items by Couturat, oConnor, John J. Robertson, Edmund F. Louis Couturat, MacTutor History of Mathematics archive, University of St Andrews. Louis Couturat at the Mathematics Genealogy Project fr. wikisource Auteur Couturat Works by Louis Couturat at Project Gutenberg Works by or about Louis Couturat at Internet Archive
22.
Truth table
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In particular, truth tables can be used to show whether a propositional expression is true for all legitimate input values, that is, logically valid. A truth table has one column for each variable. Each row of the table contains one possible configuration of the input variables. See the examples below for further clarification, the Com row indicates whether an operator, op, is commutative - P op Q = Q op P. The L id row shows the operators left identities if it has any - values I such that I op Q = Q, the R id row shows the operators right identities if it has any - values I such that P op I = P. The four combinations of values for p, q, are read by row from the table above. The output function for p, q combination, can be read, by row. Key, The following table is oriented by column, rather than by row, There are four columns rather than four rows, to display the four combinations of p, q, as input. P, T T F Fq, T F T F There are 16 rows in this key, the output row for ↚ is thus 2, F F T F and the 16-row key is Logical operators can also be visualized using Venn diagrams. Logical conjunction is an operation on two values, typically the values of two propositions, that produces a value of true if both of its operands are true. The truth table for p AND q is as follows, In ordinary language terms, for all other assignments of logical values to p and to q the conjunction p ∧ q is false. The truth table for p OR q is as follows, Stated in English, if p, then p ∨ q is p, otherwise p ∨ q is q. The truth table associated with the conditional if p then q. Logical equality is an operation on two values, typically the values of two propositions, that produces a value of true if both operands are false or both operands are true. The truth table for p XNOR q is as follows, So p EQ q is true if p and q have the truth value. Exclusive disjunction is an operation on two values, typically the values of two propositions, that produces a value of true if one but not both of its operands is true. The truth table for p XOR q is as follows, For two propositions, XOR can also be written as ∨. The logical NAND is an operation on two values, typically the values of two propositions, that produces a value of false if both of its operands are true
23.
Emil Post
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Emil Leon Post was a Polish-born American mathematician and logician. He is best known for his work in the field eventually became known as computability theory. Post was born in Augustów, Suwałki Governorate, Russian Empire into a Polish-Jewish family that immigrated to New York City in May 1904 and his parents were Arnold and Pearl Post. Post had been interested in astronomy, but at the age of twelve lost his arm in a car accident. This loss was a significant obstacle to being a professional astronomer and he decided to pursue mathematics, rather than astronomy. Post attended the Townsend Harris High School and continued on to graduate from City College of New York in 1917 with a B. S. in Mathematics. After completing his Ph. D. in mathematics at Columbia University, supervised by Cassius Jackson Keyser, Post then became a high school mathematics teacher in New York City. Post married Gertrude Singer in 1929, with whom he had a daughter, Post spent at most three hours a day on research on the advice of his doctor in order to avoid manic attacks, which he had been experiencing since his year at Princeton. In 1936, he was appointed to the department at the City College of New York. He died in 1954 of an attack following electroshock treatment for depression. Post also devised truth tables independently of Wittgenstein and C. S. Peirce, jean Van Heijenoorts well-known source book on mathematical logic reprinted Posts classic article setting out these results. While at Princeton, Post came very close to discovering the incompleteness of Principia Mathematica, Post initially failed to publish his ideas as he believed he needed a complete analysis for them to be accepted. In 1936, Post developed, independently of Alan Turing, a model of computation that was essentially equivalent to the Turing machine model. Intending this as the first of a series of models of equivalent power but increasing complexity, Post devised a method of auxiliary symbols by which he could canonically represent any Post-generative language, and indeed any computable function or set at all. The unsolvability of his Post correspondence problem turned out to be exactly what was needed to obtain unsolvability results in the theory of formal languages and this question, which became known as Posts problem, stimulated much research. It was solved in the affirmative in the 1950s by the introduction of the priority method in recursion theory. Post made a fundamental and still-influential contribution to the theory of polyadic, or n-ary and his major theorem showed that a polyadic group is the iterated multiplication of elements of a normal subgroup of a group, such that the quotient group is cyclic of order n −1. He also demonstrated that a group operation on a set can be expressed in terms of a group operation on the same set
24.
