: 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 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 set of integers, SUBSET-SUM is the problem of finding whether there exists a subset summing to zero.

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 that this etymology does not appear in the usual scholarly works on etymology, this claim is usually dismissed as a false etymology.

Given a syscall that does nothing, a full round-trip under BSD would require about 40μs, whereas on a user-space Mach system it would take just under 500μs.

Given unlimited resources, a classical computer can simulate an arbitrary quantum algorithm so quantum computation does not violate the Church – Turing thesis.

Given a functor U and an object X as above, there may or may not exist an initial morphism from X to U. If, however, an initial morphism ( A, φ ) does exist then it is essentially unique.

Given an object X, a functor G ( taking for simplicity the first functor to be the identity ) and an isomorphism proof of unnaturality is most easily shown by giving an automorphism that does not commute with this isomorphism ( so ).

Given the well-publicized hoopla, some sources still consider this as a nomination for Davis ; however, the Academy does not officially record this as a nomination.

:: See Committee Note to Interim Rule 8001 ( f ) (" Given the short time limit to file the petition with the circuit clerk, subdivision ( f )( 1 ) provides that entry of a certification on the docket does not occur until an effective appeal is taken under Rule 8003 ( a ) or ( b ).

Given real life prisons are same-sex communities, this fetish does lend itself to male on male or female on female activities and settings.

Given: that Race B, in its present condition, does not develop fast enough to suit Race A.

Given the lack of concrete historical knowledge about one of the most potent figures in British mythology, it is unlikely that any definitive conclusions about the claims for these places will ever be established ; nevertheless it is both interesting and important to try to evaluate the body of evidence which does exist and examine it critically.

Given the time reversal operator T, it does nothing to the x-operator, TxT < sup >− 1 </ sup > =

Given two points A and B, with A not lower than B, there is just one upside down cycloid that passes through A with infinite slope, passes also through B and does not have maximum points between A and B.

Given the extensive linguistic, religious and ethnic diversity of the Indian population, nationalism in India in general does not fall within the purview of a solitary variant of nationalism.

Given the test scores of two random samples of men and women, does one group differ from the other?

Given a function w on U × Y, with finite integral of its modulus for any input function u and initial state x ( 0 ) over any finite time t, called the " supply rate ", a system is said to be dissipative if there exist a continuous nonnegative function V ( x ), with x ( 0 ) = 0, called the storage function, such that for any input u and initial state x ( 0 ) the difference V ( x ( t )) − V ( x ( 0 )) does not exceed the integral of the supply over ( 0, t ) for any t ( dissipation inequality ).

#( Affine axiom of parallelism ) Given a point A and a line r, not through A, there is at most one line through A which does not meet r.

Given that " Pegasus does not exist " is true, it follows that the referent of " Pegasus " does not exist.

Given the tendency of political parties in India to make and break alliances frequently, the National Democratic Alliance does not have a formal governing structure in place, such as an executive board or politburo.

Given the facts that solar thermal power is reliable, can deliver peak load and does not cause pollution, a price of US $ 0. 10 per kWh starts to become competitive, although a price of US $ 0. 06 has been claimed With some operational cost a simple target is 1 dollar ( or lower ) investment for 1 kWh production in a year.

Given this symmetry, it is puzzling that the universe does not have equal amounts of matter and antimatter.

Given the recency with which spoken language has emerged, it is very likely that human body language does include some more or less involuntary responses that have a similar origin to the communication we see in other animals.

Given that his reign lasted for, at best, only a few weeks after his acclamation and he does not seem to have secured significant military or political support Domitianus is more properly categorized as a Roman Usurper rather than an Emperor.

Given this preexisting pattern, having a preference for longer and more colorful tails gives an advantage just as having a longer and more colorful tail does.