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Combinatorics
Published in Sriraman Sridharan, R. Balakrishnan, Foundations of Discrete Mathematics with Algorithms and Programming, 2019
Sriraman Sridharan, R. Balakrishnan
If A and B are two given sets then the Cartesian product or simply product of A and B is denoted by A×B $ A\times B $ and defined as the set of all ordered pairs (a, b) with a∈A $ a\in A $ and b∈B $ b\in B $ . Symbolically, A×B={(a,b)∣a∈A,b∈B} $$ A\times B=\{\,(a,b)\mid a\in A, b\in B\,\} $$
Introduction to Sets and Relations
Published in Richard L. Shell, Ernest L. Hall, Handbook of Industrial Automation, 2000
By restricting the choice of coordinates of ordered pairs to elements of given sets, we arrived at the concept of Cartesian products. The name is derived from the classical method of determining the coordinates of a point in the plane by the French mathematician Rene Descartes (1596–1650). Given two sets A and B, the set of all ordered pairs (x, y) with x ∈ A and y ∈ B, is called the Cartesian product of A and B and it is denoted A × B. In symbols, A×B={(x,y):x∈Aand y∈B}
A tool for modeling systems
Published in William L. Chapman, A. Terry Bahill, A. Wayne Wymore, Engineering Modeling and Design, 2018
William L. Chapman, A. Terry Bahill, A. Wayne Wymore
The Cartesian product of sets is an abbreviated way of writing all possible combinations of elements using one element from each set. The Cartesian product of Set A with Set B is expressed as A × B. If A is {a1,a2,a3} and B is {b1 ,b2}, then
Vagueness as an epiphenomenon, and non-transitivity
Published in Journal of Applied Non-Classical Logics, 2022
Finally, is a function such that: to each ordered pair , where a is an object belonging to W and b an n-ary relation symbol, it assigns a subset of the Cartesian product to each ordered pair , where a is an object belonging to W and b a constant symbol, it assigns an object in Dto each ordered pair , where a an object belonging to W and b an object in D, it assigns b
Nonlinear separation methods and applications for vector equilibrium problems using improvement sets
Published in Applicable Analysis, 2021
Yang Dong Xu, Cheng Yu Lei, Chun Yu Sun
Now, we recall the problem of a bicriteria strategic game described in [33]. The bicriteria strategic game is a tuple , where N is the set of players, is the strategy set for player is the Cartesian product of the strategy sets , and is the utility function for player i. Let . By [27, Definition 3.4], is called an ε-Pareto equilibrium of the bicriteria strategic game Υ if, for each , where . Here, the set of ε-Pareto best answer to is the set of such that where . For each , set for all , where
On extending and optimising the direct product decomposition
Published in Molecular Physics, 2019
The basic observation that leads to a reduction in computational cost when point group symmetry is taken into account is that a generic integral, e.g. is only non-zero when certain symmetry constraints are met. The symmetry properties of wave functions, operators, and other objects are classified by their representation, denoted as for example. These representations may then be combined using the direct product operation, (not to be confused with the outer or Cartesian product). The direct product of one or more representations is itself a representation. When this overall representation contains (i.e. has non-zero projection onto) a special representation called the totally-symmetric representation, then the integral is allowed to be non-zero. Thus, any contributions to the various components of the integral which are known to lead to only non-symmetric products may be discarded. The earliest uses of this result (in LCAO-type calculations at least) used the symmetry properties of the atomic orbital basis to screen out integrals that are known to be zero or redundant, for example using the so-called ‘petite list’ approach [5]. This approach was and continues to be popular in many AO-direct (meaning that integrals are computed and consumed on-the-fly instead of being stored) computations such as SCF and CI, including multi-reference variants [6].