For finite sets X, the axiom of choice follows from the other axioms of set theory.

For example, the number of solutions of an equation over a finite field reflects the topological nature of its solutions over the complex numbers.

* For a finite field of prime order p, the algebraic closure is a countably infinite field which contains a copy of the field of order p < sup > n </ sup > for each positive integer n ( and is in fact the union of these copies ).

For example, intervals, where takes all integer values in Z, cover R but there is no finite subcover.

For example, the real line equipped with the discrete topology is closed and bounded but not compact, as the collection of all singleton points of the space is an open cover which admits no finite subcover.

For a finite group, the derived series terminates in a perfect group, which may or may not be trivial.

For an infinite group, the derived series need not terminate at a finite stage, and one can continue it to infinite ordinal numbers via transfinite recursion, thereby obtaining the transfinite derived series, which eventually terminates at the perfect core of the group.

For systems where the volume is preserved by the flow, Poincaré discovered the recurrence theorem: Assume the phase space has a finite Liouville volume and let F be a phase space volume-preserving map and A a subset of the phase space.

For continuous dynamical systems, the map τ is understood to be a finite time evolution map and the construction is more complicated.

* For every prime number p and positive integer n, there exists a finite field with p < sup > n </ sup > elements.

For example, Graham's number, though finite, is unimaginably larger than other well-known large numbers such as a googol, googolplex, and even larger than Skewes ' number and Moser's number.

For some finite n-valued logics, there is an analogous law called the law of excluded n + 1th.

For every finite dimensional matrix Lie algebra, there is a linear group ( matrix Lie group ) with this algebra as its Lie algebra.

Note: For any arbitrary number of propositional constants, we can form a finite number of cases which list their possible truth-values.

For instance, Fermat's little theorem for the nonzero integers modulo a prime generalizes to Euler's theorem for the invertible numbers modulo any nonzero integer, which generalizes to Lagrange's theorem for finite groups.

For example, the real numbers with the standard Lebesgue measure are σ-finite but not finite.

For a non-relativistic system consisting of a finite number of distinguishable particles, the component systems are the individual particles.

For example, the usual decimal representation of whole numbers gives every whole number a unique representation as a finite sequence of digits.

For a nonzero vector of finite norm in, one can assume that is nonzero, or to fix ideas.

For example, a non-identity finite group is simple if and only if it is isomorphic to all of its non-identity homomorphic images, a finite group is perfect if and only if it has no normal subgroups of prime index, and a group is imperfect if and only if the derived subgroup is not supplemented by any proper normal subgroup.

For any luminosity from a given distance L ( r ) N ( r ) proportional to r < sup > a </ sup >, is infinite for a ≥ − 1 but finite for a < − 1.

For point masses the gravitational energy decreases without limit as they approach zero separation, and it is convenient and conventional to take the potential energy as zero when they are an infinite distance apart, and then negative ( since it decreases from zero ) for smaller finite distances.

For a set of polynomial equations in several unknowns, there are algorithms to decide if they have a finite number of complex solutions.

For both economic and political reasons all segments of the population must be able to share in the growth of a country.

For purposes of sample selection only ( individual tests were given later ) we obtained group test scores of reading achievement and intelligence from school records of the entire third-grade population in each school system.

For almost one-sixth of the national population discrimination in the free selection of residence casts a considerable shadow upon these values assumed as self-evident by most Americans.

For instance, we cannot know whether even for church members the degree of conformity to Christian standards of morality increased or declined as the proportion of church members in the population rose.

For example, per capita GDP gives an approximation of the arithmetic average income of a nation's population.

For this reason a significant proportion of Australia's indigenous population lives in the Outback, in areas such as the Anangu Pitjantjatjara Yankunytjatjara lands in northern South Australia.

For this reason it's important that the ' metapopulation management plan ' occurs concurrently with conservation strategies to enhance in-situ population growth.

For much of the PRC's population, living standards have seen extremely large improvements and freedom continues to expand, but political controls remain tight and rural areas poor.

For hundreds of years this isolated population consisting now of 4. 400 inhabitants, has claimed to be the direct descendant of the Cimbri retreating in this area after the Roman aftermath.

For example, as New England became increasingly settled and the resident wolves were eliminated, the coyote population increased, filling the empty ecological niche.

For this reason international documents refer to censuses of population and housing.

For example, one population of a species of tropical lizards ( Tropidurus hispidus ) has a flattened body relative to the main populations that live in open savanna.

For example, the strongest evidence for therapeutic interventions is provided by systematic review of randomized, triple-blind, placebo-controlled trials with allocation concealment and complete follow-up involving a homogeneous patient population and medical condition.

For example, a drug may influence a secondary clinical endpoint, such as a test result ( blood pressure, glucose or cholesterol levels ), without having the power to show that it decreases overall mortality or morbidity in a population.

For a situation where the traffic to be handled is completely new traffic, the only choice is to try to model expected user behavior, estimating active user population, N, expected level of use, U ( number of calls / transactions per user per day ), busy-hour concentration factor, C ( proportion of daily activity that will fall in the busy hour ), and average holding time / service time, h ( expressed in minutes ).

( For a large user population, the Engset and the Erlang-B calculations give the same result.

For example, the population which is at risk of the severe debilitating disease kuru has significant over-representation of an immune variant of the prion protein gene G127V versus non-immune alleles.

For instance, several recent studies and surveys by sociologists and political scientists that utilize more complex definitional parameters have estimated the number of evangelicals in the U. S. at about 25 – 30 % of the population, or roughly between 70 and 80 million people.

For instance, 30 % of the world's population either has, or is at risk for developing, iodine deficiency.

For example, Ireland currently has 4 licenced operators with their own networks despite a smaller and sparser population than France's making improved coverage less economically rewarding.

For those reasons, the population decreased in the plains, while increasing on the mountains.

For example, the Rashtriya Janata Dal ( translated as National People's Party ) has a vote bank among the Yadav and Muslim population of Bihar and the All India Trinamool Congress does not have any significant support outside West Bengal.

For example, whilst the Israel Central Bureau of Statistics defines the area of Israel to include the annexed East Jerusalem and Golan Heights and to exclude the militarily controlled regions of the West Bank and Gaza Strip, it defines the population of Israel to also include Israeli settlers living in the West Bank.

For continuous wave operation it is required for the population inversion of the gain medium to be continually replenished by a steady pump source.

For example, a hypothetical population might include 10 million men and 10 million women.