Venn diagram
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, sets as regions inside closed curves. A Venn diagram consists of multiple overlapping closed curves circles, each representing a set; the points inside a curve labelled S represent elements of the set S, while points outside the boundary represent elements not in the set S. This lends to read visualizations. In Venn diagrams the curves are overlapped in every possible way, showing all possible relations between the sets, they are thus a special case of Euler diagrams, which do not show all relations. Venn diagrams were conceived around 1880 by John Venn, they are used to teach elementary set theory, as well as illustrate simple set relationships in probability, statistics and computer science. A Venn diagram in which the area of each shape is proportional to the number of elements it contains is called an area-proportional or scaled Venn diagram.
This example involves A and B, represented here as coloured circles. The orange circle, set A, represents all living creatures; the blue circle, set B, represents the living creatures. 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 in both sets, so they correspond to points in the region where the blue and orange circles overlap, it is important to note that this overlapping region would only contain those elements that are members of both set A and are members of set B Humans and penguins are bipedal, so are in the orange circle, but since they cannot fly they appear in the left part of the orange circle, where it does not overlap with the blue circle. Mosquitoes have six legs, fly, so the point for mosquitoes is in the part of the blue 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 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. Venn diagrams were introduced in 1880 by John Venn in a paper entitled On the Diagrammatic and Mechanical Representation of Propositions and Reasonings in the "Philosophical Magazine and Journal of Science", about the different ways to represent propositions by diagrams; the use of these types of diagrams in formal logic, according to Frank Ruskey and Mark Weston, is "not an easy history to trace, but it is certain that the diagrams that are popularly associated with Venn, in fact, originated much earlier. They are rightly associated with Venn, because he comprehensively surveyed and formalized their usage, was the first to generalize them".
Venn himself did not use the term "Venn diagram" and referred to his invention as "Eulerian Circles". For example, in the opening sentence of his 1880 article Venn writes, "Schemes of diagrammatic representation have been so familiarly introduced into logical treatises during the last century or so, that many readers those who have made no professional study of logic, may be supposed to be acquainted with the general nature and object of such devices. Of these schemes one only, viz. that called'Eulerian circles,' has met with any general acceptance..." Lewis Carroll includes "Venn's Method of Diagrams" as well as "Euler's Method of Diagrams" in an "Appendix, Addressed to Teachers" of his book "Symbolic Logic". The term "Venn diagram" was used by 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. M. E. Baron has noted that Leibniz in the 17th century produced similar diagrams before Euler, but much of it was unpublished.
She observes earlier Euler-like diagrams by Ramon Llull 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, he showed that such symmetric Venn diagrams exist when n is five or seven. In 2002 Peter Hamburger found symmetric Venn diagrams for n = 11 and in 2003, Griggs and Savage showed that symmetric Venn diagrams exist for all other primes, thus rotationally symmetric Venn diagrams exist. Venn diagrams and Euler diagrams were incorporated as part of instruction in set theory as part of the new math movement in the 1960s. Since they have been adopted in the curriculum of other fields such as reading. A Venn diagram is constructed with a collection of simple closed curves drawn in a plane. According to Lewis, the "principle of these diagrams is that classes be represented by regions in such relation to one another that all the possible logical relations of these classes can be indicated in the same diagram.
That is, the diagram leaves room for any possible relation
Playing card
A playing card is a piece of specially prepared heavy paper, thin cardboard, plastic-coated paper, cotton-paper blend, or thin plastic, marked with distinguishing motifs and used as one of a set for playing card games, performing magic tricks and flourishes, for cardistry, in card throwing. Playing cards are palm-sized for convenient handling, are sold together as a deck of cards or pack of cards. Playing cards were first invented in China during the Tang dynasty. Playing cards may have been invented during the Tang dynasty around the 9th century AD as a result of the usage of woodblock printing technology; the first possible reference to card games comes from a 9th-century text known as the Collection of Miscellanea at Duyang, written by Tang dynasty writer Su E. It describes Princess Tongchang, daughter of Emperor Yizong of Tang, playing the "leaf game" in 868 with members of the Wei clan, the family of the princess' husband; the first known book on the "leaf" game was called the Yezi Gexi and written by a Tang woman.