Claude Shannon
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Claude Elwood Shannon was an American mathematician, electrical engineer, and cryptographer known as the father of information theory. Shannon is noted for having founded information theory with a paper, A Mathematical Theory of Communication. Shannon contributed to the field of cryptanalysis for national defense during World War II, including his work on codebreaking. Shannon was born in Petoskey, Michigan and grew up in Gaylord and his father, Claude, Sr. a descendant of early settlers of New Jersey, was a self-made businessman, and for a while, a Judge of Probate. Shannons mother, Mabel Wolf Shannon, was a language teacher, most of the first 16 years of Shannons life were spent in Gaylord, where he attended public school, graduating from Gaylord High School in 1932. Shannon showed an inclination towards mechanical and electrical things and his best subjects were science and mathematics. At home he constructed such devices as models of planes, a model boat. While growing up, he worked under Andrew Coltrey as a messenger for the Western Union company. His childhood hero was Thomas Edison, who he learned was a distant cousin. Both were descendants of John Ogden, a leader and an ancestor of many distinguished people. Shannon was apolitical and an atheist, in 1932, Shannon entered the University of Michigan, where he was introduced to the work of George Boole. He graduated in 1936 with two degrees, one in electrical engineering and the other in mathematics. In 1936, Shannon began his studies in electrical engineering at MIT, where he worked on Vannevar Bushs differential analyzer. While studying the complicated ad hoc circuits of this analyzer, Shannon designed switching circuits based on Booles concepts, in 1937, he wrote his masters degree thesis, A Symbolic Analysis of Relay and Switching Circuits, A paper from this thesis was published in 1938. In this work, Shannon proved that his switching circuits could be used to simplify the arrangement of the electromechanical relays that were used then in telephone call routing switches, next, he expanded this concept, proving that these circuits could solve all problems that Boolean algebra could solve. In the last chapter, he presents diagrams of several circuits, using this property of electrical switches to implement logic is the fundamental concept that underlies all electronic digital computers. Shannons work became the foundation of digital design, as it became widely known in the electrical engineering community during. The theoretical rigor of Shannons work superseded the ad hoc methods that had prevailed previously, howard Gardner called Shannons thesis possibly the most important, and also the most noted, masters thesis of the century
25.
George Stibitz
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George Robert Stibitz is internationally recognized as one of the fathers of the modern first digital computer. He was a Bell Labs researcher known for his work in the 1930s and 1940s on the realization of Boolean logic digital circuits using electromechanical relays as the switching element, Stibitz was born in York, Pennsylvania. He received his bachelors degree from Denison University in Granville, Ohio, his masters degree from Union College in 1927, in November 1937, George Stibitz, then working at Bell Labs, completed a relay-based calculator he later dubbed the Model K, which calculated using binary addition. Stibitz Computer and Communications Pioneer Awards are granted, Bell Labs subsequently authorized a full research program in late 1938 with Stibitz at the helm. Their Complex Number Computer, completed in November 1939, was able to do calculations on complex numbers and it was the first computing machine ever used remotely. After the United States entered World War II in December,1941, the Labs most famous invention was the M-9 Gun Director, an ingenious analog device that directed anti-aircraft fire with uncanny accuracy. Stibitz moved to the National Defense Research Committee, a body for the government. For the next years, with his guidance, the Labs developed relay computers of ever-increasing sophistication. The first of them was used to test the M-9 Gun Director, later models had more sophisticated capabilities. They had specialized names, but later on, Bell Labs renamed them Model II, Model III, etc. all used telephone relays for logic, and paper tape for sequencing and control. Stibitz noted that the fell into two broad categories, analog and pulse. In a memo written after the meeting, he suggested that the term digital be used in place of pulse, the word digit at the time had two common meanings, the ten fingers of ones hands, and the numbers 0 through 9. The adjective digital was also in use, although it was not as common, for example, among physicians, a digital examination referred to the use of a doctors finger to palpate part of the body. Stibitzs memorandum was the first known use of the digital to refer to calculating machinery. Harry H. Goode Memorial Award in 1965 IEEE Emanuel R and he became a member of the faculty at Dartmouth College in 1964 to build bridges between the fields of computing and medicine, and retired from research in 1983. In his later years, Stibitz turned to non-verbal uses of the computer, specifically, he used a Commodore-Amiga to create computer art. The quotes are obligatory, for the result of my efforts is not to create important art but to show that activity is fun. The Mathematics and Computer Science department at Denison University has enlarged and displayed some of his artwork, the Origins of Digital Computers, Selected Papers, pp. 237–286
26.
Alan Turing
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Alan Mathison Turing OBE FRS was an English computer scientist, mathematician, logician, cryptanalyst and theoretical biologist. Turing is widely considered to be the father of computer science. During the Second World War, Turing worked for the Government Code and Cypher School at Bletchley Park, for a time he led Hut 8, the section responsible for German naval cryptanalysis. After the war, he worked at the National Physical Laboratory and he wrote a paper on the chemical basis of morphogenesis, and predicted oscillating chemical reactions such as the Belousov–Zhabotinsky reaction, first observed in the 1960s. Turing was prosecuted in 1952 for homosexual acts, when by the Labouchere Amendment and he accepted chemical castration treatment, with DES, as an alternative to prison. Turing died in 1954,16 days before his 42nd birthday, an inquest determined his death as suicide, but it has been noted that the known evidence is also consistent with accidental poisoning. In 2009, following an Internet campaign, British Prime Minister Gordon Brown made a public apology on behalf of the British government for the appalling way he was treated. Queen Elizabeth II granted him a pardon in 2013. The Alan Turing law is now a term for a 2017 law in the United Kingdom that retroactively pardons men cautioned or convicted under historical legislation that outlawed homosexual acts. Turings father was the son of a clergyman, the Rev. John Robert Turing, from a Scottish family of merchants that had based in the Netherlands. Turings mother, Julius wife, was Ethel Sara, daughter of Edward Waller Stoney, the Stoneys were a Protestant Anglo-Irish gentry family from both County Tipperary and County Longford, while Ethel herself had spent much of her childhood in County Clare. Julius work with the ICS brought the family to British India and he had an elder brother, John. At Hastings, Turing stayed at Baston Lodge, Upper Maze Hill, St Leonards-on-Sea, very early in life, Turing showed signs of the genius that he was later to display prominently. His parents purchased a house in Guildford in 1927, and Turing lived there during school holidays, the location is also marked with a blue plaque. Turings parents enrolled him at St Michaels, a day school at 20 Charles Road, St Leonards-on-Sea, the headmistress recognised his talent early on, as did many of his subsequent educators. From January 1922 to 1926, Turing was educated at Hazelhurst Preparatory School, in 1926, at the age of 13, he went on to Sherborne School, an independent school in the market town of Sherborne in Dorset. Turings natural inclination towards mathematics and science did not earn him respect from some of the teachers at Sherborne and his headmaster wrote to his parents, I hope he will not fall between two stools. If he is to stay at school, he must aim at becoming educated
27.
Claude E. Shannon
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Claude Elwood Shannon was an American mathematician, electrical engineer, and cryptographer known as the father of information theory. Shannon is noted for having founded information theory with a paper, A Mathematical Theory of Communication. Shannon contributed to the field of cryptanalysis for national defense during World War II, including his work on codebreaking. Shannon was born in Petoskey, Michigan and grew up in Gaylord and his father, Claude, Sr. a descendant of early settlers of New Jersey, was a self-made businessman, and for a while, a Judge of Probate. Shannons mother, Mabel Wolf Shannon, was a language teacher, most of the first 16 years of Shannons life were spent in Gaylord, where he attended public school, graduating from Gaylord High School in 1932. Shannon showed an inclination towards mechanical and electrical things and his best subjects were science and mathematics. At home he constructed such devices as models of planes, a model boat. While growing up, he worked under Andrew Coltrey as a messenger for the Western Union company. His childhood hero was Thomas Edison, who he learned was a distant cousin. Both were descendants of John Ogden, a leader and an ancestor of many distinguished people. Shannon was apolitical and an atheist, in 1932, Shannon entered the University of Michigan, where he was introduced to the work of George Boole. He graduated in 1936 with two degrees, one in electrical engineering and the other in mathematics. In 1936, Shannon began his studies in electrical engineering at MIT, where he worked on Vannevar Bushs differential analyzer. While studying the complicated ad hoc circuits of this analyzer, Shannon designed switching circuits based on Booles concepts, in 1937, he wrote his masters degree thesis, A Symbolic Analysis of Relay and Switching Circuits, A paper from this thesis was published in 1938. In this work, Shannon proved that his switching circuits could be used to simplify the arrangement of the electromechanical relays that were used then in telephone call routing switches, next, he expanded this concept, proving that these circuits could solve all problems that Boolean algebra could solve. In the last chapter, he presents diagrams of several circuits, using this property of electrical switches to implement logic is the fundamental concept that underlies all electronic digital computers. Shannons work became the foundation of digital design, as it became widely known in the electrical engineering community during. The theoretical rigor of Shannons work superseded the ad hoc methods that had prevailed previously, howard Gardner called Shannons thesis possibly the most important, and also the most noted, masters thesis of the century
28.
M.I.T.
