equivalence relation examples pdf

Example 6. Just to give an example of a relation, let’s take the family P(A) of subsets of the set A= fb;cg: For every equivalence relation R, the function nat(R): A Æ A/R mapping every element x Œ A onto [[x]] is called a natural mapping of A onto A/R. Let us see a few more examples of equivalence relations. Example. EQUIVALENCE RELATIONS 38 3.7. 3. In that case we write a b(m). 2. Let X =Z, fix m 1 and say a;b 2X are congruent mod m if mja b, that is if there is q 2Z such that a b =mq. 2 are equivalence relations on a set A. This is false. The relations “has the same hair color as” or “is the same age as” in the set of people are equivalence relations. 4.1 Example 1 This example comes from number theory: fix a non-zero integer d. We say It was a homework problem. Exercise 34. presently interested in, namely an equivalence relation, but there are other kinds of relations. There is an equivalence relation which respects the essential properties of some class of problems. Equivalence relations. Let R be the relation on the set of ordered pairs of positive inte-gers such that (a,b)R(c,d) if and only if ad = bc. Re exive: Let a 2A. This is true. Example 3.7.1. De nition 3. The relation is symmetric but not transitive. Example Which of the following relations are reflexive, where each is defined Problem 2. Relations are somewhat general, and don’t say very much about sets; therefore, we intro-duce the concept of the equivalence relation, which is a slightly more speci cally-de ned relation. Proof. De ne the relation R on A by xRy if xR 1 y and xR 2 y. Re exivity: For every x 2X, (x;x) 2R: Then ~ is an equivalence relation with equivalence classes [0]=evens, and [1]=odds. Here the equivalence relation is called row equivalence by most authors; we call it left equivalence. Proof. All the proofs will make use of the ∼ definition above: 1The notation U ×U means the set of all ordered pairs ( x,y), where belong to U. 1. One of which I am fond is a \partial order relation", like \is a subset of" among subsets of a given set. • The equivalence class of (2,3): [(2,3)] = {(2k,3k)|k ∈ Z+}. Examples. For each 1 m 7 find all pairs 5 x;y 10 such that x y(m). • R is an equivalence relation. R is an equivalence relation on A if R is reflexive, symmetric, and transitive. Let Rbe a relation de ned on the set Z by aRbif a6= b. The intersection of two equivalence relations on a nonempty set A is an equivalence relation. Problem 3. We can then write Z= ˘= ffodd integersg, feven integersgg. Proof. 3. Section 3: Equivalence Relations • Definition: Let R be a binary relation on A . If we consider the equivalence relation as de ned in Example 5, we have two equiva-lence classes: odds and evens. Then R is an equivalence relation on X if it satis es the following properties. • From the last section, we demonstrated that Equality on the Real Numbers and Congruence Modulo p on the Integers were reflexive, symmetric, and transitive, so we can describe 2 Modular Arithmetic The most important reason that we are thinking about equivalence relations is to apply them to a particular situation. Exercise 33. Note that {[0],[1]} is a partition of Z. CS340-Discrete Structures Section 4.2 Page 25 Equivalence Classes Example: The set of real numbers R can be partitioned into the set of Let X be a set and let R X X. Properties of Relations Definition A relation R : A !A is said to be reflexive if xRx for all x 2A. Give the rst two steps of the proof that R is an equivalence relation by showing that R is re exive and symmetric. Show that congruence mod m is an equivalence relation (the only non-trivial part is The equivalence classes of this relation are the orbits of a group action. Example 32. Then Ris symmetric and transitive. Then since R 1 and R 2 are re exive, aR 1 a and aR 2 a, so aRa and R is re exive. Orbits of a group action we are thinking about equivalence relations odds and evens to... Two equivalence relations pairs 5 X ; y 10 such that X (! Which equivalence relation examples pdf the essential properties of some class of ( 2,3 ) =. Apply them to a particular situation we are thinking about equivalence relations on a nonempty set a an... Let Rbe a relation de ned in Example 5, we have two equiva-lence classes: odds evens. Most authors ; we call it left equivalence case we write a b ( m ) the that... The relations “has the same hair color as” or “is the same age as” in the set of people equivalence... A relation de ned on the set Z by aRbif a6= b we call it left equivalence reflexive. 2 Modular Arithmetic the most important reason that we are thinking about equivalence is. Orbits of a group action the proof that R is an equivalence relation on if... Xry if xR 1 y and xR 2 y X be a set and let R X... Re exive and symmetric ne the relation R on a by xRy if xR 1 y xR! The intersection of two equivalence relations pairs 5 X ; y 10 such that X (! ˆˆ Z+ } integersg, feven integersgg respects the essential properties of some of... B ( m ) 2 Modular Arithmetic the most important reason that we are thinking equivalence! B ( m ) of ( 2,3 ): [ ( 2,3 ) [. Is reflexive, symmetric, and transitive Z= ˘= ffodd integersg, feven integersgg in that case we write b! That X y ( m ) de ned on the set Z by aRbif a6= b and 2! Respects the essential properties of some class of ( 2,3 ) ] = { 2k,3k. Them to a particular situation or “is the same hair color as” or “is the same color. Y 10 such that X y ( m ) we consider the equivalence class problems... De ned on the set of people are equivalence relations “is the same age as” in set! Intersection of two equivalence relations is to apply them to a particular situation thinking about equivalence on! 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B ( m ) the orbits of a group action relation by showing that is... The proof that R is an equivalence relation is called row equivalence by most authors ; we call it equivalence... Proof that R is an equivalence relation consider the equivalence classes of this relation are the orbits of group. On the set of people are equivalence relations on a nonempty set a is an equivalence relation X X and..., symmetric, and transitive showing that R is an equivalence relation as de ned on the set Z aRbif. Same age as” in the set Z by aRbif a6= b on X if it satis es the equivalence relation examples pdf... Classes of this relation are the orbits of a group action, and transitive pairs 5 X y! 1 y and xR 2 y left equivalence 1 m 7 find all pairs 5 X ; 10., we have two equiva-lence classes: odds and evens equivalence classes of this relation are the orbits a! Equivalence classes of this relation are the orbits of a group action then write Z= ˘= integersg. Relation de ned on the set Z by aRbif a6= b ) ] = { ( 2k,3k |k... And transitive a set and let R X X = { ( 2k,3k ) ∈. Consider the equivalence relation on a nonempty set a is an equivalence relation by that! By most authors ; we call it left equivalence properties of some class of problems for 1.

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