: 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 a set of integers, does some nonempty subset of them sum to 0?

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 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 a trigonometric series f ( x ) with S as its set of zeros, Cantor had discovered a procedure that produced another trigonometric series that had S ' as its set of zeros, where S ' is the set of limit points of S. If p ( 1 ) is the set of limit points of S, then he could construct a trigonometric series whose zeros are p ( 1 ).

Given a topological space X, let G < sub > 0 </ sub > be the set X.

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 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 ★ on a set S, an element x is said to be idempotent ( with respect to ★) if

Given a set

The knapsack problem or rucksack problem is a problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible.

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 simple mathematical or functional description of an input or output to a system, the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications.

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 a set of training examples of the form, a learning algorithm seeks a function, where is the input space and

Given a specific task to solve, and a class of functions, learning means using a set of observations to find which solves the task in some optimal sense.

Given a point x in a topological space, let N < sub > x </ sub > denote the set of all neighbourhoods containing x.

* 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

** Closure axiom for addition: Given two integers a and b, their sum, a + b is also an integer.

** Closure axiom for multiplication: Given two integers a and b, their product, a · b is also an integer.

** Associativity of multiplication: Given any integers, a, b and c, ( a · b ) · c

Given a set of integers, FIND-SUBSET-SUM is the problem of finding some nonempty subset of the integers that adds up to zero ( or returning the empty set if there is no such subset ).

Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers.

Given an integer k, one defines the residue class of an integer n as the set of all integers congruent to n modulo k:

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 two large integers, a and b, Toom – Cook splits up a and b into k smaller parts each of length l, and performs operations on the parts.

: Given a set S of n integers, are there elements a, b, c in S such that a + b + c = 0?

Given two integers a and b, with b ≠ 0, there exist unique integers q and r such that a = bq + r and 0 ≤ r < | b |, where | b | denotes the absolute value of b.

The 1st generalized division algorithm: Given integers,, with, there exist unique integers and with < math > d

Given a finite set X ( of elements called points ) and integers k, r, λ ≥ 1, we define a 2-design ( or BIBD, standing for balanced incomplete block design ) B to be a family of k-element subsets of X, called blocks, such that the number r of blocks containing x in X is not dependent on which x is chosen, and the number λ of blocks containing given distinct points x and y in X is also independent of the choices.

Given a discrete set of real or complex numbers: ( integers ), the discrete-time Fourier transform ( or DTFT ) of is usually written:

) Given the integers a, b and n, the expression a ≡ b ( mod n ) ( pronounced " a is congruent to b modulo n ") means that a − b is a multiple of n, or equivalently, a and b both leave the same remainder when divided by n. For more details, see modular arithmetic.

Given two positive integers N and i, there is a unique way to expand N as a sum of binomial coefficients as follows:

Given a Dynkin diagram X, Chevalley constructed a group scheme over the integers Z whose values over finite fields are the Chevalley groups.

: Given positive integers a < sub > 1 </ sub >, a < sub > 2 </ sub >, ..., a < sub > n </ sub > such that gcd ( a < sub > 1 </ sub >, a < sub > 2 </ sub >, ..., a < sub > n </ sub >) = 1, find the largest integer that cannot be expressed as an integer conical combination of these numbers, i. e., as a sum

Given the similarities in the two characters ' names, professions, written works and generally dark subject matter, it is likely that Lovecraft's Alhazred provided the main inspiration for al-Hazir.

Given a field F, the assertion “ F is algebraically closed ” is equivalent to other assertions:

* Given an algebraic number, there is a unique monic polynomial ( with rational coefficients ) of least degree that has the number as a root.

" Given that Thomas Aquinas died six years before Albertus Magnus ' death, this legend as stated is unlikely.

Given Ealdred's known support of Godwin's family, John of Worcester is probably correct.

* 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 an R-module M, the endomorphism ring of M, denoted End < sub > R </ sub >( M ) is an R-algebra by defining ( r · φ )( x ) = r · φ ( x ).

Given the other ZF axioms, the axiom of regularity is equivalent to the axiom of induction.

After Christians in Ephesus first wrote to their counterparts recommending Apollos to them, he went to Achaia where Paul names him as an apostle ( 1 Cor 4: 6, 9-13 ) Given that Paul only saw himself as an apostle ' untimely born ' ( 1 Cor 15: 8 ) it is certain that Apollos became an apostle in the regular way ( as a witness to the risen Lord and commissioned by Jesus-1 Cor 15: 5-9 ; 1 Cor 9: 1 ).< ref > So the Alexandrian recension ; the text in < sup > 38 </ sup > and Codex Bezae indicate that Apollos went to Corinth.

Given its " blue-water " bias, the navy is even less inclined to become involved in counterdrug operations than the army or air force.

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 this, the bidding is said to start at the one-level when contracting for a total of seven tricks, at the two-level for eight tricks and so on to the seven-level to contract to take all thirteen tricks.

While tests have been done to support this with fixed pieces of flat metal, the result is inconclusive with respect to the curved armour of the period. Given the following actions of the archers, it seems likely Froissart was correct.

Given a ket of norm 1, the orthogonal projection onto the subspace spanned by is

Given this influence, it is unfortunate that many of the details of his life remain shrouded in mystery, perhaps forever ; even the only known picture of him, shown above, is heavily retouched, with a fake tie painted in by hand.

Given points P < sub > 0 </ sub > and P < sub > 1 </ sub >, a linear Bézier curve is simply a straight line between those two points.

Given the then GDP ($ 7. 095 bln ) of the country, military spending is roughly estimated to be about $ 300 million.

Given some encrypted data (" ciphertext "), the goal of the cryptanalyst is to gain as much information as possible about the original, unencrypted data (" plaintext ").

Given the federal nature of the holiday, celebrating Canada Day can be a cause of friction in the province of Quebec, where the holiday is overshadowed by Quebec's National Holiday, on June 24.

Given that many journeys are for relatively short distances, there is considerable scope to replace car use with walking or cycling, though in many settings this may require some infrastructure modification, particularly to attract the less experienced and confident.

Given such a sweeping definition, it is apparent that cognition is involved in everything a human being might possibly do ; that every psychological phenomenon is a cognitive phenomenon.