It received commentary by writers of subsequent dynasties. The Song dynasty scholar Ouyang Xiu asserts that the "leaf" game existed at least since the mid-Tang dynasty and associated its invention with the development of printed sheets as a writing medium. However, Ouyang claims that the "leaves" were pages of a book used in a board game played with dice, that the rules of the game were lost by 1067. Other games revolving around alcoholic drinking involved using playing cards of a sort from the Tang dynasty onward. However, these cards did not contain numbers. Instead, they were printed with forfeits for whomever drew them; the earliest dated instance of a game involving cards with suits and numerals occurred on 17 July 1294 when "Yan Sengzhu and Zheng Pig-Dog were caught playing cards and that wood blocks for printing them had been impounded, together with nine of the actual cards."William Henry Wilkinson suggests that the first cards may have been actual paper currency which doubled as both the tools of gaming and the stakes being played for, similar to trading card games.
Using paper money was inconvenient and risky so they were substituted by play money known as "money cards". One of the earliest games in which we know the rules is madiao, a trick-taking game, which dates to the Ming Dynasty. 15th-century scholar Lu Rong described it is as being played with 38 "money cards" divided into four suits: 9 in coins, 9 in strings of coins, 9 in myriads, 11 in tens of myriads. The two latter suits had Water Margin characters instead of pips on them with Chinese characters to mark their rank and suit; the suit of coins is in reverse order with 9 of coins being the lowest going up to 1 of coins as the high card. Despite the wide variety of patterns, the suits show a uniformity of structure; every suit contains twelve cards with the top two being the court cards of king and vizier and the bottom ten being pip cards. Half the suits use reverse ranking for their pip cards. There are many motifs for the suit pips but some include coins, clubs and swords which resemble Mamluk and Latin suits.
Michael Dummett speculated that Mamluk cards may have descended from an earlier deck which consisted of 48 cards divided into four suits each with ten pip cards and two court cards. By the 11th century, playing cards were spreading throughout the Asian continent and came into Egypt; the oldest surviving cards in the world are four fragments found in the Keir Collection and one in the Benaki Museum. They are dated to the 13th centuries. A near complete pack of Mamluk playing cards dating to the 15th century and of similar appearance to the fragments above was discovered by Leo Aryeh Mayer in the Topkapı Palace, Istanbul, in 1939, it is not a complete set and is composed of three different packs to replace missing cards. The Topkapı pack contained 52 cards comprising four suits: polo-sticks, coins and cups; each suit contained ten pip cards and three court cards, called malik, nā'ib malik, thānī nā'ib. The thānī nā ` ib is a non-existent title. In fact, the word "Kanjifah" appears in Arabic on the king of swords and is still used in parts of the Middle East to describe modern playing cards.
Influence from further east can explain why the Mamluks, most of whom were Central Asian Turkic Kipchaks, called their cups tuman which means myriad in Turkic and Jurchen languages. Wilkinson postulated that the cups may have been derived from inverting the Chinese and Jurchen ideogram for myriad; the Mamluk court cards showed abstract designs or calligraphy not depicting persons due to religious proscription in Sunni Islam, though they did bear the ranks on the cards. Nā'ib would be borrowed into French and Spanish, the latter word still in common usage. Panels on the pip cards in two suits show they had a reverse ranking, a feature found in madiao and old European card games like ombre and maw. A fragment of two uncut sheets of Moorish-styled cards of a similar but plainer style were found in Spain and dated to the early 15th century. Export of these cards, ceased after the fall of the Mamluks in the 16th century; the rules to play these games are lost but they are believed to be plain trick games without trumps.
Four-suited playing cards ar
Outcome (probability)
In probability theory, an outcome is a possible result of an experiment. Each possible outcome of a particular experiment is unique, different outcomes are mutually exclusive. All of the possible outcomes of an experiment form the elements of a sample space. For the experiment where we flip a coin twice, the four possible outcomes that make up our sample space are, where "H" represents a "heads", "T" represents a "tails". Outcomes should not be confused with events, which are sets of outcomes. For comparison, we could define an event to occur when "at least one'heads'" is flipped in the experiment - that is, when the outcome contains at least one'heads'; this event would contain all outcomes in the sample space except the element. Since individual outcomes may be of little practical interest, or because there may be prohibitively many of them, outcomes are grouped into sets of outcomes that satisfy some condition, which are called "events." The collection of all such events is a sigma-algebra.
