In this image, the universal set U (the entire rectangle) is dichotomized into the two sets A (in pink) and its complement Ac (in grey).
If there is a concept A, and it is split into parts B and not-B, then the parts form a dichotomy: they are mutually exclusive, since no part of B is contained in not-B and vice versa, and they are jointly exhaustive, since they cover all of A, and together again give A.
Such a partition is also frequently called a bipartition.
The two parts thus formed are complements. In logic, the partitions are opposites if there exists a proposition such that it holds over one and not the other.
The term dichotomy is from the Greek language Greek: διχοτομίαdichotomía "dividing in two" from δίχα dícha "in two, asunder" and τομή tomḗ "a cutting, incision".
Usage and examples
In set theory, a dichotomous relation R is such that either aRb, bRa, but not both.[1]
A false dichotomy is an informal fallacy consisting of a supposed dichotomy which fails one or both of the conditions: it is not jointly exhaustive and/or not mutually exclusive. In its most common form, two entities are presented as if they are exhaustive, when in fact other alternatives are possible. In some cases, they may be presented as if they are mutually exclusive although there is a broad middle ground [2](see also undistributed middle).
One type of dichotomy is dichotomous classification – classifying objects by recursively splitting them into two groups. As Lewis Carroll explains, "After dividing a Class, by the Process of Dichotomy, into two smaller Classes, we may sub-divide each of these into two still smaller Classes; and this Process may be repeated over and over again, the number of Classes being doubled at each repetition. For example, we may divide “books” into “old” and “new” (i.e. “not-old”): we may then sub-divide each of these into “English” and “foreign” (i.e. “not-English”), thus getting four Classes."[3]
In statistics, dichotomous data may only exist at first two levels of measurement, namely at the nominal level of measurement (such as "British" vs "American" when measuring nationality) and at the ordinal level of measurement (such as "tall" vs "short", when measuring height). A variable measured dichotomously is called a dummy variable.
In computer science, more specifically in programming-language engineering, dichotomies are fundamental dualities in a language's design. For instance, C++ has a dichotomy in its memory model (heap versus stack), whereas Java has a dichotomy in its type system (references versus primitive data types).
In astronomy dichotomy is when the Moon or an inferior planet is exactly half-lit as viewed from Earth. For the Moon, this occurs slightly before one quarter Moon orbit and slightly after the third quarter of the Moon's orbit at 89.85° and 270.15°, respectively. Dichotomy occurs when the Sun-Moon-Earth angle is 90°. (This is not to be confused with Quadrature_(astronomy) which is when the Sun-Earth-Moon/superior planet angle is 90°.)
^Komjath, Peter; Totik, Vilmos (2006). Problems and Theorems in Classical Set Theory. Springer Science & Business Media. p. 497. ISBN978-0-387-30293-5.
^Baronett, Stan (2013). Logic. Oxford University Press. p. 134.
^Carroll, Lewis (1897), Symbolic Logic, vol. 1.3.2 (4th ed.), London: Macmillan and Co., Ltd.
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March 08, 2023
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This article is about dichotomy in logic and related topics For usage of dichotomous in botany see Glossary of botanical terms dichotomous For the album by Becoming the Archetype see Dichotomy album This article needs additional citations for verification Please help improve this article by adding citations to reliable sources Unsourced material may be challenged and removed Find sources Dichotomy news newspapers books scholar JSTOR September 2014 Learn how and when to remove this template message A dichotomy d aɪ ˈ k ɒ t e m i is a partition of a whole or a set into two parts subsets In other words this couple of parts must bejointly exhaustive everything must belong to one part or the other and mutually exclusive nothing can belong simultaneously to both parts In this image the universal set U the entire rectangle is dichotomized into the two sets A in pink and its complement Ac in grey If there is a concept A and it is split into parts B and not B then the parts form a dichotomy they are mutually exclusive since no part of B is contained in not B and vice versa and they are jointly exhaustive since they cover all of A and together again give A Such a partition is also frequently called a bipartition The two parts thus formed are complements In logic the partitions are opposites if there exists a proposition such that it holds over one and not the other Treating continuous variables or multicategorical variables as binary variables is called dichotomization The discretization error inherent in dichotomization is temporarily ignored for modeling purposes Contents 1 Etymology 2 Usage and examples 3 See also 4 ReferencesEtymology EditThe term dichotomy is from the Greek language Greek dixotomia dichotomia dividing in two from dixa dicha in two asunder and tomh tomḗ a cutting incision Usage and examples EditIn set theory a dichotomous relation R is such that either aRb bRa but not both 1 A false dichotomy is an informal fallacy consisting of a supposed dichotomy which fails one or both of the conditions it is not jointly exhaustive and or not mutually exclusive In its most common form two entities are presented as if they are exhaustive when in fact other alternatives are possible In some cases they may be presented as if they are mutually exclusive although there is a broad middle ground 2 see also undistributed middle One type of dichotomy is dichotomous classification classifying objects by recursively splitting them into two groups As Lewis Carroll explains After dividing a Class by the Process of Dichotomy into two smaller Classes we may sub divide each of these into two still smaller Classes and this Process may be repeated over and over again the number of Classes being doubled at each repetition For example we may divide books into old and new i e not old we may then sub divide each of these into English and foreign i e not English thus getting four Classes 3 In statistics dichotomous data may only exist at first two levels of measurement namely at the nominal level of measurement such as British vs American when measuring nationality and at the ordinal level of measurement such as tall vs short when measuring height A variable measured dichotomously is called a dummy variable In computer science more specifically in programming language engineering dichotomies are fundamental dualities in a language s design For instance C has a dichotomy in its memory model heap versus stack whereas Java has a dichotomy in its type system references versus primitive data types In astronomy dichotomy is when the Moon or an inferior planet is exactly half lit as viewed from Earth For the Moon this occurs slightly before one quarter Moon orbit and slightly after the third quarter of the Moon s orbit at 89 85 and 270 15 respectively Dichotomy occurs when the Sun Moon Earth angle is 90 This is not to be confused with Quadrature astronomy which is when the Sun Earth Moon superior planet angle is 90 See also Edit Linguistics portal Philosophy portal Psychology portal The dictionary definition of dichotomy at Wiktionary Binary opposition Bipartite disambiguation Class set theory Dichotomy paradox Dilemma Law of excluded middle which in logic asserts the existence of a dichotomy Polychotomy Taxonomy Trichotomy disambiguation References Edit Komjath Peter Totik Vilmos 2006 Problems and Theorems in Classical Set Theory Springer Science amp Business Media p 497 ISBN 978 0 387 30293 5 Baronett Stan 2013 Logic Oxford University Press p 134 Carroll Lewis 1897 Symbolic Logic vol 1 3 2 4th ed London Macmillan and Co Ltd Wikimedia Commons has media related to Dichotomies Retrieved from https en wikipedia org w index php title Dichotomy amp oldid 1126556747, wikipedia, wiki, book, books, library,
