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Page "Binary function" ¶ 12
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Given and any
: 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: XX 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 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 XX.
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 and element
Given a set S with a partial order ≤, an infinite descending chain is a chain V that is a subset of S upon which ≤ defines a total order such that V has no least element, that is, an element m such that for all elements n in V it holds that m ≤ n.
* Given a binary operation, an idempotent element ( or simply an idempotent ) for the operation is a value for which the operation, when given that value for both of its operands, gives the value as the result.
Given a binary operation ★ on a set S, an element x is said to be idempotent ( with respect to ★) if
Given a basis of a vector space, every element of the vector space can be expressed uniquely as a finite linear combination of basis vectors.
Given an element φ of H *, the orthogonal complement of the kernel of φ is a one-dimensional subspace of H. Take a non-zero element z in that subspace, and set.
Given a natural transformation Φ from h < sup > A </ sup > to F, the corresponding element of F ( A ) is.
# Given any element a of G, a ~ a ( reflexivity );
: Given any set A, there is a set B such that, for any element c, c is a member of B if and only if there is a set D such that c is a member of D and D is a member of A.
Given the slow speed and the coarse resolution this was not a feasible technique for printing large images, but could usefully print a small logo onto a letterhead and then the following letter, all in a single unattended print run without changing the print element.
Given any group G, the group consisting of only the identity element is a trivial group and being a subgroup of G is called the trivial subgroup of G.
Given a set S of matrices, each of which is diagonalizable, and any two of which commute, it is always possible to simultaneously diagonalize all of the elements of S. Equivalently, for any set S of mutually commuting semisimple linear transformations of a finite-dimensional vector space V there exists a basis of V consisting of simultaneous eigenvectors of all elements of S. Each of these common eigenvectors v ∈ V, defines a linear functional on the subalgebra U of End ( V ) generated by the set of endomorphisms S ; this functional is defined as the map which associates to each element of U its eigenvalue on the eigenvector v. This " generalized eigenvalue " is a prototype for the notion of a weight.
Given a function and an element g ∈ G,
* Given a maximal torus T in G, every element g ∈ G is conjugate to an element in T.
Given an integral domain, let be an element of, the polynomial ring with coefficients in.
Given an element of a disjoint union A + B, it is possible to determine whether it came from A or B.
Given then a normal extension L of K, with automorphism group Aut ( L / K ) = G, and containing α, any element g ( α ) for g in G will be a conjugate of α, since the automorphism g sends roots of p to roots of p. Conversely any conjugate β of α is of this form: in other words, G acts transitively on the conjugates.
Given a partially ordered class P, we can form a category with P's elements as objects and a single arrow going from one object ( element ) to another if the first is less than or equal to the second.
Given a cyclic group of order, a generator of the group and a group element, the problem is to find an integer such that
Given a subset A of G, the measure can be thought of as answering the question: what is the probability that a random element of G is in A?

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