An event containing one outcome is called an elementary event. The event that contains all possible outcomes of an experiment is its sample space. A single outcome can be a part of many different events; when the sample space is finite, any subset of the sample space is an event. However, this approach does not work well in cases where the sample space is uncountably infinite. So, when defining a probability space it is possible, necessary, to exclude certain subsets of the sample space from being events. Outcomes may occur with probabilities that are between zero and one. In a discrete probability distribution whose sample space is finite, each outcome is assigned a particular probability. In contrast, in a continuous distribution, individual outcomes all have zero probability, non-zero probabilities can only be assigned to ranges of outcomes; some "mixed" distributions contain both stretches of continuous outcomes and some discrete outcomes. Under the measure-theoretic definition of a probability space, the probability of an outcome need not be defined.
In particular, the set of events on which probability is defined may be some σ-algebra on S and not the full power set. In some sample spaces, it is reasonable to estimate or assume that all outcomes in the space are likely. For example, when tossing an ordinary coin, one assumes that the outcomes "head" and "tail" are likely to occur. An implicit assumption that all outcomes are likely underpins most randomization tools used in common games of chance. Of course, players in such games can try to cheat by subtly introducing systematic deviations from equal likelihood; some treatments of probability assume that the various outcomes of an experiment are always defined so as to be likely. However, there are experiments that are not described by a set of likely outcomes— for example, if one were to toss a thumb tack many times and observe whether it landed with its point upward or downward, there is no symmetry to suggest that the two outcomes should be likely. Event Sample space Probability distribution Probability space Realization Media related to Outcome at Wikimedia Commons
Cartesian product
In set theory, a Cartesian product is a mathematical operation that returns a set from multiple sets. That is, for sets A and B, the Cartesian product A × B is the set of all ordered pairs where a ∈ A and b ∈ B. Products can be specified using e.g.. A × B =. A table can be created by taking the Cartesian product of a set of columns. If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form. More a Cartesian product of n sets known as an n-fold Cartesian product, can be represented by an array of n dimensions, where each element is an n-tuple. An ordered pair is a couple; the Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to the concept, further generalized in terms of direct product. An illustrative example is the standard 52-card deck; the standard playing card ranks form a 13-element set. The card suits form a four-element set; the Cartesian product of these sets returns a 52-element set consisting of 52 ordered pairs, which correspond to all 52 possible playing cards.
Ranks × Suits returns a set of the form. Suits × Ranks returns a set of the form. Both sets are distinct disjoint; the main historical example is the Cartesian plane in analytic geometry. In order to represent geometrical shapes in a numerical way and extract numerical information from shapes' numerical representations, René Descartes assigned to each point in the plane a pair of real numbers, called its coordinates; such a pair's first and second components are called its x and y coordinates, respectively. The set of all such pairs is thus assigned to the set of all points in the plane. A formal definition of the Cartesian product from set-theoretical principles follows from a definition of ordered pair; the most common definition of ordered pairs, the Kuratowski definition, is =. Under this definition, is an element of P, X × Y is a subset of that set, where P represents the power set operator. Therefore, the existence of the Cartesian product of any two sets in ZFC follows from the axioms of pairing, power set, specification.
Since functions are defined as a special case of relations, relations are defined as subsets of the Cartesian product, the definition of the two-set Cartesian product is prior to most other definitions. Let A, B, C, D be sets; the Cartesian product A × B is not commutative, A × B ≠ B × A, because the ordered pairs are reversed unless at least one of the following conditions is satisfied: A is equal to B, or A or B is the empty set. For example: A =. × C ≠ A × If for example A = × A = ≠ = A ×. The Cartesian product behaves nicely with respect to intersections. × = ∩. × ≠ ∪ In fact, we have that: ∪ = ∪ ∪ [ ( B
Playing card suit
In playing cards, a suit is one of the categories into which the cards of a deck are divided. Most each card bears one of several pips showing to which suit it belongs; the rank for each card is determined by the number of pips except on face cards. Ranking indicates which cards within a suit are better, higher or more valuable than others, whereas there is no order between the suits unless defined in the rules of a specific card game. In a single deck, there is one card of any given rank in any given suit. A deck may include special cards that belong to no suit called jokers. Various languages have different terminology for suits such as signs, or seeds. Modern Western playing cards are divided into two or three general suit-systems; the older Latin suits are subdivided into the Spanish suit-systems. The younger Germanic suits are subdivided into the Swiss suit-systems; the French suits are a derivative of the German suits but are considered a separate system on its own. The card suits originated in China.
