A Venn diagram (also called main diagram , set diagram or logic diagram ) is a diagram showing all possible logical connections between a finite set of different sets. This diagram illustrates the element as a point on the plane, and sets it as the area inside the closed curve. The Venn diagram consists of several overlapping closed curves, usually circles, each representing a set. The points in the curve labeled S represent the elements of the S set, while the point outside the boundary represents an element that does not exist in the set S . It helps to easily read visualization; for example, the set of all elements that are members of both sets of S and T , S Ã,? T , represented visually by the overlapping areas of the S and T regions. In the Venn diagram, the curve overlaps in every possible way, showing all possible relationships between sets. Therefore, they are a special case of Euler diagrams, which do not necessarily indicate all relationships. The Venn diagram was made sometime around 1880 by John Venn. They are used to teach basic set theory, as well as describe simple set relationships in probability, logic, statistics, linguistics, and computer science.
Venn diagram where the area of ​​each form is proportional to the number of elements it contains is called proportional plane or Venn diagram scale .
Video Venn diagram
Contoh
This example involves two sets, A and B, represented here as a colored circle. The orange circle, set A, represents all two-legged creatures. The blue circle, arranging B, represents a living creature that can fly. Each type of a separate being 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 the point in the region where blue and orange circles overlap. It is important to note that this overlapping area will contain only those elements (in this creature's example) which are members of both set A (bipedal beings) as well as members of set A (flying creature)
Humans and bipedal penguins, and so are in orange circles, but because they can not fly they appear on the left side of the orange circle, where it does not overlap with the blue circle. Mosquitoes have six legs, and fly, so the point for mosquitoes is in the blue circle that does not overlap with the orange ones. Non-biped and non-flying creatures (eg, whales and spiders) will all be represented by dots outside the two circles.
The combined regions of sets A and B are called union A and B, denoted by A? B . Unions in this case contain all two-legged or flying creatures (or both).
Areas in A and B, where they overlap, are called intersections A and B, denoted by A? B . For example, the intersection of the two sets is not empty, since there is the point representing the creature that is in both orange and blue circles.
Maps Venn diagram
History
The Venn Diagram was introduced in 1880 by John Venn in a paper entitled On the Diagrammatics and Mechanics Representation of Proposition and Reasoning in "Philosophical Magazines and Journal of Science", on various ways to represent propositions with diagrams. The use of this type of diagram in formal logic, according to Frank Ruskey and Mark Weston, is "not an easy history to trace, but it is certain that the charts that are popularly associated with Venn are, in fact, originally coming in. They are true to Venn, however, as it is comprehensively surveyed and formalized in its use, and is the first to generalize them ".
Venn himself does not use the term "Venn diagram" and refers to his invention as the "Eulerian Circle". For example, in the opening sentence of his 1880 Venn article, "Schematic representation diagrams have been so familiarly introduced into logical treatises over the last century or so, that many readers, even those who do not make the study of professional logic, should perhaps be acquainted with the general nature and objects of the device.Of this scheme only one, the so-called 'Eulerian circle,' has met with general acceptance... "Lewis Carroll (Charles Dodgson) includes" Venn Diagram Method "as well as" Euler Chart Method "in" Appendix, Addressed to Teachers "from his book" Symbolic Logic "(4th edition published in 1896). The term "Venn diagram" was then used by Clarence Irving Lewis in 1918, in his book "A Survey of Symbolic Logic".
The Venn diagram is very similar to Euler's diagram, discovered by Leonhard Euler in the 18th century. M. E, E. Baron has noted that Leibniz (1646-1716) in the 17th century produced a similar diagram before Euler, but much of it was not published. He also observed Euler's diagram-as before by Ramon Lull in the 13th century.
In the 20th century, Venn diagrams were developed further. D.Ã, W.Ã, Henderson pointed out in 1963 that the existence of a diagram n -Venn with n -fold rotation symmetry implied that n is a prime number. He also points out that such symmetrical Venn diagrams exist when n is 5 or 7. In 2002 Peter Hamburger discovered the symmetrical Venn diagram for n = 11 and in 2003, Griggs , Killian, and Savage show that symmetrical Venn diagram exists for all other prime numbers. Thus a symmetrical Venn diagram exists if and only if n is a prime number.
