: 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 a set of points in the Euclidean plane, selecting any one of them to be called 0 and another to be called 1, together with an arbitrary choice of orientation allows us to consider the points as a set of complex numbers.

Given a set of points in Euclidean space, the first principal component corresponds to a line that passes through the multidimensional mean and minimizes the sum of squares of the distances of the points from the line.

Given a vector a in Euclidean space R < sup > n </ sup >, the formula for the reflection in the hyperplane through the origin, orthogonal to a, is given by

The original problem was stated in the form that has become known as the Euclidean Steiner tree problem or geometric Steiner tree problem: Given N points in the plane, the goal is to connect them by lines of minimum total length in such a way that any two points may be interconnected by line segments either directly or via other points and line segments.

Given the Cantor – Dedekind axiom, this algorithm can be regarded as an algorithm to decide the truth of any statement in Euclidean geometry.

Given straight lines l and m, the following descriptions of line m equivalently define it as parallel to line l in Euclidean space:

Given a fixed oriented line L in the Euclidean plane R < sup > 2 </ sup >, a meander of order n is a non-self-intersecting closed curve in R < sup > 2 </ sup > which transversally intersects the line at 2n points for some positive integer n. Two meanders are said to be equivalent if they are homeomorphic in the plane.

Given an element a and a non-zero element b in a Euclidean domain R equipped with a Euclidean function d, there exist q and r in R such that and either or.

The realization problem for Euclidean minimum spanning trees is stated as follows: Given a tree T = ( V, E ), find a location D ( u ) for each vertex u ∈ V so that T is a minimum spanning tree of D ( u ): u ∈ V, or determine that no such locations exist.

Given two positive numbers, ( the dividend ) and ( the divisor ), a modulo n ( abbreviated as a mod n ) is the remainder of the Euclidean division of a by n. For instance, the expression " 5 mod 2 " would evaluate to 1 because 5 divided by 2 leaves a quotient of 2 and a remainder of 1, while " 9 mod 3 " would evaluate to 0 because the division of 9 by 3 has a quotient of 3 and leaves a remainder of 0 ; there is nothing to subtract from 9 after multiplying 3 times 3.

Given any natural numbers l, m, n > 1 exactly one of the classical two-dimensional geometries ( Euclidean, spherical, or hyperbolic ) admits a triangle with the angles ( π / l, π / m, π / n ), and the space is tiled by reflections of the triangle.

Given an imbedding of M in Euclidean space E, we set

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 triangle ABC, the following conditions would need to be fulfilled for the case to be ambiguous:

Given a triangle ABC, let the lines AO, BO and CO be drawn from the vertices to a common point O to meet opposite sides at D, E and F respectively.

* Given a triangle OAB in which O is the center of a circle k, and points A ' and B ' inverses of A and B with respect to k, then

Given a fundamental domain for the group action ( for the full, orientation-reversing symmetry group, a ( 2, 3, 7 ) triangle ), the reflection domains ( images of this domain under the group ) give a tiling of the quartic such that the automorphism group of the tiling equals the automorphism group of the surface – reflections in the lines of the tiling correspond to the reflections in the group ( reflections in the lines of a given fundamental triangle give a set of 3 generating reflections ).

Given a triangle ABC, and a triangulation T of the triangle.

Given a counterfactual conditional, e. g., ' If there had been a circle on the blackboard then there would have been a triangle ', and the subsequent information ' in fact there was no triangle ', participants make the modus tollens inference ' there was no circle ' more often than they do from an indicative conditional ( Byrne and Tasso, 1999 ).

Given a unit sphere, a " triangle " on the surface of the sphere is defined by the great circles connecting three points u, v, and w on the sphere.

Given a point inside a triangle it is also desirable to obtain the barycentric coordinates, and at this point.

Given a triangle ABC and its three Malfatti circles, let D, E, and F be the points where two of the circles touch each other, opposite vertices A, B, and C respectively.

Given a triangle ABC, and a transversal line that crosses BC, AC and AB at points D, E and F respectively, with D, E, and F distinct from A, B and C, then

Given An equilateral triangle inscribed on a circle and a point on the circle.

The chords are of considerable historical importance because, along with the sides of the triangle and tetragon ( square ), they enable the generation of a table of half chords ( effectively sine values ) Given the unit circle with sector ABO subtending an arc of, we may write − a relationship expressed in words by Copernicus:

Given any triangle ABC, and any point M on BC, construct the incircle and circumcircle of the triangle.

The has a crossing point at the origin if there is a triangle with sides a, b and c. Given the previous conditions, this means that the curve crosses the origin if and only if b < a + c.