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

Given the similarities in the two characters ' names, professions, written works and generally dark subject matter, it is likely that Lovecraft's Alhazred provided the main inspiration for al-Hazir.

Given a field F, the assertion “ F is algebraically closed ” is equivalent to other assertions:

* Given an algebraic number, there is a unique monic polynomial ( with rational coefficients ) of least degree that has the number as a root.

" Given that Thomas Aquinas died six years before Albertus Magnus ' death, this legend as stated is unlikely.

Given Ealdred's known support of Godwin's family, John of Worcester is probably correct.

* 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 an R-module M, the endomorphism ring of M, denoted End < sub > R </ sub >( M ) is an R-algebra by defining ( r · φ )( x ) = r · φ ( x ).

Given the other ZF axioms, the axiom of regularity is equivalent to the axiom of induction.

After Christians in Ephesus first wrote to their counterparts recommending Apollos to them, he went to Achaia where Paul names him as an apostle ( 1 Cor 4: 6, 9-13 ) Given that Paul only saw himself as an apostle ' untimely born ' ( 1 Cor 15: 8 ) it is certain that Apollos became an apostle in the regular way ( as a witness to the risen Lord and commissioned by Jesus-1 Cor 15: 5-9 ; 1 Cor 9: 1 ).< ref > So the Alexandrian recension ; the text in < sup > 38 </ sup > and Codex Bezae indicate that Apollos went to Corinth.

Given its " blue-water " bias, the navy is even less inclined to become involved in counterdrug operations than the army or air force.

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 this, the bidding is said to start at the one-level when contracting for a total of seven tricks, at the two-level for eight tricks and so on to the seven-level to contract to take all thirteen tricks.

While tests have been done to support this with fixed pieces of flat metal, the result is inconclusive with respect to the curved armour of the period. Given the following actions of the archers, it seems likely Froissart was correct.

Given a ket of norm 1, the orthogonal projection onto the subspace spanned by is

Given this influence, it is unfortunate that many of the details of his life remain shrouded in mystery, perhaps forever ; even the only known picture of him, shown above, is heavily retouched, with a fake tie painted in by hand.

Given points P < sub > 0 </ sub > and P < sub > 1 </ sub >, a linear Bézier curve is simply a straight line between those two points.

Given the then GDP ($ 7. 095 bln ) of the country, military spending is roughly estimated to be about $ 300 million.

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 the federal nature of the holiday, celebrating Canada Day can be a cause of friction in the province of Quebec, where the holiday is overshadowed by Quebec's National Holiday, on June 24.

Given that many journeys are for relatively short distances, there is considerable scope to replace car use with walking or cycling, though in many settings this may require some infrastructure modification, particularly to attract the less experienced and confident.

Given such a sweeping definition, it is apparent that cognition is involved in everything a human being might possibly do ; that every psychological phenomenon is a cognitive phenomenon.

Given the need for such fine regulation of Ca < sup > 2 +</ sup > signaling, it is perhaps unsurprising that dysregulated mitochondrial Ca < sup > 2 +</ sup > has been implicated in several neurodegenerative diseases, while the catalogue of tumor suppressors includes a few that are enriched at the MAM.

Given that the U. S. is still home to the world's largest economy, there's no reason it shouldn't have the most vibrant equity markets — unless regulation is holding back the creation of new public companies.

Given the scope of modern regulation, however, legislatures frequently find areas where it is impractical for lawmakers to apply the level of detail or expertise required to establish complete standards.

Given a rapidly changing technological and market environment, many in the public policy area question the government's ability to make and maintain meaningful regulation that doesn't cause more harm than good.

Given the limits of treaties or custom as sources of international law, Article 38 ( 1 ) may be looked upon as a directive to the Court to fill any gap in the law and prevent a non liquet by reference to the general principles.

Given the tide of militant nationalism that swept through India and the resentment and revolts it inspired, it is arguable that its overarching aim: to germinate public resentment and revolts within the Indian forces of the British Indian Army to overthrow the Raj was ultimately successful.

Given this reliance on literary sources, it is doubtful whether Nostradamus used any particular methods for entering a trance state, other than contemplation, meditation and incubation.

Given the structural similarities between bacterial flagella and bacterial secretory systems, it is thought that bacterial flagella may have evolved from the type three secretion system ; however, it is not known for certain whether these pores are derived from the bacterial flagella or the bacterial secretory system.

Given a set of integers, SUBSET-SUM is the problem of finding whether there exists a subset summing to zero.

Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers.

Given the choice, whether one believes that " the lives of entire peoples and societies have been decisively influenced by exceptional individuals " or, conversely, that whatever happens occurs as a result of impersonal forces oblivious to human intentions – Berlin rejected both options and the choice itself as nonsensical.

Given the morphological distinctness of the Cape Verde birds and the fact that the Cape Verde population was isolated from other populations of Red Kites, it cannot be conclusively resolved at this time whether the Cape Verde population was not a distinct subspecies ( as M. migrans fasciicauda ) or even species that frequently absorbed stragglers from the migrating European populations into its gene pool.

Given that totemistic belief systems had proved to be relatively durable over the course of human history, many scholars asked whether it was useful, as Lévi-Strauss had advocated, to dispose of totemism as a " mere " social construct.

Given a set of training examples, each marked as belonging to one of two categories, an SVM training algorithm builds a model that predicts whether a new example falls into one category or the other.

Given a model of a system, exhaustively and automatically check whether this model meets a given specification.

Given a decimal longitudinal coordinate such as-87. 728055 it will be necessary to know whether it is a latitudinal or longitudinal coordinate in order to fully convert it.

Given that it is impossible for civilians to visit the British Indian Ocean Territory, it is not clear whether the flag is actually used.

Given the very narrow margin of his defeat in the 2006 election, some have speculated that Valeri may seek a return to politics in the Hamilton East — Stoney Creek riding, although Valeri himself has not indicated whether he has an interest in returning to the House of Commons.

: Given a description of a Turing machine and its initial input, determine whether the program, when executed on this input, ever halts ( completes ).

Given an element of a disjoint union A + B, it is possible to determine whether it came from A or B.

Given a partial solution to the puzzle, they use dynamic programming within each row or column to determine whether the constraints of that row or column force any of its squares to be white or black, and whether any two squares in the same row or column can be connected by an implication relation.

Given that a positive power lens will magnify an object and a negative power lens will minify it, it is often possible to tell whether a lens is positive or negative by looking through it.

Lessig traces the simultaneous legal environment that permitted its genious: Given the challenge of deciding whether photographers would need to get per mission before taking aim, the legal system decided " in favor of the pirates .....

Given an integer n ( throughout this article, n refers to " the integer to be factored "), trial division consists of testing whether n is divisible by any number.

Given the value of p, and no other information relevant to the question of whether the sun will rise tomorrow, the probability that the sun will rise tomorrow is p. But we are not " given the value of p ".

Given these factors, it was initially uncertain whether the offensive would go forward.

Given that her lord's kinsmen ( þæt monnes magas ) are described as taking measures to separate the speaker from him, a probable interpretation of the speaker's initial circumstances is that she has been entered into an exogamous relationship typical within the Anglo-Saxon heroic tradition, and her marital status has left her isolated among her husband's people, who are hostile to her, whether due to her actions or merely due to political strife beyond her control.

Given an odd number n we can contemplate whether or not the congruence

Given a set of edges that connect some pairs of the points, the problem to determine whether they contain a triangulation is NP-complete ( Lloyd, 1977 ).