Euclid poses the problem: " Given two numbers not prime to one another, to find their greatest common measure ".

: 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 one cannot be motivated by reason alone, requiring the input of the passions, Hume argued that reason cannot be behind morality.

Given the limited success of the Charity School, however, Wheelock intended his new College as one primarily for whites.

Given also a measure on set, then, sometimes also denoted or, has as its vectors equivalence classes of measurable functions whose absolute value's-th power has finite integral, that is, functions for which one has

Given good circumstances one might be able to discern the result of some human activity such as the changing of the Netherlands ' coast or the partial drying out of the Aral Sea, but even that would not be easy.

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 our formula φ, we group strings of quantifiers of one kind together in blocks:

Given an equilateral triangle, the counterclockwise rotation by 120 ° around the center of the triangle " acts " on the set of vertices of the triangle by mapping every vertex to another one.

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 a complete set of axioms ( see below for one such set ), modus ponens is sufficient to prove all other argument forms in propositional logic, and so we may think of them as derivative.

Given significant distance from the magnetic poles, one can figure which hand is which using a magnetic compass and the sun.

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 the same set of verifiable facts, some societies or individuals will have a fundamental disagreement about what one ought to do based on societal or individual norms, and one cannot adjudicate these using some independent standard of evaluation.

Given an arithmetic function, one can generate a bi-infinite sequence of other arithmetic functions by repeatedly applying the first summation.

Given a field ordering ≤ as in Def 1, the elements such that x ≥ 0 forms a positive cone of F. Conversely, given a positive cone P of F as in Def 2, one can associate a total ordering ≤< sub > P </ sub > by setting x ≤ y to mean y − x ∈ P. This total ordering ≤< sub > P </ sub > satisfies the properties of Def 1.

Given the above fact, one can ask:

Given a preorder on S one may define an equivalence relation ~ on S such that a ~ b if and only if a b and b a.

# Given any two distinct points, there is exactly one line incident with both of them.

# Given any two distinct lines, there is exactly one point incident with both of them.

# Given an array of values to be sorted, set up an auxiliary array of initially empty " pigeonholes ," one pigeonhole for each key through the range of the original array.

Given the unusual casting and production demands, Pacific Overtures remains one of the least-performed musicals by Stephen Sondheim.

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 the situation in Spain for several years regarding terrorist attacks, these ties can be considered objectively as a threat to democracy ," the court said, also adding that these parties " contradicted the concept of a ' democratic society ' and presented a major danger to Spain's democracy ".

Given this connection, she suggests the major war should have been called " The Great Cheyenne War ".

Given glam metal's lack of a major format presence on radio, bands were left without a clear way to reach their audience.

Given that the high latitudes in this period were warm and tropical, it is hypothesized that the deciduous habit evolved in response to the unusual light availability patterns, not to major seasonal variations in temperature.

Given its widespread touch on the entirety of a major city, it's dubbed one of, along with the Old Firm, the Rome derby and the Istanbul derby, the most heated rivalries in European football.

Given since 1940, this prize recognizes major contributions to reproductive biology.

Given his major interests were domestic, and that the war interfered with his domestic focus, he was eager to free his programs from the war — if he could find a way that he considered politically acceptable.

Also Given the fact that the neighbourhood is well maintained, it can be assumed that the residents are very proud of where they live and thus major changes in the future are very unlikely.

Given that duels were generally fought at short distances ( was typical ) between stationary opponents, accuracy was not a major problem.

Given their perceived piety, The Times noted the irony when a major investigation by their reporters uncovered a link between Islamic Jihadis and child pornography ; a discovery that, according to the London paper, " is expected to improve understanding of the mindsets of both types of criminals and has been hailed as a potentially vital intelligence tool to undermine future terrorist plots .".

Given the unusually high demand for his performances, his health deteriorated and worsened by an episode of a major stroke in the late 1950s, he died on 1 May 1965.

Given Japan had surrendered to the Allies at the end of World War II, the Japanese forces in Indochina stepped aside and allowed nationalist groups to take over public buildings in most of the major cities.

Given that California's automotive market is one of the largest in the world, CARB wields enormous influence over the emissions requirements that major automakers must meet if they wish to sell into that market.

Given how critical transportation is to every human culture, it is unsurprising that in an interstellar culture, which must deal with distances orders of magnitude greater than terrestrial cultures, the unique ways in which interstellar travel is described in various fictional universes tends to create major plot elements in that universe.

Given Goldman's major investments in California state finances, all these ties to Goldman Sachs led to considerable controversy during the gubernatorial campaign.

Given a major key tonic ( I ), the related keys are:

Given so few bits to work with, one major design challenge of RDS-TMC was to find a way of using 16 bits ( about 65000 locations ) to describe an entire state or country.

Given his small number of games for Queensland, Healy ’ s selection for the Australian team to tour Pakistan in late 1988 was a major surprise.

Given its size, and the access to a major port as well as Binh Dinh's major river, it has long been the place where most of Binh Dinh's people and economic activities concentrate.

Given that Liverpool was a major trans-Atlantic port and due to the building's proximity to the River Mersey, the lower floors of the Cunard Building were allocated to provide space for cruise liner passengers, both prior to and after sailing.

Given the deadline for universal access set by world leaders, AIDS 2010 coincided with a major push for expanded access to HIV prevention, treatment, care and support.

Given the strategic and ( usually ) tactical importance of control of the center, the pawn is important enough that the player must consistently defend it, but with no pawns available for its defense, the player must " tie down " a minor or major piece ( in the opening, usually the king's knight or the queen ) with that task.

Given the high spectacularity of the freeride and downhill action, an increasing number of films are released every year, covering the annual major competitions and promoting particular trails and champions.