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The Massachusetts Institute of Technology is a private research university in Cambridge, Massachusetts, often cited as one of the worlds most prestigious universities. Researchers worked on computers, radar, and inertial guidance during World War II, post-war defense research contributed to the rapid expansion of the faculty and campus under James Killian. The current 168-acre campus opened in 1916 and extends over 1 mile along the bank of the Charles River basin. The Institute is traditionally known for its research and education in the sciences and engineering, and more recently in biology, economics, linguistics. Air Force and 6 Fields Medalists have been affiliated with MIT, the school has a strong entrepreneurial culture, and the aggregated revenues of companies founded by MIT alumni would rank as the eleventh-largest economy in the world. In 1859, a proposal was submitted to the Massachusetts General Court to use newly filled lands in Back Bay, Boston for a Conservatory of Art and Science, but the proposal failed. A charter for the incorporation of the Massachusetts Institute of Technology, Rogers, a professor from the University of Virginia, wanted to establish an institution to address rapid scientific and technological advances. The Rogers Plan reflected the German research university model, emphasizing an independent faculty engaged in research, as well as instruction oriented around seminars, two days after the charter was issued, the first battle of the Civil War broke out. After a long delay through the war years, MITs first classes were held in the Mercantile Building in Boston in 1865, in 1863 under the same act, the Commonwealth of Massachusetts founded the Massachusetts Agricultural College, which developed as the University of Massachusetts Amherst. In 1866, the proceeds from sales went toward new buildings in the Back Bay. MIT was informally called Boston Tech, the institute adopted the European polytechnic university model and emphasized laboratory instruction from an early date. Despite chronic financial problems, the institute saw growth in the last two decades of the 19th century under President Francis Amasa Walker. Programs in electrical, chemical, marine, and sanitary engineering were introduced, new buildings were built, the curriculum drifted to a vocational emphasis, with less focus on theoretical science. The fledgling school still suffered from chronic financial shortages which diverted the attention of the MIT leadership, during these Boston Tech years, MIT faculty and alumni rebuffed Harvard University president Charles W. Eliots repeated attempts to merge MIT with Harvard Colleges Lawrence Scientific School. There would be at least six attempts to absorb MIT into Harvard, in its cramped Back Bay location, MIT could not afford to expand its overcrowded facilities, driving a desperate search for a new campus and funding. Eventually the MIT Corporation approved an agreement to merge with Harvard, over the vehement objections of MIT faculty, students. However, a 1917 decision by the Massachusetts Supreme Judicial Court effectively put an end to the merger scheme, the neoclassical New Technology campus was designed by William W. Bosworth and had been funded largely by anonymous donations from a mysterious Mr. Smith, starting in 1912. In January 1920, the donor was revealed to be the industrialist George Eastman of Rochester, New York, who had invented methods of production and processing
29.
Karnaugh map
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The Karnaugh map, also known as the K-map, is a method to simplify boolean algebra expressions. Maurice Karnaugh introduced it in 1953 as a refinement of Edward Veitchs 1952 Veitch diagram, the Karnaugh map reduces the need for extensive calculations by taking advantage of humans pattern-recognition capability. It also permits the identification and elimination of potential race conditions. Optimal groups of 1s or 0s are identified, which represent the terms of a form of the logic in the original truth table. These terms can be used to write a minimal boolean expression representing the required logic, Karnaugh maps are used to simplify real-world logic requirements so that they can be implemented using a minimum number of physical logic gates. A sum-of-products expression can always be implemented using AND gates feeding into an OR gate, Karnaugh maps can also be used to simplify logic expressions in software design. Boolean conditions, as used for example in conditional statements, can get very complicated, once minimised, canonical sum-of-products and product-of-sums expressions can be implemented directly using AND and OR logic operators. Karnaugh maps are used to facilitate the simplification of Boolean algebra functions, for example, consider the Boolean function described by the following truth table. Following are two different notations describing the function in unsimplified Boolean algebra, using the Boolean variables A, B, C, D. F = ∑ m i, i ∈ where m i are the minterms to map, F = ∏ M i, i ∈ where M i are the maxterms to map. In the example above, the four input variables can be combined in 16 different ways, so the table has 16 rows. The Karnaugh map is arranged in a 4 ×4 grid. The row and column indices are ordered in Gray code rather than binary numerical order, Gray code ensures that only one variable changes between each pair of adjacent cells. Each cell of the completed Karnaugh map contains a binary digit representing the output for that combination of inputs. After the Karnaugh map has been constructed, it is used to one of the simplest possible forms — a canonical form — for the information in the truth table. Adjacent 1s in the Karnaugh map represent opportunities to simplify the expression, the minterms for the final expression are found by encircling groups of 1s in the map. Minterm groups must be rectangular and must have an area that is a power of two, minterm rectangles should be as large as possible without containing any 0s. Groups may overlap in order to each one larger
30.
