Determine which of these relations are transitive. The variables x, y, x', y'represent integers.A. (x, y) ~ (x', y') if and only if xy' = x'y.B. ~ y if and only if x + y is positive.C.~ y if and only if x + y is odd.ID. (x, y) ~ (x', y') if and only if x - y = x'- y.E. ~ y if and only if x - y is negative.OF.~ y if and only if x + y is even.

Note: As multiple questions have been posted, so only A is solved according to QnA guidelines.…

Answered: Determine which of these relations are…

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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9.25. Let A = {1, 2, 3, 4, 5, 6}. The relation R = {(1,1), (1, 5), (2, 2), (2, 3), (2, 6), (3, 2), (3, 3), (3, 6), (4, 4), (5, 1), (5, 5), (6, 2), (6, 3), (6, 6)} is an equivalence relation on A. Determine the distinct equivalence classes.
II. Identify the correspondence in the following relations 8 3 1.) 5 2.) 5 4 3 4 8 8 4.) -2 -3 -4 -5 3.) 2 1. -1 1. 5.) 4 6.) 3 2 1. 2 3 1 3 4 2/4 13
2. Let A = {1,2, 3}, B = {4, 5, 6}, C = {7,8, 9}. Consider the relations R from A to B, S from B to C, and T on A. R = {(1, 5), (2, 4), (1, 6)} S = {(4, 8), (5, 7), (6, 8), (6, 9)} Т 3 {(1,2), (2, 3), (3, 1), (1, 1), (2, 2), (3, 3)} Find the following relations (a) Find the compositions Ro S, T o R, T o T. (b) Find the matrices MR, Ms, MT, MTOT, MTOR and MRos of the respective relations R, S, т, То Т, То R, and Ro S.
17. Let R and S be relations on A= {1, 2, 3, 4} defined by R={(a, b) :b=5-a} and S = {(a, b): a
Determine which of these relations are transitive. The variables x, y, x', y' represent integers. A.x~ y if and only if x + y is positive. B.x~ y if and only if x + y is odd. C. (x, y) ~ (x', y') if and only if xy = x'y. D. (x, y) (x', y') if and only if x - y = x - y'. E.x~ y if and only if x -y is positive. F.x~ y if and only if x - y is a multiple of 10.
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…
{(а, 1), (ь, 3), (с, 4), (а, 1), (а, 2)} dan Rz {(w, 2), (w, 4), (y, 1), (z, 4)}. Determine Given two relations R1 R1 x R2.
Determine which of these relations are symmetric. The variables x, y, x', y represent integers. A. ~ y if and only if x + 2y is positive. B. x ~ y if and only if x + y is even. C. ~ y if and only if x + y is positive. D.I~ y if and only if x - y is positive. E.I~ y if and only if x - y is a multiple of 10.

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