Let S be the set of binary strings {010, 100, 110, 111,1110,01110,10111,11110,11111}. A relation R on S is defined as fol- lows: (a,b) ∈ R if and only if a ∈ S has the same number of 0’s as b ∈ S. Prove that R is an equivalence relation. Determine the partition of S that corresponds to R. (That is, determine all equivalence classes that form the partition)

Let S be the set of binary strings {010, 100, 110, 111,1110,01110,10111,11110,11111}. A relation R on S is defined as fol- lows: (a,b) ∈ R if and only if a ∈ S has the same number of 0’s as b ∈ S. Prove that R is an equivalence relation. Determine the partition of S that corresponds to R. (That is, determine all equivalence classes that form the partition)

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 3TFE: True or False Label each of the following statements as either true or false. If is an equivalence...

See similar textbooks

Similar questions

Give an example of a relation R on a nonempty set A that is symmetric and transitive, but not reflexive.
In Exercises , a relation is defined on the set of all integers. In each case, prove that is an equivalence relation. Find the distinct equivalence classes of and list at least four members of each. 10. if and only if .
Prove Theorem 1.40: If is an equivalence relation on the nonempty set , then the distinct equivalence classes of form a partition of .
Label each of the following statements as either true or false. Let R be a relation on a nonempty set A that is symmetric and transitive. Since R is symmetric xRy implies yRx. Since R is transitive xRy and yRx implies xRx. Hence R is alsoreflexive and thus an equivalence relation on A.
In Exercises 610, a relation R is defined on the set Z of all integers. In each case, prove that R is an equivalence relation. Find the distinct equivalence classes of R and list at least four members of each. xRy if and only if x+3y is a multiple of 4.
Label each of the following statements as either true or false. If R is an equivalence relation on a nonempty set A, then any two equivalence classes of R contain the same number of element.
In Exercises 610, a relation R is defined on the set Z of all integers, In each case, prove that R is an equivalence relation. Find the distinct equivalence classes of R and least four members of each. xRy if and only if x2+y2 is a multiple of 2.
23. Let be the equivalence relation on defined by if and only if there exists an element in such that .If , find , the equivalence class containing.
True or False Label each of the following statements as either true or false. 2. Every relation on a nonempty set is as mapping.
True or False Label each of the following statements as either true or false. If is an equivalence relation on a nonempty set, then the distinct equivalence classes of form a partition of.