Enumeration in the context of Cantor's theorem

Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a one-to-one correspondence (bijection) with natural numbers), rather than "continuous" (analogously to continuous functions). Objects studied in discrete mathematics include integers, graphs, and statements in logic. By contrast, discrete mathematics excludes topics in "continuous mathematics" such as real numbers, calculus or Euclidean geometry. Discrete objects can often be enumerated by integers; more formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets (finite sets or sets with the same cardinality as the natural numbers). However, there is no exact definition of the term "discrete mathematics".

The set of objects studied in discrete mathematics can be finite or infinite. The term finite mathematics is sometimes applied to parts of the field of discrete mathematics that deal with finite sets, particularly those areas relevant to business.

Counting is the process of determining the number of elements of a finite set of objects; that is, determining the size of a set. The traditional way of counting consists of continually increasing a (mental or spoken) counter by a unit for every element of the set, in some order, while marking (or displacing) those elements to avoid visiting the same element more than once, until no unmarked elements are left; if the counter was set to one after the first object, the value after visiting the final object gives the desired number of elements. The related term enumeration refers to uniquely identifying the elements of a finite (combinatorial) set or infinite set by assigning a number to each element.

Counting sometimes involves numbers other than one; for example, when counting money, counting out change, "counting by twos" (2, 4, 6, 8, 10, 12, ...), or "counting by fives" (5, 10, 15, 20, 25, ...).

The 1990 United States census, conducted by the Census Bureau, determined the resident population of the United States to be 248,709,873, an increase of 9.8 percent over the 226,545,805 persons enumerated during the 1980 census.

Approximately 16 percent of households received a "long form" of the 1990 census, which contained more than 100 questions. Full documentation on the 1990 census, including census forms and a procedural history, is available from the Integrated Public Use Microdata Series.

Repetition is the simple repeating of a word, within a short space of words (including in a poem), with no particular placement of the words to secure emphasis, within a short space of words. It is a multilinguistic written or spoken device, frequently used in English and several other languages, such as Hindi and Chinese, and so rarely termed a figure of speech, making it a multilinguistic written or spoken device. Repetition in some cases is seen as undesirable.

Its forms, many of which are listed below, have varying resonances to listing (forms of enumeration, such as "Firstly, Secondly, Thirdly, Firstly and lastly..."), as a matter of trite logic often similar in effect.

The 1950 United States census, conducted by the Census Bureau, determined the resident population of the United States to be 151,325,798, an increase of 14.5 percent over the 131,669,275 persons enumerated during the 1940 census.

In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, nth, etc.) aimed to extend enumeration to infinite sets.

A finite set can be enumerated by successively labeling each element with the least natural number that has not been previously used. To extend this process to various infinite sets, ordinal numbers are defined more generally using linearly ordered Greek letter variables that include the natural numbers and have the property that every set of ordinals has a least or "smallest" element (this is needed for giving a meaning to "the least unused element"). This more general definition allows us to define an ordinal number ${\displaystyle \omega }$ (omega) to be the least element that is greater than every natural number, along with ordinal numbers ⁠${\displaystyle \omega +1}$⁠, ⁠${\displaystyle \omega +2}$⁠, etc., which are even greater than ⁠${\displaystyle \omega }$⁠.

In computing, an opcode (abbreviated from operation code) is an enumerated value that specifies the operation to be performed. Opcodes are employed in hardware devices such as arithmetic logic units (ALUs), central processing units (CPUs), and software instruction sets. In ALUs, the opcode is directly applied to circuitry via an input signal bus. In contrast, in CPUs, the opcode is the portion of a machine language instruction that specifies the operation to be performed.

The 1910 United States census, conducted by the Census Bureau on April 15, 1910, determined the resident population of the United States to be 92,228,496, an increase of 21 percent over the 76,212,168 persons enumerated during the 1900 census. The 1910 census switched from a portrait page orientation to a landscape orientation.

This was the last census in which Texas did not record any top 50 largest cities by population, despite being the 5th most populous state at the time. It was also the first census in which all the top 50 largest cities had population over 100,000.

The 1900 United States census, conducted by the Census Office on June 1, 1900, determined the resident population of the United States to be 76,212,168, an increase of 21.01% from the 62,979,766 persons enumerated during the 1890 census. It was the last census to be conducted before the founding of the permanent United States Census Bureau.

The census saw the nation's largest city, New York City, more than double in size due to the consolidation with Brooklyn, becoming in the process the first U.S. city to record a population growth of over three million — Brooklyn was previously the 4th largest city in the United States on its own.

The 1980 United States census, conducted by the Census Bureau, determined the resident population of the United States to be 226,545,805, an increase of 11.4% over the 203,184,772 persons enumerated during the 1970 census. It was the first census in which a state—California—recorded a population of 20 million people, as well as the first in which all states recorded populations of over 400,000. This was the first census to count Hispanic and Latino Americans as an ethnicity.

The 1850 United States census was the seventh decennial United States CensusConducted by the Census Office, it determined the resident population of the United States to be 23,191,876—an increase of 35.9 percent over the 17,069,453 persons enumerated during the 1840 census. The total population included 3,204,313 enslaved people.

Although the official date of the census date was June 1, 1850, completed census forms indicate that the surveys continued to be made throughout the rest of the year.

The 1860 United States census was the eighth census conducted in the United States, starting June 1, 1860, and lasting five months. It determined the population of the United States to be 31,443,321 in 33 states and 10 organized territories. This was an increase of 35.6 percent over the 23,191,876 persons enumerated during the 1850 census. The total population included 3,953,760 slaves.

By the time the 1860 census returns were ready for tabulation, the nation was sinking into the American Civil War. As a result, census superintendent Joseph C. G. Kennedy and his staff produced only an abbreviated set of public reports, without graphic or cartographic representations. The statistics did allow the census staff to produce a cartographic display, including preparing maps of Southern states, for Union field commanders. These maps displayed militarily vital topics, including the white population, slave population, predominant agricultural products (by county), and rail and post road transportation routes.

The 1970 United States census, conducted by the Census Bureau, determined the resident population of the United States to be 203,392,031, an increase of 13.4 percent over the 179,323,175 persons enumerated during the 1960 census.

This was the first census since 1820 in which New York was not the most populous state—California overtook it in population in January 1963. This was also the first census in which all states recorded a population of over 300,000, and the first in which a city in the geographic South—Houston—recorded a population of over 1 million.

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 related to classification are the following.

A search suggest drop-down list is a query feature used in computing to show the searcher shortcuts, while the query is typed into a text box. Before the query is complete, a drop-down list with the suggested completions appears to provide options to select. The suggested queries then enable the searcher to complete the required search quickly. As a form of autocompletion, the suggestion list is distinct from search history in that it attempts to be predictive even when the user is searching for the first time. Data may come from popular searches, sponsors, geographic location or other sources. These lists are used by operating systems, web browsers and various websites, particularly search engines. Search suggestions are common with a 2014 survey finding that over 80% of e-commerce websites included them.

The computing science of syntax and algorithms are used to form search results from a database. Content management systems and frequent searches can assist software engineers in optimizing more refined queries with methods of parameters and subroutines. Suggestions can be results for the current query or related queries by words, time and dates, categories and tags. The suggestion list may be reordered by other options, as enumerative, hierarchical or faceted.