Venn diagrams and Euler diagrams were included as part of the instruction in set theory as part of a new mathematical movement in the 1960s. Since then, they have also been adopted in the curriculum of other fields such as reading.
Overview
Venn diagrams are built with a collection of simple closed curves drawn in planes. According to Lewis, "the principle of this diagram is that the class [or set ] is represented by the region in relation to each other that all possible logical connections of these classes can be shown in the same diagram, the diagram originally leaves room for every possible class relation, and the actual or given relation, then can be determined by indicating that some particular region is null or non-null.
Venn diagrams usually consist of overlapping circles. The inner part of the circle symbolically represents the elements of the set, while the outside represents elements that are not members of the set. For example, in a two-set Venn diagram, a loop can represent a group of all wood objects, while another circle may represent the set of all tables. Overlapping areas or intersections will then represent the set of all wooden tables. A shape other than a circle can be used as shown below by the higher Venn own set diagram. Venn diagrams generally do not contain information about the relative or absolute size (cardinality) of the set; ie they are schematic diagrams.
Venn diagram is similar to Euler diagram. However, the Venn diagram for the n set of components should contain all possible 2 hypothetical zones that correspond to some combination of inclusions or exceptions in each set of components. Euler diagram contains only the actual zone that may be in certain contexts. In the Venn diagram, the shaded zone can represent an empty zone, whereas in the Euler diagram, the corresponding zone is not in the diagram. For example, if one set represents milk products and the other cheese , Venn diagrams contain zones for cheese that are not dairy products. Assuming that in the context of cheese means some kind of dairy product, Euler diagram has a cheese zone entirely contained in milk-dairy zone - no zone for non-dairy cheese. This means that when the number of contours increases, the Euler diagram is usually less visually complex than the equivalent Venn diagram, especially if the number of non-empty intersections is small.
Perbedaan antara diagram Euler dan Venn dapat dilihat pada contoh berikut. Ambil tiga set:
Venn and Euler diagrams of the set are:
< span id = "10"> Extension to higher set number
Venn diagrams usually represent two or three sets, but there are possible forms for higher numbers. Below, four intersecting balls form the highest order Venn diagrams that have simplex symmetry and can be represented visually. 16 intersections correspond to the vertices of the tesseract (or cells of 16-cells respectively).
For the higher number of sets, some loss of symmetry in the diagram can not be avoided. Venn was anxious to find "symmetrical figures... elegant in themselves," representing a higher set number, and he designed a graceful four-set diagram using ellipses (see below). He also provides construction for Venn diagrams for a set number, where each successive curve that confines a set of interleaves to the previous curve, begins with a three-loop diagram.
Edwards-Venn diagram
Anthony William Fairbank Edwards constructed a series of Venn diagrams for a higher number of sets by segmenting the surface of the sphere, which became known as the Edwards-Venn diagram. For example, three sets can be easily represented by taking three spheres at right angles ( x Ã, = Ã, 0, y Ã, = Ã, 0 and z Ã, = Ã, 0). A fourth set can be added to the representation by taking a curve similar to the stitches on the tennis ball, which swings up and down around the equator, and so on. The resulting set can then be projected back to the plane to give a cogwheel diagram with an increase in the number of teeth, as shown here. These diagrams are made when designing colored glass windows to commemorate Venn.
Other diagrams
The Edwards-Venn diagram is the topological equivalent of a diagram designed by Branko GrÃÆ'¼nbaum, which is based around polygons intersected by more and more sides. They are also two dimensional representations of hypercubes.
Henry John Stephen Smith menyusun diagram n -set yang serupa menggunakan kurva sinus dengan rangkaian persamaan
Charles Lutwidge Dodgson (aka Lewis Carroll) designed a five-set diagram known as the Carroll square.
Related concepts
Source of the article : Wikipedia
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