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Elements of Algebra
Published in Gerhard X. Ritter, Gonzalo Urcid, Introduction to Lattice Algebra, 2021
Gerhard X. Ritter, Gonzalo Urcid
The Cartesian product is also known as the cross product. The element (x,y) of the set X×Y indicates the order in that the first component x of the pair must be an element of X while the second component y must be an element of Y. Additionally, these ordered pairs are subject to the condition: (a,b)=(c,d)⇔a=c and b=d. The components of the ordered pair are also known as coordinates.
Distribution Shapes
Published in Wendy L. Martinez, Angel R. Martinez, Jeffrey L. Solka, Exploratory Data Analysis with MATLAB®, 2017
Wendy L. Martinez, Angel R. Martinez, Jeffrey L. Solka
Cleveland [1993] describes another version of a quantile-based plot. He calls this the quantile plot, where we have the u¡ values (Equation 9.12) along the horizontal axis and the ordered data x(;) along the vertical. These ordered pairs are plotted as points, joined by straight lines. Of course, this does not have the same interpretation or provide the same information as the probability plot or q-q plot described previously. This quantile plot provides an initial look at the distribution of the data, so we can search for unusual structure or behavior. We also obtain some information about the sample quantiles and their relationship to the rest of the data. This type of display is discussed in the next example.
Sets, Relations and Functions
Published in Sriraman Sridharan, R. Balakrishnan, Foundations of Discrete Mathematics with Algorithms and Programming, 2019
Sriraman Sridharan, R. Balakrishnan
In the ordered pair (x, y), the order of x and y is important whereas the unordered pairs (x, y) and (y, x) are equal. As ordered pairs they are equal if and only if x=y $ x=y $ . For instance, the pairs (1, 2) and (2, 1) are not equal as ordered pairs, while they are equal as unordered pairs.
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
Consensus in multi-agent systems over time-varying networks
Published in Cyber-Physical Systems, 2020
Magdi S. Mahmoud, Mojeed Oyedeji
A graph is defined as an ordered pair of a set of vertices and edges . Graphs can be directed or undirected. In an undirected graph, there’s an edge orientation between any pair of agents, that is, any pair of agents can exchange information in either directions. Conversely, in a directed graph or digraph, each edge is directed between any pair of vertices, that is, information flow between any pair of vertices can be bidirectional or unidirectional. Graphs can also be defined as simple, complete or bipartite. A walk in a graph is a finite sequence of alternate vertex-edge mapping defined as such that for , the edge has ends and . A path is a walk with distinct vertex set. A cycle is a path starting and terminating on the same vertex. Connected graphs (or subgraphs) without cycles are called trees. A spanning tree is a tree which includes all elements of the vertex set of .
Logics for extended distributive contact lattices
Published in Journal of Applied Non-Classical Logics, 2018
We will define special sets and with their help we will obtain a Boolean algebra . Let and , , , . Then we call a special set, determined by the ordered pair .