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Introduction to Graph Theory
Published in Sriraman Sridharan, R. Balakrishnan, Foundations of Discrete Mathematics with Algorithms and Programming, 2019
Sriraman Sridharan, R. Balakrishnan
Intuitively, a graph is a diagram consisting of a finite set of points (called vertices) together with a finite set of arrows, each arrow joining a certain ordered pair of vertices. Here, vertices and arrows are undefined terms like points and lines in geometry. Many real-world situations can be abstracted by means of the concept of the graph. Graphs can be used as mathematical models for systems involving binary relations. A mathematical model is a reflection of some real-world situation. A model must mirror certain essential features of a real situation, but not all. A mathematical model is also a simplification that allows us to study the relevant properties and ignore the others. A model can be smaller or larger or roughly the same size as the thing it represents.
Introduction to Sets and Relations
Published in Richard L. Shell, Ernest L. Hall, Handbook of Industrial Automation, 2000
Order relations constitute another common type of relations. Once again, we begin by introducing several definitions. A binary relation ℜ in X is said to be antisymmetrical if for all x, y ∈ X, xℜy and yℜx imply x = y.A binary relation ℜ in X is asymmetrical if for any x, y ∈ X, xℜy implies that yℜx does not hold. In other words, we can not have xℜy an yℜx both true.A binary relation ℜ in X is a partial ordering of X if and only if it is reflexive, antisymmetrical, and transitive. The pair (X, ℜ) is called and ordered set.A binary relation in X is a strict (or total) ordering of X if and only if it is asymmetrical and transitive.
Basic concepts of systemics
Published in Alfredo Pereira, William Alfred Pickering, Ricardo Ribeiro Gudwin, Systems, Self-Organization and Information, 2018
Ettore Bresciani Filho, Itala M. Loffredo D’Ottaviano
An n-ary relation on a given set is any subset whose elements are finite (n-uple) sequences of elements of the set. As a particular case, binary relations correspond to subsets of the set of ordered pairs of elements of the initial set; if an ordered pair belongs to a binary relation, it is said that the pair satisfies the given relation – the first element of the pair is in relation to the second.
Jaccard matrix for nonlinear filter statistics
Published in SICE Journal of Control, Measurement, and System Integration, 2023
On the other hand, as the context from a general view of mathematical formulation, we pay attention to mathematical operability and analyzability. A general association of data of two variables, including nonlinear correlation, can be regarded as a binary relation R defined as a subset of a direct product of data sets A and B. A function is a binary relation such that any element of A is in a relation with one and only one element of B. In this sense, data mining for two variables implies knowledge discovery of binary relations or functions from observed data with random noises. The studies on machine learning to estimate such relations or functions suggest the effectivity of matrix representation in terms of operability and analyzability for the mathematical models. The rectangles of the grids of MIC are not congruent to each other, and it might have a disadvantage in terms of mathematical formulation and operation via matrix representation.
Preference relations and coradiants in ℝ n
Published in Journal of Control and Decision, 2023
Alireza Hosseini Dehmiry, Abbas Askarizadeh
A binary relation R on a set S is a collection of ordered pairs of elements of S. In other words, it is a subset of the Cartesian product . The statement is read ‘ is R-related to ’, and is denoted by or . Binary relations are used in many branches of mathematics to model preference relations. In particular, the concept of function is defined as a special kind of binary relation. In what follows, some important attributes of a binary relation are given. Most of these definitions and related propositions are taken from Ehrgott (2013) and Noghin (1997).
Human resource optimisation through semantically enriched data
Published in International Journal of Production Research, 2018
Damiano Arena, Apostolos Charalampos Tsolakis, Stylianos Zikos, Stelios Krinidis, Chrysovalantou Ziogou, Dimosthenis Ioannidis, Spyros Voutetakis, Dimitrios Tzovaras, Dimitris Kiritsis
Regardless of the knowledge representation paradigms (frames, description logics, logic) used to formally represent knowledge modelling components (concepts, roles, etc.), and the languages used to implement the ontologies under a given knowledge representation paradigm (Perdikakis 2015), all of them share the following minimal set of components:Classes represent general elements, which are taken in a broad sense.Relations represent a type of association between elements of the ontology, usually binary relations where the first argument is known as the domain of the relation, and the second is the range.Individuals are used to represent the instances of a class or objects.Formal axioms serve to state something about classes, relations, and individuals that is always true, which are used to verify the consistency of the ontology itself or the knowledge base (KB).