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There are many equivalent definitions of a category. [1] One commonly used definition is as follows. A category C consists of a class ob(C) of objects, a class mor(C) of morphisms or arrows, a domain or source class function dom: mor(C) → ob(C), a codomain or target class function cod: mor(C) → ob(C),
Ordinal data is a categorical, statistical data type where the variables have natural, ordered categories and the distances between the categories are not known. [ 1 ] : 2 These data exist on an ordinal scale , one of four levels of measurement described by S. S. Stevens in 1946.
For example, the ordinal 42 is generally identified as the set {0, 1, 2, ..., 41}. Conversely, any set S of ordinals that is downward closed — meaning that for any ordinal α in S and any ordinal β < α, β is also in S — is (or can be identified with) an ordinal. This definition of ordinals in terms of sets allows for infinite ordinals.
A categorical variable that can take on exactly two values is termed a binary variable or a dichotomous variable; an important special case is the Bernoulli variable. Categorical variables with more than two possible values are called polytomous variables; categorical variables are often assumed to be polytomous unless otherwise specified.
Each category is distinguished by properties that all its objects have in common, such as the empty set or the product of two topologies, yet in the definition of a category, objects are considered atomic, i.e., we do not know whether an object A is a set, a topology, or any other abstract concept. Hence, the challenge is to define special ...
The following table classifies the various simple data types, associated distributions, permissible operations, etc. Regardless of the logical possible values, all of these data types are generally coded using real numbers, because the theory of random variables often explicitly assumes that they hold real numbers.
Among the exponential functions of the form α x, setting α = e 2/e = 2.0870652... results in a sharp upper bound; the slightly smaller choice α = 2 fails to produce an upper bound, since then α 3 = 8 < 3 2. In applied fields the word "tight" is often used with the same meaning. [2] smooth
Categorical logic is the branch of mathematics in which tools and concepts from category theory are applied to the study of mathematical logic. It is also notable for its connections to theoretical computer science. [1] In broad terms, categorical logic represents both syntax and semantics by a category, and an interpretation by a functor.