Hypercube
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In geometry, a hypercube is an n-dimensional analogue of a square and a cube. A unit hypercubes longest diagonal in n-dimensions is equal to n, an n-dimensional hypercube is also called an n-cube or an n-dimensional cube. The term measure polytope is also used, notably in the work of H. S. M. Coxeter, the hypercube is the special case of a hyperrectangle. A unit hypercube is a hypercube whose side has one unit. Often, the hypercube whose corners are the 2n points in Rn with coordinates equal to 0 or 1 is called the unit hypercube, a hypercube can be defined by increasing the numbers of dimensions of a shape,0 – A point is a hypercube of dimension zero. 1 – If one moves this point one unit length, it will sweep out a line segment,2 – If one moves this line segment its length in a perpendicular direction from itself, it sweeps out a 2-dimensional square. 3 – If one moves the square one unit length in the perpendicular to the plane it lies on. 4 – If one moves the cube one unit length into the fourth dimension and this can be generalized to any number of dimensions. The 1-skeleton of a hypercube is a hypercube graph, a unit hypercube of n dimensions is the convex hull of the points given by all sign permutations of the Cartesian coordinates. It has a length of 1 and an n-dimensional volume of 1. An n-dimensional hypercube is also regarded as the convex hull of all sign permutations of the coordinates. This form is chosen due to ease of writing out the coordinates. Its edge length is 2, and its volume is 2n. Every n-cube of n >0 is composed of elements, or n-cubes of a dimension, on the -dimensional surface on the parent hypercube. A side is any element of -dimension of the parent hypercube, a hypercube of dimension n has 2n sides. The number of vertices of a hypercube is 2 n, the number of m-dimensional hypercubes on the boundary of an n-cube is E m, n =2 n − m, where = n. m. and n. denotes the factorial of n. For example, the boundary of a 4-cube contains 8 cubes,24 squares,32 lines and 16 vertices and this identity can be proved by combinatorial arguments, each of the 2 n vertices defines a vertex in a m-dimensional boundary. There are ways of choosing which lines that defines the subspace that the boundary is in, but, each side is counted 2 m times since it has that many vertices, we need to divide with this number
31.
Boolean algebra (logic)
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In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0 respectively. It is thus a formalism for describing logical relations in the way that ordinary algebra describes numeric relations. Boolean algebra was introduced by George Boole in his first book The Mathematical Analysis of Logic, according to Huntington, the term Boolean algebra was first suggested by Sheffer in 1913. Boolean algebra has been fundamental in the development of digital electronics and it is also used in set theory and statistics. Booles algebra predated the modern developments in algebra and mathematical logic. In an abstract setting, Boolean algebra was perfected in the late 19th century by Jevons, Schröder, Huntington, in fact, M. H. Stone proved in 1936 that every Boolean algebra is isomorphic to a field of sets. Shannon already had at his disposal the abstract mathematical apparatus, thus he cast his switching algebra as the two-element Boolean algebra, in circuit engineering settings today, there is little need to consider other Boolean algebras, thus switching algebra and Boolean algebra are often used interchangeably. Efficient implementation of Boolean functions is a problem in the design of combinational logic circuits. Logic sentences that can be expressed in classical propositional calculus have an equivalent expression in Boolean algebra, thus, Boolean logic is sometimes used to denote propositional calculus performed in this way. Boolean algebra is not sufficient to capture logic formulas using quantifiers, the closely related model of computation known as a Boolean circuit relates time complexity to circuit complexity. Whereas in elementary algebra expressions denote mainly numbers, in Boolean algebra they denote the truth values false and these values are represented with the bits, namely 0 and 1. Addition and multiplication then play the Boolean roles of XOR and AND respectively, Boolean algebra also deals with functions which have their values in the set. A sequence of bits is a commonly used such function, another common example is the subsets of a set E, to a subset F of E is associated the indicator function that takes the value 1 on F and 0 outside F. The most general example is the elements of a Boolean algebra, as with elementary algebra, the purely equational part of the theory may be developed without considering explicit values for the variables. The basic operations of Boolean calculus are as follows, AND, denoted x∧y, satisfies x∧y =1 if x = y =1 and x∧y =0 otherwise. OR, denoted x∨y, satisfies x∨y =0 if x = y =0, NOT, denoted ¬x, satisfies ¬x =0 if x =1 and ¬x =1 if x =0. Alternatively the values of x∧y, x∨y, and ¬x can be expressed by tabulating their values with truth tables as follows, the first operation, x → y, or Cxy, is called material implication. If x is then the value of x → y is taken to be that of y
32.