The earliest card games were trick-taking games and the invention of suits increased the level of strategy and depth in these games. A card of one suit cannot beat a card from another regardless of its rank; the concept of suits predate playing cards and can be found in Chinese dice and domino games such as Tien Gow. Chinese money-suited cards are believed to be the oldest ancestor to the Latin suit-system; the money-suit system is based on denominations of currency: Coins, Strings of Coins, Myriads of Strings, Tens of Myriads. Old Chinese coins had holes in the middle to allow them to be strung together. A string of coins could be misinterpreted as a stick to those unfamiliar with them. By the Islamic world had spread into Central Asia and had contacted China, had adopted playing cards; the Muslims renamed the suit of myriads as cups. The Chinese numeral character for Ten on the Tens of Myriads suit may have inspired the Muslim suit of swords. Another clue linking these Chinese and European cards are the ranking of certain suits.
In many early Chinese games like Madiao, the suit of coins was in reverse order so that the lower ones beat the higher ones. In the Indo-Persian game of Ganjifa, half the suits were inverted, including a suit of coins; this was true for the European games of Tarot and Ombre. The inverting of suits had no purpose in regards to gameplay but was an artifact from the earliest games; these Turko-Arabic cards, called Kanjifa, used the suits coins, clubs and swords, but the clubs represented polo sticks. The Latin suits are coins, clubs and swords, they are the earliest suit-system in Europe, were adopted from the cards imported from Mamluk Egypt and Moorish Granada in the 1370s. There are four types of Latin suits: Italian, Portuguese, an extinct archaic type; the systems can be distinguished by the pips of their long suits: clubs. Northern Italian swords are curved outward and the clubs appear to be batons, they intersect one another. Southern Italian and Spanish swords are straight, the clubs appear to be knobbly cudgels.
They do not cross each other. Portuguese pips are like the Spanish, they sometimes have dragons on the aces. This system lingers on only in the Unsun Karuta of Japan; the archaic system is like the Northern Italian one, but the swords are curved inward so they touch each other without intersecting. Minchiate used a mixed system of Portuguese swords. Despite a long history of trade with China, Japan was introduced to playing cards with the arrival of the Portuguese in the 1540s. Early locally made cards, were similar to Portuguese decks. Increasing restrictions by the Tokugawa shogunate on gambling, card playing, general foreign influence, resulted in the Hanafuda card deck that today is used most for fishing-type games; the role of rank and suit in organizing cards became switched, so the hanafuda deck has 12 suits, each representing a month of the year, each suit has 4 cards, most two normal, one Ribbon and one Special. During the 15th-century, manufacturers in German speaking lands experimented with various new suit systems to replace the Latin suits.
One early deck had the Latin ones with an extra suit of shields. The Swiss-Germans developed their own suits of shields, roses and bells around 1450. Instead of roses and shields, the Germans settled with hearts and leaves around 1460; the French derived their suits of trèfles, carreaux, cœurs, piques from the German suits around 1480. French suits correspond with German suits with the exception of the tiles with the bells but there is one early French deck that had crescents instead of tiles; the English names for the French suits of clubs and spades may have been carried over from the older Latin suits. Beginning around 1440 in northern Italy, some decks started to include of an extra suit of 21 numbered cards known as trionfi or trumps, to play tarot card games. Always included in tarot decks is one card, the Fool or Excuse, which may be part of the trump suit depending on the game or region; these cards do not have pips or f
Element (mathematics)
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. Sets can themselves be elements. For example, consider the set B =; the elements of B are not 1, 2, 3, 4. Rather, there are only three elements of B, namely the numbers 1 and 2, the set; the elements of a set can be anything. For example, C =, is the set whose elements are the colors red and blue; the relation "is an element of" 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"; the expressions "A includes x" and "A contains x" are used to mean set membership, however some authors use them to mean instead "x is a subset of A". Logician George Boolos urged that "contains" be used for membership only and "includes" for the subset relation only.
