: 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 his generous notion of what constituted an Aryan race ( Evola was convinced of the Hyperborean origins of most Europeans, the indigenous peoples of North and South America, as well as those of the Indian subcontinent ), those candidate races Evola considered to be truly ' alien ' were never explicitly catalogued — except in terms of Semites and the deeply pigmented peoples of sub-Saharan Africa ( see Evola, ' Psicologia criminale ebraica ,' Difesa della Razza 2, no.

Given a 200-seat Chamber and an 81-seat Senate, a successful first round candidate requires 101 deputies and 41 senators.

Given his increasing profile and popularity within the party due to his role as the manifesto chair, he was seen as a probable candidate for the social liberal grouping's vote in the future leadership election.

Given the grievances with ruling regime in the Northeast and Rio Grande do Sul, Liberal Alliance opposition leader Getúlio Vargas chose as his vice-presidential candidate in 1930 João Pessoa of the Northeast state of Paraíba.

Given the largely pro-China status of the Election Committee, it became common for the favoured candidate of the Chinese government to get elected unopposed.

Given such a specification, it is possible to use formal verification techniques to demonstrate that a candidate system design is correct with respect to the specification.

Given the crowded field, many believed a runoff election was likely between the top two finishers — which polls suggested would be Beshear and Lunsford — if no candidate was able to obtain at least 40 % of the vote.

Given the complexity of the calculations involved and the convoluted structure that a convertible bond can have, an arbitrageur often relies on sophisticated quantitative models in order to identify bonds that are trading cheap versus their theoretical value.

Given the bias of the Victorian age, female students were denied direct study in anatomy and could not attend drawing classes with live models ( who were often prostitutes ) until a decade later.

Given the many available models, it appears that the material removal mechanism in EDM is not yet well understood and that further investigation is necessary to clarify it, especially considering the lack of experimental scientific evidence to build and validate the current EDM models.

Given a desired backbone conformation, side chain packing can be designed using variations of the dead-end elimination algorithm ; however, attempts to design proteins of novel folds have difficulty using this method due to an absence of plausible backbone models.

:" Given that the structure of an econometric model consists of optimal decision rules of economic agents, and that optimal decision rules vary systematically with changes in the structure of series relevant to the decision maker, it follows that any change in policy will systematically alter the structure of econometric models.

Given a model selection problem in which we have to choose between two models, on the basis of observed data D, the plausibility of the two different models M < sub > 1 </ sub > and M < sub > 2 </ sub >, parametrised by model parameter vectors and is assessed by the Bayes factor K given by

Given the model designation VQ, these new luxury models utilised the long-wheelbase chassis taken from the VN Commodore station wagon, as opposed to the VN sedan's short-wheelbase.

Given a surface temperature of 7, 740 Kelvin, stellar models yield a total luminosity for the star of about 17 times the luminosity of the Sun.

Given their age difference, Hoffman considered his brothers more role models than playmates.

Given this success, it may come as a nasty shock that it seems to be very difficult, mathematically speaking, to construct rotating stellar models in which a perfect fluid interior is matched to an asymptotically flat vacuum exterior.

Given their lighter weight, Pony models were known for their ability to deliver excellent fuel economy — mileage upwards of in highway driving was not uncommon.

Given any two estimated models, the model with the lower value of BIC is the one to be preferred.

Given a propositional formula, its circumscription is the formula having only the models of that do not assign a variable to true unless necessary.

Given two models and represented this way, the condition is equivalent to setting to true every variable that sets to true.

Given a first-order logic formula containing a predicate, circumscribing this predicate amounts to selecting only the models of in which is assigned to true on a minimal set of tuples of values.