: Given any family of nonempty sets, their Cartesian product is a nonempty set.

: 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 John of Worcester wrote his chronicle after the eruption of the Canterbury – York supremacy struggle, the story of Ealdred renouncing any claims to Worcester needs to be considered suspect.

Given the absolute magnitude, for objects within our galaxy you can also calculate the apparent magnitude from any distance ( in parsecs ):

* Given any Banach space X, the continuous linear operators A: X → X form a unitary associative algebra ( using composition of operators as multiplication ); this is a Banach algebra.

* Given any topological space X, the continuous real-or complex-valued functions on X form a real or complex unitary associative algebra ; here the functions are added and multiplied pointwise.

Given any element x of X, there is a function f < sup > x </ sup >, or f ( x ,·), from Y to Z, given by f < sup > x </ sup >( y ) := f ( x, y ).

Given any expression involving complex numbers, bras, kets, inner products, outer products, and / or linear operators ( but not addition ), written in bra-ket notation, the parenthetical groupings do not matter ( i. e., the associative property holds ).

* Given any combination of complex numbers, bras, kets, inner products, outer products, and / or linear operators, written in bra-ket notation, its Hermitian conjugate can be computed by reversing the order of the components, and taking the Hermitian conjugate of each.

Given x ∈ A, the holomorphic functional calculus allows to define ƒ ( x ) ∈ A for any function ƒ holomorphic in a neighborhood of Furthermore, the spectral mapping theorem holds:

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 any vector space V over a field F, the dual space V * is defined as the set of all linear maps ( linear functionals ).

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

Given a left neutral element and for any given then A4 ’ says there exists an such that.

Given a groupoid G, the vertex groups or isotropy groups or object groups in G are the subsets of the form G ( x, x ), where x is any object of G. It follows easily from the axioms above that these are indeed groups, as every pair of elements is composable and inverses are in the same vertex group.

Given that Gable and Cukor had worked together before, in Manhattan Melodrama and Gable had no objection to working with him then, and given Selznick's desperation to get Gable for Rhett Butler, if Gable had any objections to Cukor, certainly they would have been expressed before he signed his contract for the film.

On poverty, Hoover said that " Given the chance to go forward with the policies of the last eight years, we shall soon with the help of God, be in sight of the day when poverty will be banished from this nation ", and promised, " We in America today are nearer to the final triumph over poverty than ever before in the history of any land ," but within months, the Stock Market Crash of 1929 occurred, and the world's economy spiraled downward into the Great Depression.

Given the state at some initial time ( t = 0 ), we can solve it to obtain the state at any subsequent time.

We have seen a few cavities of the appropriate size and shape for ivory-bills, but these can be old, or exceptionally large Pileated Woodpecker cavities, or mammal-enlarged Pileated Woodpecker cavities .… Given the results, it is unlikely a population of any meaningful size of Ivory-billed Woodpeckers exists in south Florida.

Given a general algorithm for integer factorization, one can factor any integer down to its constituent prime factors by repeated application of this algorithm.

Given the above-mentioned problems, regulators face the challenging task of regulating a market that is changing very rapidly, without stifling any type of innovation, and without improperly disadvantaging any competitor.

Given that the cost of replacing an executive can run over 100 % of his or her annual salary, any investment of time and energy in re-recruitment will likely pay for itself many times over if it helps a business retain just a handful of key players that would have otherwise left.

: Given any positive number ε, there is a sequence

Given the truth of the assumptions, a valid deduction guarantees the truth of the conclusion.

Given any such interpretation of a set of points as complex numbers, the points constructible using valid compass and straightedge constructions alone are precisely the elements of the smallest field containing the original set of points and closed under the complex conjugate and square root operations ( to avoid ambiguity, we can specify the square root with complex argument less than π ).

Given such a ciphertext, an adversary can compute, which is a valid encryption of, for any.

Given the logic of arguments concerning principles of uncertainty and randomness, skepticism towards knowledge merely becomes skepticism towards valid justification.

Given this equivalence, it is valid to assume “ risk neutrality ” when pricing derivatives.

Given two valid inputs

Given these facts and other considerations, the actions of PNG could be deemed to be valid in that:

Given the high stakes nature of many of these assessment purposes, it is crucial that assessment practices be guided by sound principles to insure that they are valid, fair, and appropriate to the context and purposes for which they designed.

Given the magnitude of the disaster, reconstruction and economic recovery will remain the primary focus of the Salvadoran Government for some time to come.

Given Guatemala's large expatriate community in the United States, it is the top remittance recipient in Central America, with inflows serving as a primary source of foreign income equivalent to nearly two-thirds of exports.

Given two secondary stations, the time difference ( TD ) between the primary and first secondary identifies one curve, and the time difference between the primary and second secondary identifies another curve, the intersections of which will determine a geographic point in relation to the position of the three stations.

Given that the authors were the primary developers of the Unix operating system, it is natural that QED had a strong influence on the classic UNIX text editors ed, sed and their descendants such as ex and sam.

Given the opposition to the mission when it began, the primary U. S. concern was to do its limited job and avoid casualties among its forces.

Given that no majority supported either a leader or a split, a registration campaign, enabling membership for only 20 euros, and a closed primary was organized, which Ségolène Royal won.

Given the extremely distorted shape of the primary, the relative orbital motion may be notably altered with respect to the two-body purely Keplerian scenario because of non-negligible long-term orbital perturbations affecting, for example, its orbital period.

Given that NASA and USAF ( as well as government programs in other countries ) are the primary customers and sources of development capital, government regulatory requirements for oversight, parwork, quality, safety, and other documentation tend to inflate the operational costs of any such system.

Given that the Oostvaardersplassen is below sea level, many of its primary processes have been regulated.

Given this function, a transformer sold for isolation is often built with special insulation between primary and secondary, and is tested, specified, and marked to withstand a high voltage between windings, typically in the 1000 to 5000 volt range.

Given this poor performance, the USAAF placed its most experienced and accurate bombardiers aboard each formation's lead bombers ( primary and deputy lead ) and had the remaining airplanes open their bomb bays and drop their bombs when the leader did.

Given that the same basic design appears in several primary sources, it seems likely that this is not a stylistic coincidence but a depiction of a design of whip in current use at the time the articles were made.

Given this additional support the scheme has expanded its reach to cover most children in primary schools in India.

Given his poor relationship with the White House, Walsh anticipated that the administration might even support an opponent in a Democratic primary when he next ran for reelection.

Given the propensity for cardiovascular and cerebrovascular complications, minimizing vascular risk factors and implementing therapy for primary or secondary prevention of stroke and myocardial infarction seems prudent.

Given command of the Corpus Christi district, he protected the primary trade route between Texas and Mexico from outlaws.

Given a target optimal product mix, an independent oil refiner can attempt to hedge itself against adverse price movements by buying oil futures and selling futures for its primary refined products according to the proportions of its optimal mix.

Given that labor is the individual's primary commodity in the market, decommodification refers to activities and efforts ( generally by the government ) that reduce individuals ' reliance on the market ( and their labor ) for their well-being.

Given its primary military role of reliably identifying friends, IFF has much more secure ( encrypted ) messages to prevent " spoofing " by the enemy, and also is used on all kinds of military platforms including air, sea and land vehicles.

Given the Mesoamerican influence on Casas Grandes Nahuatl was probably widely spoken but it was not the primary language of the people.

* James Fowler Given 1825-1867 American Episcopalian evangelist, primary co-founder and organizer of the Christian Union denomination in 1864