For the relation ∈, the converse relation ∈T may be written A ∋ x, meaning "A contains x". The negation of set membership is denoted by the symbol "∉". Writing x ∉ A means that "x is not an element of A"; the symbol ∈ was first used by Giuseppe Peano 1889 in his work Arithmetices principia, nova methodo exposita. Here he wrote on page X: Signum ∈ significat est. Ita a ∈ b legitur a est quoddam b. So a ∈ b is read; every relation R: U → V is subject to two involutions: complementation R → R ¯ and conversion RT: V → U. The relation ∈ has for its domain a universal set U, has the power set P for its codomain or range; the complementary relation ∈ ¯ = ∉ expresses the opposite of ∈. An element x ∈ U may have x ∉ A, in which case x ∈ U \ A, the complement of A in U; the converse relation ∈ T = ∋ swaps the domain and range with ∈. For any A in P, A ∋ x is true when x ∈ A; the number of elements in a particular set is a property known as cardinality. In the above examples the cardinality of the set A is 4, while the cardinality of either of the sets B and C is 3.
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 ∈ B 3,4 ∉ B is a member of B Yellow ∉ C The cardinality of D = is finite and equal to 5; 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 axiomatized, not that it is silly or easy. Jech, Thomas, "Set Theory", Stanford Encyclopedia of Philosophy Suppes, Axiomatic Set Theory, NY: Dover Publications, Inc. ISBN 0-486-61630-4 - Both the notion of set, membership or element-hood, the axiom of extension, the axiom of separation, the union axiom are needed for a more thorough understanding of "set element". Weisstein, Eric W. "Element". MathWorld
Probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms; these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of these outcomes is called an event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, stochastic processes, which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion. Although it is not possible to predict random events, much can be said about their behavior. Two major results in probability theory describing such behaviour are the law of large numbers and the central limit theorem.
As a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of data. Methods of probability theory apply to descriptions of complex systems given only partial knowledge of their state, as in statistical mechanics. A great discovery of twentieth-century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics; the mathematical theory of probability has its roots in attempts to analyze games of chance by Gerolamo Cardano in the sixteenth century, by Pierre de Fermat and Blaise Pascal in the seventeenth century. Christiaan Huygens published a book on the subject in 1657 and in the 19th century, Pierre Laplace completed what is today considered the classic interpretation. Probability theory considered discrete events, its methods were combinatorial. Analytical considerations compelled the incorporation of continuous variables into the theory; this culminated on foundations laid by Andrey Nikolaevich Kolmogorov.
Kolmogorov combined the notion of sample space, introduced by Richard von Mises, measure theory and presented his axiom system for probability theory in 1933. This became the undisputed axiomatic basis for modern probability theory. Most introductions to probability theory treat discrete probability distributions and continuous probability distributions separately; the measure theory-based treatment of probability covers the discrete, continuous, a mix of the two, more. Consider an experiment that can produce a number of outcomes; the set of all outcomes is called the sample space of the experiment. The power set of the sample space is formed by considering all different collections of possible results. For example, rolling an honest die produces one of six possible results. One collection of possible results corresponds to getting an odd number. Thus, the subset is an element of the power set of the sample space of die rolls; these collections are called events. In this case, is the event that the die falls on some odd number.
If the results that occur fall in a given event, that event is said to have occurred. Probability is a way of assigning every "event" a value between zero and one, with the requirement that the event made up of all possible results be assigned a value of one. To qualify as a probability distribution, the assignment of values must satisfy the requirement that if you look at a collection of mutually exclusive events, the probability that any of these events occurs is given by the sum of the probabilities of the events; the probability that any one of the events, or will occur is 5/6. This is the same as saying that the probability of event is 5/6; this event encompasses the possibility of any number except five being rolled. The mutually exclusive event has a probability of 1/6, the event has a probability of 1, that is, absolute certainty; when doing calculations using the outcomes of an experiment, it is necessary that all those elementary events have a number assigned to them. This is done using a random variable.
A random variable is a function that assigns to each elementary event in the sample space a real number. This function is denoted by a capital letter. In the case of a die, the assignment of a number to a certain elementary events can be done using the identity function; this does not always work. For example, when flipping a coin the two possible outcomes are "heads" and "tails". In this example, the random variable X could assign to the outcome "heads" the number "0" and to the outcome "tails" the number "1". Discrete probability theory deals with events. Examples: Throwing dice, experiments with decks of cards, random walk, tossing coins Classical definition: Initially the probability of an event to occur was defined as the number of cases favorable for the event, over the number of total outcomes possible in an equiprobable sample space: see Classical definition of probability. For example, if the event is "occurrence of an number when a die is
Media related to Sample space at Wikimedia Commons