1.prove that for an equivalence relation R on A, both sets, the class ofequivalence classes of R -A/R– is a set.Specifically show that this follows by Principles 0-2 implicitlysubclass-theorem.-via theHint. You will need, of course, to find a superset of A/R, that is, a classX that demonstrably is a set, and satisfies A/RCX.

Given: To prove that the equivalence relation R on A. Both sets, the class of equivalence classes of…

Answered: 1. prove that for an equivalence…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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A relation R on a set A is backwards transitive if, and only if, for every r, y, z E A, if xRy and yRz then z Rr. Prove that a relation R is reflexive and backwards transitive if and only if R is an equivalence relation.
(b) Consider the finite set A = equivalence classes that S partitions the set A into. {1,2,3,,,,,). Draw a graphical representation of the equivalence relation S from Part 4a as defined on the set A, and use that representation to list out the distinct
Let A be a non-empty funute set and let S = P(A). Define relation R ⊆ S × S by:xRy ⇔ |x| = |y|,where | · | denotes the set cardinality.(i) Prove that R is an equivalence relation.(ii) Find all equivalence classes for the case A = {1, 2, 3}.
Let L be a relation on R such that for all x and y in R, x L y if and only if x < y. Give a counter exampleto the statement ”L is an equivalence relation on R .
Let A be a nonempty set and R be a relation on A such that domain(R) = A. Prove that if R is symmetric and transitive then R is an equivalence.
1. Define a relation on R by a - b iff a4 – 62 = b4 - a². Show that - is an equivalence relation. (b) Determine the equivalence class [-1]. (c) Prove or disprove: Every equivalence class in R/ ~ contains exactly 2 real numbers.
(a) Define the following terms: (i) Cartesian product of two sets; (ii) relation on a set X. (b) Write down a relation on the set {1,2,3} which is reflexive and symmetric but not transitive. (c) Let S be the relation on the set R \ {0} defined by xSy if and only if y/x € Q. Prove that S is an equivalence relation.
Let T be the set {w = {0, 1}* ||w| ≤ 4}. Let R be the equivalence relation defined on T as follows: R = {(x, y) | x ≤T, yɛT, no(x) = = no(y)}, where no(r) represents the number of zeroes in the string x, and no(y) represents the number of zeroes in the string y. For example, (1011, 01) is a pair in R because the two strings 1011 and 01 have the same number of zeroes as each other. Every element in the set will appear in exactly one equivalence class and will be related to all elements in its class and not related to any elements outside of its class. What are the equivalence classes of T created by the relation R?
A car dealership sclls cars that were made in 2015 through 2020. Let the cars for sale be the domain of a relation R where two cars are related if they were made in the same year. (a) Prove that this relation is an equivalence relation. (b) Describe the partition defined by the cquivalence classes.
Let X = {−1, 0, 1} and A = ?(x) and define a relation R on A as follows: For all sets s and t in ?(x), s R t ⇔ the sum of the elements in s equals the sum of the elements in t. It is a fact that R is an equivalence relation on A. Use set-roster notation to list the distinct equivalence classes of R. (Enter your answer as a comma-separated list of sets. Enter EMPTY or ∅ for the empty set.)
Find the number of different partions of a set (a) with one element (b) with two elements (c) with four elements Prove the second part of the “Equivalence Relations and Partions” Theorem. That is, prove that given a partition of a set S the relation defined as follows “xRy if and only if x and y are in the same cell” is an equivalence relation on S. Determine, with justification, whether the following relations are equivalence relations. If so, describe the partition arising from the equivalence relation. (a) x ∼ y in R if x ≥ y (b) n ∼ m in Z if |n| = |m|. We saw in class that the residue classes modulo 3 in Z + are {1, 4, 7, 10, . . . } {2, 5, 8, 11, . . . } {3, 6, 9, 12, . . . } Write the residue classes modulo n in Z + for (a) n = 4 (b) n = 5 Simplify the given expression and write your answer in the form a + bi for a, b ∈ R. (a) i^3 (b) i^5 (c) i^23 (d) (5 + 6i)(7 − i) (e) (1 + i)^3 (f) |5 + 3i|
Let X be the set of nonempty sets of {-1, 0, 1} and define a relation R on X as follows: For all sets s and tin X, s R t⇒ the sum of the elements in s equals the sum of the elements in t. It is a fact that R is an equivalence relation on X. Use set-roster notation to list the distinct equivalence classes of R. (Enter your answer as a comma-separated list of sets. Enter EMPTY or Ø for the empty set.)

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