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A dichotomy is a partition of a whole (or a set) into two parts (subsets). In other words, this couple of parts must be * jointly exhaustive: everything must belong to one part or the other, and * mutually exclusive: nothing can belong simultaneously to both parts. 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 Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
, the partitions are opposites if there exists a proposition such that it holds over one and not the other. Treating
continuous variable In mathematics and statistics, a quantitative variable may be continuous or discrete if they are typically obtained by ''measuring'' or '' counting'', respectively. If it can take on two particular real values such that it can also take on all ...
s or multi
categorical variable In statistics, a categorical variable (also called qualitative variable) is a variable that can take on one of a limited, and usually fixed, number of possible values, assigning each individual or other unit of observation to a particular group or ...
s as
binary variable Binary data is data whose unit can take on only two possible states. These are often labelled as 0 and 1 in accordance with the binary numeral system and Boolean algebra. Binary data occurs in many different technical and scientific fields, wher ...
s is called
dichotomization In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical ...
. The discretization error inherent in dichotomization is temporarily ignored for modeling purposes.

Etymology

The term ''dichotomy'' is from the Greek language el, διχοτομία ''dichotomía'' "dividing in two" from δίχα ''dícha'' "in two, asunder" and τομή ''tomḗ'' "a cutting, incision".

Usage and examples

* In
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
, a dichotomous relation ''R'' is such that either ''aRb'', ''bRa'', but not both. * A
false dichotomy A false dilemma, also referred to as false dichotomy or false binary, is an informal fallacy based on a premise that erroneously limits what options are available. The source of the fallacy lies not in an invalid form of inference but in a false ...
is an
informal fallacy Informal fallacies are a type of incorrect argument in natural language. The source of the error is not just due to the ''form'' of the argument, as is the case for formal fallacies, but can also be due to their ''content'' and ''context''. Falla ...
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 (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." * In
statistics Statistics (from German: ''Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industria ...
, dichotomous data may only exist at first two levels of measurement, namely at the
nominal Nominal may refer to: Linguistics and grammar * Nominal (linguistics), one of the parts of speech * Nominal, the adjectival form of "noun", as in "nominal agreement" (= "noun agreement") * Nominal sentence, a sentence without a finite verb * No ...
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++ C, or c, is the third Letter (alphabet), letter in the Latin alphabet, used in the English alphabet, modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is English alphabet#Le ...
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 Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, galaxies ...
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) In spherical astronomy, quadrature is the configuration of a celestial object in which its elongation is perpendicular to the direction of the Sun. It is applied especially to the position of a superior planet or the Moon at its first and la ...
which is when the Sun-Earth-Moon/superior planet angle is 90°.)

See also

* *
Binary opposition A binary opposition (also binary system) is a pair of related terms or concepts that are opposite in meaning. Binary opposition is the system of language and/or thought by which two theoretical opposites are strictly defined and set off against one ...
* Bipartite (disambiguation) *
Class (set theory) In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. Classes act as a way to have set-like ...
*
Dichotomy paradox Zeno's paradoxes are a set of philosophical problems generally thought to have been devised by Greek philosopher Zeno of Elea (c. 490–430 BC) to support Parmenides' doctrine that contrary to the evidence of one's senses, the belief in pluralit ...
*
Dilemma A dilemma ( grc-gre, δίλημμα "double proposition") is a problem offering two possibilities, neither of which is unambiguously acceptable or preferable. The possibilities are termed the ''horns'' of the dilemma, a clichéd usage, but dis ...
* Law of excluded middle, which in logic asserts the existence of a dichotomy *
Polychotomy A polychotomy (päl′i kät′ə mē; plural ''polychotomies'') is a division or separation into many parts or classes. Polychotomy is a generalization of dichotomy, which is a polychotomy of exactly two parts. In evolutionary biology, the term p ...
*
Taxonomy Taxonomy is the practice and science of categorization or classification. A taxonomy (or taxonomical classification) is a scheme of classification, especially a hierarchical classification, in which things are organized into groups or types ...
* Trichotomy (disambiguation)

References

{{Authority control Broad-concept articles Logic