: Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains exactly one element in common with each of the sets in X.

Given that a natural language such as English contains, at any given time, a finite number of words, any comprehensive list of definitions must either be circular or rely upon primitive notions.

Given a bounded sequence, there exists a closed ball that contains the image of ( is a subset of the scalar field ).

Given a base for a topology, in order to prove convergence of a net or sequence it is sufficient to prove that it is eventually in every set in the base which contains the putative limit.

Given the discriminator PARTY TYPE ( which could be individual, partnership, C Corporation, Sub Chapter S Association, Association, Governmental Unit, Quasi-governmental agency ) the two subtype entities are PERSON, which contains individual-specific information such as first and last name and date of birth, and ORGANIZATION, which would contain such attributes as the legal name, and organizational hierarchies such as cost centers.

Given an orientable Haken manifold M, by definition it contains an orientable, incompressible surface S. Take the regular neighborhood of S and delete its interior from M. In effect, we've cut M along the surface S. ( This is analogous, in one less dimension, to cutting a surface along a circle or arc.

; Generating set: Given a field extension E / F and a subset S of E, we write F ( S ) for the smallest subfield of E that contains both F and S. It consists of all the elements of E that can be obtained by repeatedly using the operations +,-,*,/ on the elements of F and S. If E = F ( S ) we say that E is generated by S over F.

* Given a point and line there is a unique line which contains the point and is parallel to the line

Another partition of the given interval, Q, is defined as a refinement of the partition, P, when it contains all the points of P and possibly some other points as well ; the partition Q is said to be “ finer ” than P. Given two partitions, P and Q, one can always form their common refinement, denoted P ∨ Q, which consists of all the points of P and Q, re-numbered in order.

: Given a field K and a torsion-free group G, is it true that K contains no zero divisors?

# Given a subdivision of the plane into vertical slabs, determine which slab contains a given point.

# Given a slab subdivided into regions by non-intersecting segments that completely cross the slab from left to right, determine which region contains a given point.

* Given a point and line there is a unique line which contains the point and is parallel to the line ( Playfair's Axiom ).

Given a set of elements ( called the universe ) and sets whose union comprises the universe, the set cover problem is to identify the smallest number of sets whose union still contains all elements in the universe.

Given that switchgrass contains approximately 18. 8 GJ / ODT of biomass, the energy output-to-input ratio for the crop can be up to 20: 1.

* Given a space X with the indiscrete topology the neighbourhood system for any point x only contains the whole space,

Given that the beetle order ( Coleoptera ) contains about one-quarter of all known organisms, the Curculioninae represent one of the – if not the – most successful radiations of terrestrial Metazoa.

Given a certain number n of people, is it possible to assign them to sets so that each person is in at least one set, each pair of people is in exactly one set together, every two sets have exactly one person in common, and no set contains everyone, all but one person, or exactly one person?

Given two candidates, A and B, A is better than B on a constraint if A incurs fewer violations than B.

Given the smaller size of the senate, there are fewer members competing for committee and subcommittee assignments.

Given a finite set of n properties, a thing is good if it is perceived to have all of the properties, fair if it has more than < sup > n </ sup >/< sub > 2 </ sub > of them, average if < sup > n </ sup >/< sub > 2 </ sub > of them, and bad if it has fewer than < sup > n </ sup >/< sub > 2 </ sub >.

Given the number of original marchers in the putsch, the number of posthumous awards given under the 1938 extensions ( 436 ), and the awards for outstanding service under those same extensions, the total number of recipients numbered fewer than 6, 000.

Given that disadvantaged students generally have fewer opportunities to learn academic content outside of school, wasted class time due to an unfocused lesson presents a particular problem.

Given some encrypted data (" ciphertext "), the goal of the cryptanalyst is to gain as much information as possible about the original, unencrypted data (" plaintext ").

* Given any set X, there is an equivalence relation over the set of all possible functions X → X.

Given the broad scope of environmental law, no fully definitive list of environmental laws is possible.

Given the existence of a Godlike object in one world, proven above, we may conclude that there is a Godlike object in every possible world, as required.

Given sufficient travel time and engineering work, both unmanned and generational interstellar travel seem possible, though present considerable technological and economic challenges are unlikely to be met in the near future, particularly for manned probes.

Given the widespread expansion in the tenth revision, it is not possible to convert ICD-9 data sets directly into ICD-10 data sets, although some tools are available to help guide users.

The knapsack problem or rucksack problem is a problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible.

:( a ) Given player 2's strategy, the best payoff possible for player 1 is V, and

:( b ) Given player 1's strategy, the best payoff possible for player 2 is − V.

Given the notion of a lexeme, it is possible to distinguish two kinds of morphological rules.

Given this ( relative ) philosophic flexibility, variant viewpoints are possible, particularly in areas not explicitly demarcated by the Halakha.

Given a list of cities and their pairwise distances, the task is to find the shortest possible route that visits each city exactly once and returns to the origin city.

Given current trends in the UMC — with overseas churches growing, especially in Africa, and U. S. churches collectively losing about 1, 000 members a week — it has been estimated that Africans will make up at least 30 % of the delegates at the 2012 General Conference, and it is also possible that 40 % of the delegates will be from outside the U. S. One Congolese bishop has estimated that typical Sunday attendance of the UMC is higher in his country than in the entire United States.

Given the being's chaotic nature, almost anything is possible.

Given the structure of the experiment, it may or may not be possible to actually perform it, and, in the case that it is possible for it to be performed, there need be no intention of any kind to actually perform the experiment in question.

Given that hearing impairments can vary by frequency and that audiograms are plotted with a logarithmic scale, the idea of a percentage of hearing loss is somewhat arbitrary, but where decibels of loss are converted via a recognized legal formula, it is possible to calculate a standardized " percentage of hearing loss " which is suitable for legal purposes only.

Given that coconuts are ideally suited for ocean dispersal, it seems possible that individuals from one population could have floated to the other.

Given X and Y, several different embeddings of X in Y may be possible.

Given Russia's often inadequately trained conscript armies and the use of inaccurate smooth-bore muskets, Russian officers preferred to use the bayonet charge in lieu of musket volley fire where possible.

* Aggregate signature-a signature scheme that supports aggregation: Given n signatures on n messages from n users, it is possible to aggregate all these signatures into a single signature whose size is constant in the number of users.

Given a set A of N people, it has 2 < sup > N </ sup > possible subsets.

Given Zelazny's academic interest in the Medieval European period, it is not a stretch to see a possible influence in Henry Adams ' 1905 work Mont Saint-Michel and Chartres, wherein he discusses the building of Chartres Cathedral, and the tidal-islet of Mont Saint-Michel, on the Normandy coast of France.

Given any input, GEN generates an infinite number of candidates, or possible realizations of that input.

Given that the Liber historiae Francorum admits four years for Chlothar's reign, it remains possible that Chlothar began to exert some real authority when his mentors deemed him an adult, 669.