Characterization (mathematics)
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a characterization of an object is a set of conditions that, while different from the definition of the object, is logically equivalent to it. To say that "Property ''P'' characterizes object ''X''" is to say that not only does ''X'' have
property Property is a system of rights that gives people legal control of valuable things, and also refers to the valuable things themselves. Depending on the nature of the property, an owner of property may have the right to consume, alter, share, r ...
''P'', but that ''X'' is the ''only'' thing that has property ''P'' (i.e., ''P'' is a defining property of ''X''). Similarly, a set of properties ''P'' is said to characterize ''X'', when these properties distinguish ''X'' from all other objects. Even though a characterization identifies an object in a unique way, several characterizations can exist for a single object. Common mathematical expressions for a characterization of ''X'' in terms of ''P'' include "''P'' is
necessary and sufficient In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...
for ''X''", and "''X'' holds
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...
''P''". It is also common to find statements such as "Property ''Q'' characterizes ''Y''
up to Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' wi ...
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
". The first type of statement says in different words that the
extension Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (predicate logic), the set of tuples of values that satisfy the predicate * E ...
of ''P'' is a
singleton Singleton may refer to: Sciences, technology Mathematics * Singleton (mathematics), a set with exactly one element * Singleton field, used in conformal field theory Computing * Singleton pattern, a design pattern that allows only one instance ...
set, while the second says that the extension of ''Q'' is a single
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
(for isomorphism, in the given example — depending on how ''
up to Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' wi ...
'' is being used, some other
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation ...
might be involved). A reference on mathematical terminology notes that ''characteristic'' originates from the Greek term ''kharax'', "a pointed stake":
"From Greek ''kharax'' came ''kharakhter'', an instrument used to mark or engrave an object. Once an object was marked, it became distinctive, so the character of something came to mean its distinctive nature. The Late Greek suffix ''-istikos'' converted the noun ''character'' into the adjective ''characteristic'', which, in addition to maintaining its adjectival meaning, later became a noun as well."
Just as in chemistry, the characteristic property of a material will serve to identify a sample, or in the study of materials, structures and properties will determine
characterization Characterization or characterisation is the representation of persons (or other beings or creatures) in narrative and dramatic works. The term character development is sometimes used as a synonym. This representation may include direct methods ...
, in mathematics there is a continual effort to express properties that will distinguish a desired feature in a theory or system. Characterization is not unique to mathematics, but since the science is abstract, much of the activity can be described as "characterization". For instance, in ''
Mathematical Reviews ''Mathematical Reviews'' is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science. The AMS also pu ...
'', as of 2018, more than 24,000 articles contain the word in the article title, and 93,600 somewhere in the review. In an arbitrary context of objects and features, characterizations have been expressed via the
heterogeneous relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and in ...
''aRb'', meaning that object ''a'' has feature ''b''. For example, ''b'' may mean abstract or concrete. The objects can be considered the
extension Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (predicate logic), the set of tuples of values that satisfy the predicate * E ...
s of the world, while the features are expression of the
intension In any of several fields of study that treat the use of signs — for example, in linguistics, logic, mathematics, semantics, semiotics, and philosophy of language — an intension is any property or quality connoted by a word, phrase, or anoth ...
s. A continuing program of characterization of various objects leads to their
categorization Categorization is the ability and activity of recognizing shared features or similarities between the elements of the experience of the world (such as Object (philosophy), objects, events, or ideas), organizing and classifying experience by a ...
.


Examples

* A
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
, generally defined as a
ratio In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
of two integers, can be characterized as a number with finite or repeating
decimal expansion A decimal representation of a non-negative real number is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator: r = b_k b_\ldots b_0.a_1a_2\ldots Here is the decimal separator, is ...
. *A
parallelogram In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equa ...
is a
quadrilateral In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...
whose opposing sides are parallel. One of its characterizations is that its diagonals bisect each other. This means that the diagonals in all parallelograms bisect each other, and conversely, that any quadrilateral whose diagonals bisect each other must be a parallelogram. The latter statement is only true if inclusive definitions of quadrilaterals are used (so that, for example,
rectangle In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containi ...
s count as parallelograms), which is the dominant way of defining objects in mathematics nowadays. * "Among
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
s on the interval from 0 to ∞ on the real line,
memorylessness In probability and statistics, memorylessness is a property of certain probability distributions. It usually refers to the cases when the distribution of a "waiting time" until a certain event does not depend on how much time has elapsed already ...
characterizes the
exponential distribution In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average ...
s." This statement means that the exponential distributions are the only probability distributions that are memoryless, provided that the distribution is continuous as defined above (see Characterization of probability distributions for more). * "According to
Bohr–Mollerup theorem In mathematical analysis, the Bohr–Mollerup theorem is a theorem proved by the Danish mathematicians Harald Bohr and Johannes Mollerup. The theorem characterizes the gamma function, defined for by :\Gamma(x)=\int_0^\infty t^ e^\,dt as the '' ...
, among all functions ''f'' such that ''f''(1) = 1 and ''x f''(''x'') = ''f''(''x'' + 1) for ''x'' > 0, log-convexity characterizes the
gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
." This means that among all such functions, the gamma function is the ''only'' one that is log-convex.A function ''f'' is ''log-convex''
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...
log(''f'') is a
convex function In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of a function, graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigra ...
. The base of the logarithm does not matter as long as it is more than 1, but mathematicians generally take "log" with no subscript to mean the
natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
, whose base is ''e''.
* The circle is characterized as a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
by being one-dimensional,
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
and
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
; here the characterization, as a smooth manifold, is
up to Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' wi ...
diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two m ...
.


See also

* Characterization of probability distributions *
Characterizations of the category of topological spaces In the mathematical field of topology, a topological space is usually defined by declaring its open sets. However, this is not necessary, as there are many equivalent axiomatic foundations, each leading to exactly the same concept. For instance, a t ...
*
Characterizations of the exponential function In mathematics, the exponential function can be characterized in many ways. The following characterizations (definitions) are most common. This article discusses why each characterization makes sense, and why the characterizations are independent o ...
*
Characteristic (algebra) In mathematics, the characteristic of a ring (mathematics), ring , often denoted , is defined to be the smallest number of times one must use the ring's identity element, multiplicative identity (1) in a sum to get the additive identity (0). If th ...
* Characteristic (exponent notation) *
Classification theorem In mathematics, a classification theorem answers the classification problem "What are the objects of a given type, up to some equivalence?". It gives a non-redundant enumeration: each object is equivalent to exactly one class. A few issues relate ...
*
Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
*
Character (mathematics) In mathematics, a character is (most commonly) a special kind of function from a group to a field (such as the complex numbers). There are at least two distinct, but overlapping meanings. Other uses of the word "character" are almost always qualifie ...


References

{{DEFAULTSORT:Characterization (Mathematics) Mathematical terminology Equivalence (mathematics)