: 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 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 a recursively enumerable set A of positive integers that has insoluble membership problem, ⟨ a, b, c, d | a < sup > n </ sup > ba < sup > n </ sup > = c < sup > n </ sup > dc < sup > n </ sup >: n ∈ A ⟩ is a finitely generated group with a recursively enumerable presentation whose word problem is insoluble

Given a Hilbert space L < sup > 2 </ sup >( m ), m being a finite measure, the inner product < ·, · > gives rise to a positive functional φ by

Given the abundance of such optimization problems in everyday life, a positive answer to the " P vs. NP " question would likely have profound practical and philosophical consequences.

Given an encoding of the known background knowledge and a set of examples represented as a logical database of facts, an ILP system will derive a hypothesised logic program which entails all the positive and none of the negative examples.

Given the ( period, cash flow ) pairs (, ) where is a positive integer, the total number of periods, and the net present value, the internal rate of return is given by in:

Given that the absolute vorticity is positive ( negative ) in the Northern ( Southern ) hemisphere, the threshold value should be taken as positive ( negative ) north ( south ) of the equator.

Given any neighborhood of, we can choose a small positive and a small of either sign to get values both greater and less than.

Given names are chosen based on a range of factors, including possession of pleasing sound and tonal qualities, as well as bearing positive associations or a beautiful shape.

" Dietrich Bonhoeffer of the German Confessing Church framed the same characterization in less positive terms when he called Pietism the last attempt to save Christianity as a religion: Given that for him religion was a negative term, more or less an opposite to revelation, this constitutes a rather scathing judgment.

Given a sequence of positive integers, the Gödel encoding of the sequence is the product of the first n primes raised to their corresponding values in the sequence:

Given an encoding of the known background knowledge and a set of examples represented as a logical database of facts, an ILP system will derive a hypothesized logic program which entails all the positive and none of the negative examples.

* Given a positive real number ε, an ε-isometry or almost isometry ( also called a Hausdorff approximation ) is a map between metric spaces such that

Given the prevalence of disapproval as a tool of government, including the criminal law and diplomatic relations, some fail to see voting as a positive and voluntary choice of a desirable outcome.

* Given a positive integer n, the set of all positive divisors of n forms a distributive lattice, again with the greatest common divisor as meet and the least common multiple as join.

Given an appropriately designed spam filtering algorithm, outbound spam filtering can be implemented with a near zero false positive rate, which keeps customer related issues with blocked legitimate email down to a minimum.

Given a positive integer n, it is not at all a routine matter to determine how many isomorphism types of groups of order n there are.

Given that bankruptcies and real estate prices did not fare as negatively in Central Canada as in the rest of Canada and the United States during the NEP, it is possible that the NEP had a positive effect in Central Canada.

Given the integral representation of a principal branch of γ, the following equation holds for all positive real s, x:

Given two nondecreasing positive functions a and b one can say that

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.