: 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 first four rolls turn up heads, the probability that the next toss is a head is in fact,

Given this assumption, there are four categories in which a firm's profit may be considered to be.

Given a year to complete the work, Gibson undertook the actual writing out of " blind animal terror " at the obligation to write an entire novel – a feat which he felt he was " four or five years away from ".

In the account books of Johanna, Duchess of Brabant and Wenceslaus I, Duke of Luxemburg, an entry dated May 14, 1379 reads: " Given to Monsieur and Madame four peters, two forms, value eight and a half moutons, wherewith to buy a pack of cards ".

Given the above expression, evidently the result of her ( local ) measurement is that the three-particle state would collapse to one of the following four states ( with equal probability of obtaining each ):

Given that the four cups are in reference to the four expressions of redemption in Exodus 6: 6-7, some rabbis felt that it was important to include a fifth cup for the fifth expression of redemption in Exodus 6: 8.

# Given any elements a and a ' of G, if a ~ a ', then a < sup >− 1 </ sup > ~ a ' < sup >− 1 </ sup > ( this can actually be proven from the other four, so is strictly redundant ).

Given the choice between playing a game at an arcade three or four times ( perhaps 15 minutes of play for a typical arcade game ), and renting, at about the same price, exactly the same game — for a video game console — the console became the preferred choice.

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.

Given a sample of DNA, the instrument determines the order of the four bases — adenine, guanine, cytosine, and thymine — composing pieces of the sample.

Given that the four trials under consideration can indeed be consistently considered and experimentally realized, the assumptions concerning hidden variables which lead to the indicated mathematical contradiction are therefore collectively unsuitable to represent all experimental results ; namely the assumption of local hidden variables which occur equally in distinct trials.

Given a non-empty totally ordered set R with the following four properties:

The specific problems are: Given two, three or four points on a straight line, find another point on it such that its distances from the given points satisfy the condition that the square on one or the rectangle contained by two has a given ratio either ( 1 ) to the square on the remaining one or the rectangle contained by the remaining two or ( 2 ) to the rectangle contained by the remaining one and another given straight line.

Given how different the four foundational styles of Kajukenbo are it is impossible to fully incorporate everything and some specialization is inevitable.

Given four points in general linear position ( no three collinear ; in particular, no two coincident ), there are exactly three pairs of lines ( degenerate conics ) passing through them, which will in general be intersecting, unless the points form a trapezoid ( one pair is parallel ) or a parallelogram ( two pairs are parallel ).

Given this lengthy support it was reasonable for Tyrconnell and Tyrone to try to solicit help from Philip III, but Spanish policy was to maintain its recent ( 1604 ) peace with England, and its European fleet had anyway been destroyed by the Dutch over four months earlier.

Given four circles with curvatures k < sub > i </ sub > and centers z < sub > i </ sub > ( for i = 1 ... 4 ), the following equality holds in addition to equation ( 1 ):

Given the short distance ( 90 miles ) between Axis airfields on Sicily and the planned drop zones over Malta, it was possible for the motorized transports to make four round-trips per day.

Internazionale scored four times past Given as Newcastle lost 1 – 4 in their opening match in the second group stage.

James Berardinelli gave the film three stars out of four, writing: " Given the expectations that constrain it, Lethal Weapon 4 is probably the best motion picture that could possibly result from another teaming of Martin Riggs and Roger Murtaugh.

Given two actors, oneself and someone else, there are four types of possible behavior directly impacting the welfare of the actors: selfishness, altruism, spite, and cooperation.

Given four operators ( F, G, U, and V ) together with a product operation (∙) defined for any pair of these four operators, and given that the following four pairs of operators commute: