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 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 ".

Given the influence of the temperature of water, as well as the lack of complete adjustment for age, alcohol consumption and smoking, the study concludes that there is limited evidence, but hot mate drinking is " probably carcinogenic to humans ".

Given the dependence of Hittite chronology on Egyptian chronology, a lowering of Egyptian dates would result in a lowering of the end of the Hittite New Kingdom and a resulting reduction ( or complete removal ) of the Anatolian Dark Age.

Given the precarious situation, Hitler moved away from his " no step backward " agenda and agreed to an OKW plan for the complete withdrawal of Army Group G and B.

Given this, sabermetricians generally regard Cy Young's total of 749 complete games as the career baseball record most unlikely to ever be broken.

Given my time again, I wouldn't repeat it-but the great legend is complete, bloody rubbish.

Given a sequence ( X < sub > n </ sub >, p < sub > n </ sub >) of locally compact complete length metric spaces with distinguished points, it converges to ( Y, p ) if for any R > 0 the closed R-balls around p < sub > n </ sub > in X < sub > n </ sub > converge to the closed R-ball around p in Y in the usual Gromov – Hausdorff sense.

Given an arbitrary window in a spatial file manager, it must be possible to determine with complete certainty which folder that window represents.

Given the isolation of this region, the inhabitants tend towards self-sufficiency, and until recently lived in almost complete autarky.

Given the situation there was no reason for the U. S. forces to stay in the field, their mission was complete and arguably a success.

" Given the complete lack of citations to this research in GeoRef, JSTOR, Web of Knowledge, and other scientific bibliographic databases, neither the data collected from nor the interpretations and conclusions made as a result of this expedition very likely have been publicly reported in a scientific venue where their credibility can be openly evaluated.

Given the absolute necessity of a complete text for liturgical use, it is likely that Mozart spliced in movements from his earlier Masses for the premiere.

Given the total mass and the scalars r and v at a single point of the orbit, one can compute the specific orbital energy, allowing an object orbiting a larger object to be classified as having not enough energy to remain in orbit, hence being " suborbital " ( a ballistic missile, for example ), having enough energy to be " orbital ", but without the possibility to complete a full orbit anyway because it eventually collides with the other body, or having enough energy to come from and / or go to infinity ( as a meteor, for example ).

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 the almost complete elimination of 15th century music manuscripts in England, largely by Henry VIII during the Dissolution of the Monasteries in the 1530s, it is not surprising that most of Morton's music survives in sources from the Continent, and if he was ever active as a musician in his native land, all trace is lost.

Given the large number of articles transferred between servers and the large size of individual articles, their complete propagation to any one server farm is not guaranteed.

Given the success of the earlier DVDs, it was hoped that the sales would encourage Sony to produce a final direct-to-DVD movie to complete the original Homefront Campaign story arc, but there have been no new announcements regarding the Roughneck series.

Given the complete name of " Lapis Lazuli ," " Lapis " refers to the time when the protagonist is male and " Lazuli " when female.

Given that the understudy for the part – a member of the choir – was, in fact, a doctor and therefore looking after Allen, his offer was accepted and McCarthy went onstage to complete the performance.

Given their Christian cultural background and high level of European-style education, the new emigrants were less likely to follow the example of the 1492 expellees by settling in the Ottoman Empire, where a complete culture change would be required.

Given the conditions, the organizers declared that each team member only needed to complete a minimum of 1 lap to qualify for a placing ( normally a minimum of 2 laps are required ) and that teams retiring early ( ie before the full 24 hours had elapsed ) would be placed according to the number of laps.

Given a forcing poset P, there is a corresponding complete Boolean algebra B, often obtained as the collection of regular open subsets of P, where the topology on P is generated by cones ( sets of the form

Given a Boolean ring R, for x and y in R we can define

Given a Boolean algebra B, we turn B into a Boolean ring by using the symmetric difference as addition and the meet operation as multiplication.

Given two interior algebras A and B, a map f: A → B is an interior algebra homomorphism if and only if f is a homomorphism between the underlying Boolean algebras of A and B, that also preserves interiors and closures.

Given a topological space X = 〈 X, T 〉 one can form the power set Boolean algebra of X:

Given an interior algebra we can form the Stone representation of its underlying Boolean algebra and then extend this to a topological field of sets by taking the topology generated by the complexes corresponding to the open elements of the interior algebra ( which form a base for a topology ).

Given a set X, one can form the free Boolean algebra A generated by this set and then take its completion B.

Given how these structures interact / collide with each other and with static signals, they can be used to store information as states and implement different Boolean functions.

* 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 a finite dimensional real quadratic space with quadratic form, the geometric algebra for this quadratic space is the Clifford algebra Cℓ ( V, Q ).

Given two Lie algebras and, their direct sum is the Lie algebra consisting of the vector space

Given a vector space V and a quadratic form g an explicit matrix representation of the Clifford algebra can be defined as follows.

Given any vector space V over K we can construct the tensor algebra T ( V ) of V. The tensor algebra is characterized by the fact:

Given a list of operations and axioms in universal algebra, the corresponding algebras and homomorphisms are the objects and morphisms of a category.

A Clifford algebra Cℓ ( V, Q ) is a unital associative algebra over K together with a linear map satisfying for all defined by the following universal property: Given any associative algebra A over K and any linear map such that

So, a collection of functions with given signatures generate a free algebra, the term algebra T. Given a set of equational identities ( the axioms ), one may consider their symmetric, transitive closure E. The quotient algebra T / E is then the algebraic structure or variety.

Given his animosity to infinitesimals it is fitting that the result was couched in terms of algebra rather than analysis.

Given a manifold and a Lie algebra valued 1-form, over it, we can define a family of p-forms:

Given a bounded lattice with largest and smallest elements 1 and 0, and a binary operation, these together form a Heyting algebra if and only if the following hold:

Given a set with three binary operations and, and two distinguished elements 0 and 1, then is a Heyting algebra for these operations ( and the relation defined by the condition that when ) if and only if the following conditions hold for any elements and of:

* Given the action of a Lie algebra g on a manifold M, the set of g-invariant vector fields on M is a Lie algebroid over the space of orbits of the action.

Given an algebra, a homomorphism h thus defines two algebras homomorphic to, the image h () and The two are isomorphic, a result known as the homomorphic image theorem.

Let X be any Lie algebra over K. Given a unital associative K-algebra U and a Lie algebra homomorphism: h: X → U < sub > L </ sub >, ( notation as above ) we say that U is the universal enveloping algebra of X if it satisfies the following universal property: for any unital associative K-algebra A and Lie algebra homomorphism f: X → A < sub > L </ sub > there exists a unique unital algebra homomorphism g: U → A such that: f (-) = g < sub > L </ sub > ( h (-)).