Propositional calculus
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Logical connectives are found in natural languages. In English for example, some examples are and, or, not”, the following is an example of a very simple inference within the scope of propositional logic, Premise 1, If its raining then its cloudy. Both premises and the conclusion are propositions, the premises are taken for granted and then with the application of modus ponens the conclusion follows. Not only that, but they will also correspond with any other inference of this form, Propositional logic may be studied through a formal system in which formulas of a formal language may be interpreted to represent propositions. A system of rules and axioms allows certain formulas to be derived. These derived formulas are called theorems and may be interpreted to be true propositions, a constructed sequence of such formulas is known as a derivation or proof and the last formula of the sequence is the theorem. The derivation may be interpreted as proof of the represented by the theorem. When a formal system is used to represent formal logic, only statement letters are represented directly, usually in truth-functional propositional logic, formulas are interpreted as having either a truth value of true or a truth value of false. Truth-functional propositional logic and systems isomorphic to it, are considered to be zeroth-order logic, although propositional logic had been hinted by earlier philosophers, it was developed into a formal logic by Chrysippus in the 3rd century BC and expanded by his successor Stoics. The logic was focused on propositions and this advancement was different from the traditional syllogistic logic which was focused on terms. However, later in antiquity, the propositional logic developed by the Stoics was no longer understood, consequently, the system was essentially reinvented by Peter Abelard in the 12th century. Propositional logic was eventually refined using symbolic logic, the 17th/18th-century mathematician Gottfried Leibniz has been credited with being the founder of symbolic logic for his work with the calculus ratiocinator. Although his work was the first of its kind, it was unknown to the larger logical community, consequently, many of the advances achieved by Leibniz were recreated by logicians like George Boole and Augustus De Morgan completely independent of Leibniz. Just as propositional logic can be considered an advancement from the earlier syllogistic logic, one author describes predicate logic as combining the distinctive features of syllogistic logic and propositional logic. Consequently, predicate logic ushered in a new era in history, however, advances in propositional logic were still made after Frege, including Natural Deduction. Natural deduction was invented by Gerhard Gentzen and Jan Łukasiewicz, Truth-Trees were invented by Evert Willem Beth. The invention of truth-tables, however, is of controversial attribution, within works by Frege and Bertrand Russell, are ideas influential to the invention of truth tables. The actual tabular structure, itself, is credited to either Ludwig Wittgenstein or Emil Post
33.
Boolean equation
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In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0 respectively. It is thus a formalism for describing logical relations in the way that ordinary algebra describes numeric relations. Boolean algebra was introduced by George Boole in his first book The Mathematical Analysis of Logic, according to Huntington, the term Boolean algebra was first suggested by Sheffer in 1913. Boolean algebra has been fundamental in the development of digital electronics and it is also used in set theory and statistics. Booles algebra predated the modern developments in algebra and mathematical logic. In an abstract setting, Boolean algebra was perfected in the late 19th century by Jevons, Schröder, Huntington, in fact, M. H. Stone proved in 1936 that every Boolean algebra is isomorphic to a field of sets. Shannon already had at his disposal the abstract mathematical apparatus, thus he cast his switching algebra as the two-element Boolean algebra, in circuit engineering settings today, there is little need to consider other Boolean algebras, thus switching algebra and Boolean algebra are often used interchangeably. Efficient implementation of Boolean functions is a problem in the design of combinational logic circuits. Logic sentences that can be expressed in classical propositional calculus have an equivalent expression in Boolean algebra, thus, Boolean logic is sometimes used to denote propositional calculus performed in this way. Boolean algebra is not sufficient to capture logic formulas using quantifiers, the closely related model of computation known as a Boolean circuit relates time complexity to circuit complexity. Whereas in elementary algebra expressions denote mainly numbers, in Boolean algebra they denote the truth values false and these values are represented with the bits, namely 0 and 1. Addition and multiplication then play the Boolean roles of XOR and AND respectively, Boolean algebra also deals with functions which have their values in the set. A sequence of bits is a commonly used such function, another common example is the subsets of a set E, to a subset F of E is associated the indicator function that takes the value 1 on F and 0 outside F. The most general example is the elements of a Boolean algebra, as with elementary algebra, the purely equational part of the theory may be developed without considering explicit values for the variables. The basic operations of Boolean calculus are as follows, AND, denoted x∧y, satisfies x∧y =1 if x = y =1 and x∧y =0 otherwise. OR, denoted x∨y, satisfies x∨y =0 if x = y =0, NOT, denoted ¬x, satisfies ¬x =0 if x =1 and ¬x =1 if x =0. Alternatively the values of x∧y, x∨y, and ¬x can be expressed by tabulating their values with truth tables as follows, the first operation, x → y, or Cxy, is called material implication. If x is then the value of x → y is taken to be that of y
34.
