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

Given a particular query, the DBMS ' query optimizer is responsible for devising the most efficient strategy for finding matching data.

Given a particular query, the DBMS ' query optimizer is responsible for devising the most efficient strategy for finding matching data.

Given this kind of data, the estimated coefficient on Years of Education in the equation above reflects both the effect of education on wages and the effect of other variables on wages, if those other variables were correlated with education.

Given the widespread expansion in the tenth revision, it is not possible to convert ICD-9 data sets directly into ICD-10 data sets, although some tools are available to help guide users.

Given the repeated failures and frustrations of workers ' revolutions and movements, Marx also sought to understand capitalism, and spent a great deal of time in the reading room of the British Museum studying and reflecting on the works of political economists and on economic data.

Given fixed resources, it is often better to spend more time collecting additional training data and more informative features than it is to spend extra time tuning the learning algorithms.

Given a model of some atomic positions, these positions and their respective Debye-Waller factors ( or B-factors, accounting for the thermal motion of the atom ) can be refined to fit the observed diffraction data, ideally yielding a better set of phases.

Given that radio communication channels are allocated by agencies such as the Federal Communication Commission giving a prescribed ( maximum ) bandwidth, the advantage of QPSK over BPSK becomes evident: QPSK transmits twice the data rate in a given bandwidth compared to BPSK-at the same BER.

Given the CIA's purposeful destruction of most records, its failure to follow informed consent protocols with thousands of participants, the uncontrolled nature of the experiments, and the lack of follow-up data, the full impact of MKUltra experiments, including deaths, will never be known.

Given inconsistencies and the lack of reliable data and completely contradicting and unpredictable behavior, the true nature of ball lightning is still unknown and was often regarded as a fantasy or a hoax.

Given some training data, a set of n points of the form

Formally, the problem of supervised pattern recognition can be stated as follows: Given an unknown function ( the ground truth ) that maps input instances to output labels, along with training data assumed to represent accurate examples of the mapping, produce a function that approximates as closely as possible the correct mapping.

Given a table of costs and revenues at each quantity, we can either compute equations or plot the data directly on a graph.

Given a grid point field of geopotential height, storm tracks can be visualized by contouring its average standard deviation, after the data has been band-pass filtered.

* Point location: Given a partitioning of the space into cells, produce a data structure that efficiently tells in which cell a query point is located.

* Ray tracing: Given a set of objects in space, produce a data structure that efficiently tells which object a query ray intersects first.

Given inconsistencies and the lack of reliable data, the true nature of ball lightning is still unknown.

Given a set of n + 1 data points ( x < sub > i </ sub >, y < sub > i </ sub >) where no two x < sub > i </ sub > are the same, one is looking for a polynomial p of degree at most n with the property

Given a set of k + 1 data points

Given a set of k + 1 data points

Given the shortcomings of the beneficial acclimation hypothesis, researchers are continuing to search for a theory that will be supported by empirical data.

Given a set of objects in d-dimensional space, preprocess them into a data structure so that for each query ray the first object hit by the ray can be found quickly.