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- https://math.libretexts.org/Bookshelves/Linear_Algebra/Linear_Algebra_with_Applications_(Nicholson)/05%3A_Vector_Space_R/5.05%3A_Similarity_and_DiagonalizationHence the eigenvalues are λ1=i and λ2=−i, with corresponding eigenvectors x1=[1−i] and \(\mathbf{x}_2 = \lef...Hence the eigenvalues are λ1=i and λ2=−i, with corresponding eigenvectors x1=[1−i] and x2=[1i]. Hence A is diagonalizable by the complex version of Theorem [thm:016145], and the complex version of Theorem [thm:016068] shows that \(P = \left[ x1 x2 \right]= \left[ \begin{array}{rr} 1 & 1 …
- https://math.libretexts.org/Courses/Monroe_Community_College/MTH_220_Discrete_Math/6%3A_Relations/6.3%3A_Equivalence_Relations_and_PartitionsThe overall idea in this section is that given an equivalence relation on set A, the collection of equivalence classes forms a partition of set A, (Theorem 6.3.3). If R is an equivalence r...The overall idea in this section is that given an equivalence relation on set A, the collection of equivalence classes forms a partition of set A, (Theorem 6.3.3). If R is an equivalence relation on any non-empty set A, then the distinct set of equivalence classes of R forms a partition of A. if R is an equivalence relation on any non-empty set A, then the distinct set of equivalence classes of R forms a partition of A.
- https://math.libretexts.org/Bookshelves/Geometry/Geometry_with_an_Introduction_to_Cosmic_Topology_(Hitchman)/07%3A_Geometry_on_Surfaces/7.07%3A_Quotient_SpacesA relation on a set S is a subset R of S x S. In other words, a relation R consists of a set of ordered pairs of the form (a,b) where a and b are in S. A partition of a set consists of a collection...A relation on a set S is a subset R of S x S. In other words, a relation R consists of a set of ordered pairs of the form (a,b) where a and b are in S. A partition of a set consists of a collection of non-empty subsets of A that are mutually disjoint and have union equal to A. An equivalence relation on a set A serves to partition A by the equivalence classes.
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Gentle_Introduction_to_the_Art_of_Mathematics_(Fields)/06%3A_Relations_and_Functions/6.03%3A_Equivalence_RelationsThe main idea of an equivalence relation is that it is something like equality, but not quite. Usually there is some property that we can name, so that equivalent things share that property. For examp...The main idea of an equivalence relation is that it is something like equality, but not quite. Usually there is some property that we can name, so that equivalent things share that property. For example Albert Einstein and Adolf Eichmann were two entirely different human beings, if you consider all the different criteria that one can use to distinguish human beings there is little they have in common.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Rings_with_Inquiry_(Janssen_and_Lindsey)/01%3A_The_Integers/1.04%3A_The_Integers_modulo__mThe foundation for our exploration of abstract algebra is nearly complete. We need the basics of one more "number system" in order to appreciate the abstract approach developed in subsequent chapters.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Yet_Another_Introductory_Number_Theory_Textbook_-_Cryptology_Emphasis_(Poritz)/02%3A_Congruences/2.04%3A_Equivalence_ClassesIn this section, we shall consider another way to work with congruences.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Discrete_Mathematics_for_Computer_Science_(Fitch)/04%3A_Relations/4.03%3A_Equivalence_RelationsThis page explores equivalence relations in mathematics, detailing properties like reflexivity, symmetry, and transitivity. It defines equivalence classes and provides checkpoints for assessing equiva...This page explores equivalence relations in mathematics, detailing properties like reflexivity, symmetry, and transitivity. It defines equivalence classes and provides checkpoints for assessing equivalence in subsets, modular arithmetic, and integer divisibility.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/A_Spiral_Workbook_for_Discrete_Mathematics_(Kwong)/07%3A_Relations/7.03%3A_Equivalence_RelationsA relation on a set A is an equivalence relation if it is reflexive, symmetric, and transitive. We often use the tilde notation a∼b to denote an equivalence relation.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Introduction_to_Groups_and_Geometries_(Lyons)/01%3A_Preliminaries/1.04%3A_Equivalence_RelationsGiven a function f:X→Y, there is a natural equivalence relation ∼f on X given by x∼fy if and only if f(x)=f(y). The corresponding set of equivalence...Given a function f:X→Y, there is a natural equivalence relation ∼f on X given by x∼fy if and only if f(x)=f(y). The corresponding set of equivalence classes is X/∼f={f−1(y):y∈f(X)}. Furthermore, the function X/∼f→f(X) given by [x]→f(x) is a one-to-one correspondence.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_Through_Guided_Discovery_(Bogart)/zz%3A_Back_Matter/21%3A_Appendix_A%3A_RelationsA typical way to define a function f from a set S, called the domain of the function, to a set T, called the range, is that f is a relationship between S to T that relates one and only one member of T...A typical way to define a function f from a set S, called the domain of the function, to a set T, called the range, is that f is a relationship between S to T that relates one and only one member of T to each element of X. We use f(x) to stand for the element of T that is related to the element x of S. If we wanted to make our definition more precise, we could substitute the word “relation” for the word “relationship” and we would have a more precise definition.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/First-Semester_Abstract_Algebra%3A_A_Structural_Approach_(Sklar)/07%3A_The_Wonderful_World_of_Cosets/7.01%3A_Partitions_of_and_Equivalence_Relations_on_SetsThe number of partitions of a finite set of n elements gets large very quickly as n goes to infinity. Indeed, there are 52 partitions of a set containing just 5 elements! (The total number of part...The number of partitions of a finite set of n elements gets large very quickly as n goes to infinity. Indeed, there are 52 partitions of a set containing just 5 elements! (The total number of partitions of an n-element set is the Bell number. There is no trivial way of computing Bell number, in general, though the Bell number do satisfy the relatively simple recurrence relation.