Logical implication
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Logical consequence is a fundamental concept in logic, which describes the relationship between statements that holds true when one statement logically follows from one or more statements. A valid logical argument is one in which the conclusions are entailed by the premises, the philosophical analysis of logical consequence involves the questions, In what sense does a conclusion follow from its premises. And What does it mean for a conclusion to be a consequence of premises, All of philosophical logic is meant to provide accounts of the nature of logical consequence and the nature of logical truth. Logical consequence is necessary and formal, by way of examples that explain with formal proof and models of interpretation. A sentence is said to be a consequence of a set of sentences, for a given language, if and only if. The most widely prevailing view on how to best account for logical consequence is to appeal to formality and this is to say that whether statements follow from one another logically depends on the structure or logical form of the statements without regard to the contents of that form. Syntactic accounts of logical consequence rely on schemes using inference rules, for instance, we can express the logical form of a valid argument as, All A are B. All C are A. Therefore, all C are B and this argument is formally valid, because every instance of arguments constructed using this scheme are valid. This is in contrast to an argument like Fred is Mikes brothers son, if you know that Q follows logically from P no information about the possible interpretations of P or Q will affect that knowledge. Our knowledge that Q is a consequence of P cannot be influenced by empirical knowledge. Deductively valid arguments can be known to be so without recourse to experience, however, formality alone does not guarantee that logical consequence is not influenced by empirical knowledge. So the a property of logical consequence is considered to be independent of formality. The two prevailing techniques for providing accounts of logical consequence involve expressing the concept in terms of proofs, the study of the syntactic consequence is called proof theory whereas the study of semantic consequence is called model theory. A formula A is a syntactic consequence within some formal system F S of a set Γ of formulas if there is a proof in F S of A from the set Γ. Γ ⊢ F S A Syntactic consequence does not depend on any interpretation of the formal system, or, in other words, the set of the interpretations that make all members of Γ true is a subset of the set of the interpretations that make A true. Modal accounts of logical consequence are variations on the basic idea, Γ ⊢ A is true if and only if it is necessary that if all of the elements of Γ are true. Alternatively, Γ ⊢ A is true if and only if it is impossible for all of the elements of Γ to be true, such accounts are called modal because they appeal to the modal notions of logical necessity and logical possibility. Consider the modal account in terms of the argument given as an example above, the conclusion is a logical consequence of the premises because we cant imagine a possible world where all frogs are green, Kermit is a frog, and Kermit is not green
35.
Tautology (logic)
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In logic, a tautology is a formula that is true in every possible interpretation. Philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921, a formula is satisfiable if it is true under at least one interpretation, and thus a tautology is a formula whose negation is unsatisfiable. Unsatisfiable statements, both through negation and affirmation, are known formally as contradictions, a formula that is neither a tautology nor a contradiction is said to be logically contingent. Such a formula can be either true or false based on the values assigned to its propositional variables. The double turnstile notation ⊨ S is used to indicate that S is a tautology, Tautology is sometimes symbolized by Vpq, and contradiction by Opq. Tautologies are a key concept in logic, where a tautology is defined as a propositional formula that is true under any possible Boolean valuation of its propositional variables. A key property of tautologies in propositional logic is that a method exists for testing whether a given formula is always satisfied. The definition of tautology can be extended to sentences in predicate logic, in propositional logic, there is no distinction between a tautology and a logically valid formula. The set of formulas is a proper subset of the set of logically valid sentences of predicate logic. In 1800, Immanuel Kant wrote in his book Logic, The identity of concepts in analytical judgments can be explicit or non-explicit. In the former case analytic propositions are tautological, here analytic proposition refers to an analytic truth, a statement in natural language that is true solely because of the terms involved. In 1884, Gottlob Frege proposed in his Grundlagen that a truth is analytic if it can be derived using logic. But he maintained a distinction between analytic truths and tautologies, in 1921, in his Tractatus Logico-Philosophicus, Ludwig Wittgenstein proposed that statements that can be deduced by logical deduction are tautological as well as being analytic truths. Henri Poincaré had made remarks in Science and Hypothesis in 1905. It has got to be something that has some quality, which I do not know how to define. Here logical proposition refers to a proposition that is using the laws of logic. During the 1930s, the formalization of the semantics of propositional logic in terms of truth assignments was developed, the term tautology began to be applied to those propositional formulas that are true regardless of the truth or falsity of their propositional variables. Some early books on logic used the term for any proposition that is universally valid, propositional logic begins with propositional variables, atomic units that represent concrete propositions
36.
Triangle
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A triangle is a polygon with three edges and three vertices. It is one of the shapes in geometry. A triangle with vertices A, B, and C is denoted △ A B C, in Euclidean geometry any three points, when non-collinear, determine a unique triangle and a unique plane. This article is about triangles in Euclidean geometry except where otherwise noted, triangles can be classified according to the lengths of their sides, An equilateral triangle has all sides the same length. An equilateral triangle is also a polygon with all angles measuring 60°. An isosceles triangle has two sides of equal length, some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides. The latter definition would make all equilateral triangles isosceles triangles, the 45–45–90 right triangle, which appears in the tetrakis square tiling, is isosceles. A scalene triangle has all its sides of different lengths, equivalently, it has all angles of different measure. Hatch marks, also called tick marks, are used in diagrams of triangles, a side can be marked with a pattern of ticks, short line segments in the form of tally marks, two sides have equal lengths if they are both marked with the same pattern. In a triangle, the pattern is no more than 3 ticks. Similarly, patterns of 1,2, or 3 concentric arcs inside the angles are used to indicate equal angles, triangles can also be classified according to their internal angles, measured here in degrees. A right triangle has one of its interior angles measuring 90°, the side opposite to the right angle is the hypotenuse, the longest side of the triangle. The other two sides are called the legs or catheti of the triangle, special right triangles are right triangles with additional properties that make calculations involving them easier. One of the two most famous is the 3–4–5 right triangle, where 32 +42 =52, in this situation,3,4, and 5 are a Pythagorean triple. The other one is a triangle that has 2 angles that each measure 45 degrees. Triangles that do not have an angle measuring 90° are called oblique triangles, a triangle with all interior angles measuring less than 90° is an acute triangle or acute-angled triangle. If c is the length of the longest side, then a2 + b2 > c2, a triangle with one interior angle measuring more than 90° is an obtuse triangle or obtuse-angled triangle. If c is the length of the longest side, then a2 + b2 < c2, a triangle with an interior angle of 180° is degenerate
37.
