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In computer science, an enumeration algorithm is an algorithm that enumerates the answers to a computational problem. Formally, such an algorithm applies to problems that take an input and produce a list of solutions, similarly to function problems. For each input, the enumeration algorithm must produce the list of all solutions, without ...
That is, an enumeration of a set S is a bijective function from the natural numbers or an initial segment {1, ..., n} of the natural numbers to S. A set is countable if it can be enumerated, that is, if there exists an enumeration of it. Otherwise, it is uncountable. For example, the set of the real numbers is uncountable.
The example mapping f happens to correspond to the example enumeration s in the picture above. A generalized form of the diagonal argument was used by Cantor to prove Cantor's theorem: for every set S, the power set of S—that is, the set of all subsets of S (here written as P(S))—cannot be in bijection with S itself. This proof proceeds as ...
In computer programming, an enumerated type (also called enumeration, enum, or factor in the R programming language, and a categorical variable in statistics) is a data type consisting of a set of named values called elements, members, enumeral, or enumerators of the type.
Examples of category-theoretic statements which require choice include: Every small category has a skeleton. If two small categories are weakly equivalent, then they are equivalent. Every continuous functor on a small-complete category which satisfies the appropriate solution set condition has a left-adjoint (the Freyd adjoint functor theorem).
Many countries use a post-enumeration survey to adjust the raw census counts. [13] This works similarly to capture-recapture estimation for animal populations. Among census experts, this method is called dual system enumeration (DSE). A sample of households is visited by interviewers who record the details of the household as of census day.
One particularly egregious example was that over $480 million in food assistance was at risk of spoilage and diversion and, in the absence of personnel to conduct required vetting, the program was ...
Two examples of this type of problem are counting combinations and counting permutations. More generally, given an infinite collection of finite sets S i indexed by the natural numbers , enumerative combinatorics seeks to describe a counting function which counts the number of objects in S n for each n .