If R was the group knowledge relationship {0,1,2,3,4,5,6,7}is as follows: 3/x-y Ryprove that Requivalence relationship on x Then find theequivalence classes for the relationship R

A relation is said to be an equivalence relation if R is reflexive, symmetric, and transitive.

Answered: If R was the group knowledge…

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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All of the given relations are onto. True or false
2. Let A = {2,3,4} and B = { 6, 8, 10} and define a binary relation R from A to B as follows: for all (x, y) E AX B, (x, y) E R⇒ x | y. a) Is 4 R 6? b) Is 4 R 8? c) Is (3,8) € R? Is (2, 10) € R? d) Write R as a set of ordered pairs.
This question refers to unions and intersections of relations. Since relations are subsets of Cartesian products, their unions and intersections can be calculated as for any subsets. Given two relations R and S from a set A to a set B, RUS = {(x, y) E AXB I (x, y) ER or (x, y) E S} RNS = {(x, y)E AXB| (x, y) ER and (x, y) Es}. Let A = {-2, 2, 3, 5} and B = {2, 3} and define relations R and S from A to B as follows: For every (x, y) E Ax B, x R y - |x| = lyl and xsyex - y is even. Using set-roster notation, state explicitly which ordered pairs are in A x B, R, S, RU S, and RN s. (Enter your answers as comma-separated lists of ordered pairs.) AXB = R = S = RUS = ROS =
1. Let A = {1,2,3,4,5,6), and consider the following equivalence relation on A: R = {(1, 1), (2, 2), (3, 3), (4,4), (5,5), (6, 6), (2, 3), (3, 2), (4, 5), (5, 4), (4, 6), (6,4), (5,6), (6,5)} List the equivalence classes of R. 2. Let A = {a,bed el Suppose A is an equivalence relation on 4. Suppose R bes two equivaler.se alasses. Also wad, Re and aRd Write out R as a cot 3. Let A = {a,b,c,d,e}. Suppose R is an equivalence relation on A. Suppose R has three equivalence classes. Also aRd and bRc. Write out R as a set. 4. Let A {a,b,c,d,e). Suppose A is an equivalence relation on 4. Suppose also that anta and one, aRa and cite. How many equivalones classes dees At have! CIV 5. There are two different equivalence relations on the set A = {a,b}. Describe them. Diagrams will suffice.
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VIII.1. Let A = {1,2, 3, 4} and let R, S, T and U be the following relations: R= {(1,3), (3, 2), (2, 1), (4, 4)}, S = {(2,1), (3, 3), (4, 2)}, T = {(4,1), (4, 2), (3, 1), (3, 2), (1, 2)}, U = {(x, y) | x > y}. (a) For each of R, S, T and U determine whether they are functional, reflexive, symmetric, anti-symmetric or transitive. Explain your answer in each case, showing why your answer is correct. (b) What is the transitive closure of R? (c) Explain why R", the transitive closure of R, is an equivalence relation. De- scribe the equivalence classes E, into which the relation partitions the set A. VIII.2. Prove or give a counterexample to the following statement: for any relation R both R and RoR always have the same transitive closure. VIII.3. Is there a mistake in the following proof that any transitive and symmetric relation Ris reflexive? If so, what is it? Let a Rb. By symmetry, bRa. By transitivity, if aRb and bRa, then aRa. This proves reflexivity. VIII.4. Determine for the…
Let S be a nonempty subset of Z and let R be a relation defined on S by xRy if 3 | (x + 2y). If S = {−7, −6, −2, 0, 1, 4, 5, 7}, then what are the distinct equivalence classes in this case? Please show how you find them
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If R was the group knowledge relationship {0,1,2,3,4,5,6,7} is as follows: 3/x-y Ry prove that Requivalence relationship on x Then find the equivalence classes for the relationship R