Spider diagram
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In mathematics, a unitary spider diagram adds existential points to an Euler or a Venn diagram. The points indicate the existence of a described by the intersection of contours in the Euler diagram. These points may be joined together forming a shape like a spider, joined points represent an or condition, also known as a logical disjunction. A spider diagram is an expression involving unitary spider diagrams. For example, it may consist of the conjunction of two spider diagrams, the disjunction of two spider diagrams, or the negation of a spider diagram, in the image shown, the following conjunctions are apparent from the Euler diagram. The set C is only available as a subset of B, often, in complicated diagrams, singleton sets and/or conjunctions may be obscured by other set combinations. The two spiders in the example correspond to the logical expressions, Red spider, ∨ ∨ Blue spider, ∨ ∨ Howse, J. and Stapleton, G. and Taylor. Spider Diagrams London Mathematical Society Journal of Computation and Mathematics, v.8, ISSN 1461-1570 Accessed on January 8,2012 here Stapleton, G. and Howse, J. and Taylor, J. and Thompson, S. Diagrams, v.168, pgs 169-219 Accessed on January 4,2012 here Stapleton, G. and Jamnik, M. and Masthoff, on the Readability of Diagrammatic Proofs Proc. PDF Brighton and Kent University - Euler Diagrams
38.
Henry Longueville Mansel
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The Very Reverend Henry Longueville Mansel, D. D. was an English philosopher and ecclesiastic. He was born at Cosgrove, Northamptonshire and he was educated at Merchant Taylors School, London and St Johns College, Oxford. He took a double first in 1843, and became tutor of his college and he was appointed reader in moral and metaphysical philosophy at Magdalen College in 1855, and Waynflete Professor of Metaphysical Philosophy in 1859. He was an opponent of university reform and of the Hegelianism which was then beginning to take root in Oxford. In 1867 he succeeded Arthur Penrhyn Stanley as regius professor of ecclesiastical history and he died in Cosgrove on the first of July 1871. The philosophy of Mansel, like that of Sir William Hamilton, was due to Aristotle, Immanuel Kant. While denying all knowledge of the supersensuous, Mansel deviated from Kant in contending that cognition of the ego as it really is belongs among the facts of experience. Consciousness, he held — agreeing thus with the doctrine of realism which Hamilton developed from Reid — implies knowledge both of self and of the external world. The latter Mansels psychology reduces to consciousness of our organism as extended, with the former is given consciousness of free will and these lectures led Mansel to a bitter controversy with the Christian socialist theologian Frederick Maurice. A summary of Mansels philosophy is contained in his article Metaphysics in the 5th edition of the Encyclopædia Britannica. He also wrote The Philosophy of the Conditioned in reply to John Stuart Mills criticism of Hamilton, Letters, Lectures and he contributed a commentary on the first two gospels to the Speakers Commentary. 100–112, David Masson, Recent British Philosophy, pp.252 seq, kenneth D. Freeman, The Role of Reason in Religion, A Study of Henry Mansel, Leslie Stephen. Attribution This article incorporates text from a now in the public domain, Chisholm, Hugh, ed. Manael. Media related to Henry Longueville Mansel at Wikimedia Commons Works by Henry Longueville Mansel at Project Gutenberg Works by or about Henry Longueville Mansel at Internet Archive
39.
John Veitch (poet)
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John Veitch, Scottish poet, philosopher, and historian, son of a Peninsular War veteran, was born at Peebles, and educated at Edinburgh University. He was assistant lecturer successively to Sir William Hamilton and Alexander Campbell Fraser, in 1860 he was appointed to the chair of logic, metaphysics and rhetoric at St Andrews, and in 1864 to the corresponding chair at Glasgow. In philosophy an intuitionist, he dismissed the idealist arguments with some abruptness and he will be remembered chiefly for his work on Border literature and antiquities. See Memoir by his niece, Mary RL Bryce, Works written by or about John Veitch at Wikisource Works by John Veitch at Project Gutenberg Works by or about John Veitch at Internet Archive Works by John Veitch at LibriVox