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- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Discrete_Mathematics_for_Computer_Science_(Fitch)/04%3A_Relations/4.04%3A_Partially_Ordered_SetsThis page defines partially ordered sets (posets) and their properties like reflexivity, anti-symmetry, and transitivity. It explains representation through Hasse diagrams and illustrates examples suc...This page defines partially ordered sets (posets) and their properties like reflexivity, anti-symmetry, and transitivity. It explains representation through Hasse diagrams and illustrates examples such as numeric divisibility and subset relations. Key terms related to posets, including comparable elements, total ordering, and lattices, are defined, alongside practical checkpoints for assessing poset properties and constructing Hasse diagrams.
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Gentle_Introduction_to_the_Art_of_Mathematics_(Fields)/06%3A_Relations_and_Functions/6.02%3A_Properties_of_RelationsThere are two special classes of relations that we will study in the next two sections, equivalence relations and ordering relations. The prototype for an equivalence relation is the ordinary notion o...There are two special classes of relations that we will study in the next two sections, equivalence relations and ordering relations. The prototype for an equivalence relation is the ordinary notion of numerical equality, = . The prototypical ordering relation is ≤ . Each of these has certain salient properties that are the root causes of their importance. In this section, we will study a compendium of properties that a relation may or may not have.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/A_Spiral_Workbook_for_Discrete_Mathematics_(Kwong)/07%3A_Relations/7.02%3A_Properties_of_RelationsIf R is a relation from A to A , then R⊆A×A ; we say that R is a relation on A .
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Discrete_Mathematics_for_Computer_Science_(Fitch)/04%3A_Relations/4.02%3A_Mathematical_RelationsThis page discusses the concept of relations in data organization, defining them as subsets of a Cartesian product. It covers various types of relations—reflexive, symmetric, anti-symmetric, and trans...This page discusses the concept of relations in data organization, defining them as subsets of a Cartesian product. It covers various types of relations—reflexive, symmetric, anti-symmetric, and transitive—illustrated with examples involving integers and real numbers. The text highlights specific properties and contexts, such as subset inclusion, divisibility, and the less than or equal to relation.
- https://math.libretexts.org/Courses/Monroe_Community_College/MTH_220_Discrete_Math/6%3A_Relations/6.2%3A_Properties_of_RelationsNote: If we say R is a relation "on set A" this means R is a relation from A to A; in other words, R⊆A×A. Determine whether the following relation W on a non...Note: If we say R is a relation "on set A" this means R is a relation from A to A; in other words, R⊆A×A. Determine whether the following relation W on a nonempty set of individuals in a community is an equivalence relation: aWb⇔a and b have the same last name.
- 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/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.