Relations and Equivalence Relations
ثبت نشده
چکیده
In this section, we shall introduce a formal definition for the notion of a relation on a set. This is something we often take for granted in elementary algebra courses, but is a fundamental concept in mathematics i.e. the very notion of a function relies upon the definition of a relation. Following this, we shall discuss special types of relations on sets. 1. Binary Relations and Basic Definitions We start with a formal definition of a relation on a set S. Definition 1.1. A (binary) relation on a set S is a subset R of the Cartesian product S × S. If R is a relation and (x, y) ∈ R, then we say " x is related to y by R " or simply xRy. Example 1.2. The most familiar of all relations is the relation " = " (equals) which consists of all the elements (x, x) ∈ S × S. (less than with standard definition from the integers). Write down all the elements of R. (since it consists of all elements (x, y) with x < y). There are certain special properties a relation can have such as the following: Definition 1.4. Suppose R is a relation on a set S. Then we define the following: • We say R is reflexive if xRx for all x ∈ S • We say that R is symmetric if xRy implies yRx for all x, y ∈ S • We say R is transitive if xRy and yRz implies xRz for all x, y, z ∈ S We illustrate with some examples. Example 1.5. Show that the relation < (less than) on R is a transitive relation which is not symmetric or reflexive. Suppose x < y and y < z. Then clearly x < z and hence < is transitive. We do not have x < x and if x < y, then it is not the case that y < x, so it follows that it is neither reflexive or symmetric.
منابع مشابه
FUZZY SUBGROUPS AND CERTAIN EQUIVALENCE RELATIONS
In this paper, we study an equivalence relation on the set of fuzzysubgroups of an arbitrary group G and give four equivalent conditions each ofwhich characterizes this relation. We demonstrate that with this equivalencerelation each equivalence class constitutes a lattice under the ordering of fuzzy setinclusion. Moreover, we study the behavior of these equivalence classes under theaction of a...
متن کاملOn certain semigroups of transformations that preserve double direction equivalence
Let TX be the full transformation semigroups on the set X. For an equivalence E on X, let TE(X) = {α ∈ TX : ∀(x, y) ∈ E ⇔ (xα, yα) ∈ E}It is known that TE(X) is a subsemigroup of TX. In this paper, we discussthe Green's *-relations, certain *-ideal and certain Rees quotient semigroup for TE(X).
متن کاملGood strongly regular relations on weak $Gamma$-(semi)hypergroups
In this paper first we introduce the notion of weak $Gamma$-(semi)hypergroups, next some classes of equivalence relations which are called good regular and strongly good regular relations are defined. Then we investigate some properties of this kind of relations on weak $Gamma$-(semi)hypergroups.
متن کاملON THE COMPATIBILITY OF A CRISP RELATION WITH A FUZZY EQUIVALENCE RELATION
In a recent paper, De Baets et al. have characterized the fuzzytolerance and fuzzy equivalence relations that a given strict order relation iscompatible with. In this paper, we generalize this characterization by consideringan arbitrary (crisp) relation instead of a strict order relation, while payingattention to the particular cases of a reflexive or irreflexive relation. The reasoninglargely ...
متن کاملTHE CONNECTION BETWEEN SOME EQUIVALENCE RELATIONS ON FUZZY SUBGROUPS
This paper, deals with some equivalence relations in fuzzy subgroups. Further the probability of commuting two fuzzy subgroups of some finite abelian groups is defined.
متن کاملCardinality of Equivalence Relations
This entry provides formulae for counting the number of equivalence relations and partial equivalence relations over a finite carrier set with given cardinality. To count the number of equivalence relations, we provide bijections between equivalence relations and set partitions [4], and then transfer the main results of the two AFP entries, Cardinality of Set Partitions [1] and Spivey’s General...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2008