: 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 that the vast majority of all emeralds are treated as described above, and the fact that two stones that appear visually similar may actually be quite far apart in treatment level and therefore in value, a consumer considering a purchase of an expensive emerald is well advised to insist upon a treatment report from a reputable gemological laboratory.

Given its deviations from the classification's norms, this voice type is quite rare.

Given its sharp-edged beak, huge bill and wide gape, the Shoebill can hunt quite large prey, often targeting larger prey than other large wading birds.

Given that there wasn't quite a full album worth of material, the band decided to regroup to record a few new songs.

Given this, it is quite possible that the plant was pharmacologically active in the prevention or termination of pregnancy.

Given the technical limitations of daguerreotype photography which required long exposure times, this was one situation where the portrait subject remained quite still.

Given this complexity, early games using the system tended to be quite simple in layout, using the TIA to create simple symmetric playfields with players on top.

Given that many of those most actively involved in early efforts were academics ( drawn from a variety of disciplines including Anthropology, Computing Science, Development Studies, Information Science and Systems, Management, Planning, Sociology, and Social Work among others ) it is only inevitable that a process of " sense-making " with respect to these efforts would follow on quite quickly from the flurry of " tool-making " efforts.

Given that Fokás / Fuca was the family name borne by the seafarer's father and grandfather, Valeriános is likely to be a nickname used on the island which would have been quite meaningless in the Spanish Empire.

Given the questionable connotation of the ' physical ' as something opposed to the mental, it is important to keep in mind that thinkers of the Enlightenment, such as Locke, can not quite give us a metaphysical definition that gets at the complexities involved in the meaning of such a term.

Given the orbital eccentricity of this object, different epochs can generate quite different heliocentric unperturbed two-body best-fit solutions to the aphelion distance ( maximum distance ) of this object.

Given the very light weight of the " pin top ," the. 22 is quite capable of removing the target from the table with a single well-placed shot.

Given the tendency of birds that have become flightless to increase in size markedly in just a few million years, the probability of Hesperornithes deriving from an animal at least superficially similar and not too far related to Gansus is quite high.

Given Serge's personality ( a mixture of bizarre topic-hopping as his attention drifts and his penchant for brutal honesty ), this proves quite a challenge.

Given its inner-city roots, chicken and food is quite often ordered a la carte.

Given the forename Jean, she dropped it for her middle name Gareth while still quite young.

Given the small population of the region ( compared with the Central Canadian provinces or the New England states ), the regional economy is a net exporter of natural resources, manufactured goods, and services.

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 two groups G and H and a group homomorphism f: G → H, let K be a normal subgroup in G and φ the natural surjective homomorphism G → G / K ( where G / K is a quotient group ).

Given that the coiled texture is the natural state of some African Americans ' hair, or perceived as being more " African ", this simple style is now often seen as a sign of self-acceptance and an affirmation that the beauty norms of the ( eurocentric ) dominant culture are not absolute.

Given an arbitrary group G, there is a related profinite group G < sup >^</ sup >, the profinite completion of G. It is defined as the inverse limit of the groups G / N, where N runs through the normal subgroups in G of finite index ( these normal subgroups are partially ordered by inclusion, which translates into an inverse system of natural homomorphisms between the quotients ).

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 a natural transformation Φ from h < sup > A </ sup > to F, the corresponding element of F ( A ) is.

Given a morphism f: B → A the associated natural transformation is denoted Hom ( f ,–).

Given a diagram F: J → C ( thought of as an object in C < sup > J </ sup >), a natural transformation ψ: Δ ( N ) → F ( which is just a morphism in the category C < sup > J </ sup >) is the same thing as a cone from N to F. The components of ψ are the morphisms ψ < sub > X </ sub >: N → F ( X ).

Given his socialist agenda in Egypt, the Ba ' ath should have been his natural ally, but Nasser was hesitant to share power.

Given the reputed humorous character of Valencians, it was natural that they began to burn figurines depicting people and events of the past year.

Given a stochastic process, the natural filtration for ( or induced by ) this process is the filtration where is generated by all values of up to time s

Given this natural tendency to glide forward, terza rima is especially well-suited to narration and description ".

Given a series with values in a normed abelian group G and a permutation σ of the natural numbers, one builds a new series, said to be a rearrangement of the original series.

Given a natural number x, h outputs the index of the partial computable function that performs the following computation:

Given its central location, Athlone is a natural hub for transport, with frequent bus and rail schedules.

Given the finite supply of natural resources at any specific cost and location, agriculture that is inefficient or damaging to needed resources may eventually exhaust the available resources or the ability to afford and acquire them.

Given this difference, it is necessary that logistic regression take the natural logarithm of the odds ( referred to as the logit or log-odds ) to create a continuous criterion.

Given that the logit is not generally interpreted and that the inverse of the natural logarithm, the exponential function of the logit is generally interpreted instead, it is also helpful to examine this function ( denoted: ).

Given a presheaf, a natural question to ask is to what extent its sections over an open set U are specified by their restrictions to smaller open sets V < sub > i </ sub > of an open cover of U. A presheaf is separated if its sections are " locally determined ": whenever two sections over U coincide when restricted to each of V < sub > i </ sub >, the two sections are identical.

Given that some countries now ban docking, breeders are again attempting to select dogs with the genes for natural bob tails.

Given such a G-module M, it is natural to consider the subgroup of G-invariant elements:

Given its native distribution the species is also a potential natural vagrant to Western Europe and there have been records in areas such as Cornwall and the Isles of Scilly which some observers consider may relate to wild birds ; however, given the Wood Duck's popularity in captivity it would be extremely difficult to prove their provenance one way or another.

* Given a partial function f from the natural numbers into the natural numbers, f is a partial recursive function if and only if the graph of f, that is, the set of all pairs such that f ( x ) is defined, is recursively enumerable.