10.Define a relation p on S = {1, 2, 3, 4, 5, 6} byp = {(1, 1), (1,2), (2, 1), (2, 2), (3, 3), (3, 5), (3, 6), (4, 4), (5, 3), (5, 5), (5, 6), (6,3), (6,5), (6,6)}. Itcan be easily seen that p is an equivalence relation (you should understand why, but don't needto show here). Find the equivalence classes [1], [2], [3], [4], [5] and [6].

We can find the equivalence classes [1], [2], [3], [4], [5], [6] by determining which elements are…

Answered: 10. Define a relation p on S = {1, 2,…

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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(b) Let R = {(1, 1),(1, 2),(2, 1),(2, 2), (3, 3)}. be a relation on a set A= 1,2,3} , the R is %D Antisymmetric and transitive. ii. i. Not antisymmetric, and not transitive Antisymmetric, and not transitive Non of these iii. iv.
Let A = (0,1,2,3). The relations R, S and T in the set A are defined as follows…R={(0,0),(0,1),(0,3),(1,0),(1,1),(1,3),(2,2),(3,0),(3,1),(3,3)}S ={(0,0),(0,2),(0,3),(2,3)}T={(0,1),(2,3)}Find the equivalent relation is:
Consider the family of sets S= [{1,5, 6}, {2,4, 6), (4, 5, 6), {2, 5, 6}, {3,4, 5}, {1, 4, 6}]. (1.1) and let Y - (1,3, 4, 5}. Define the relation Ron S by (A, B) ER = A\Y = B\Y, where A, B are arbitrary elements of S. (1) (a) Since for every A ES A\Y = \Y, where ? = B ,the relation R (here and below in (b-d), please type is or is not in the input field below) is reflexive. (b) Given any A, BES, we have that A\Y = B\Y = B\Y= \Y where and hence the relation R is symmetric. (c) For every A, B, CES, (A\Y = B\Y) and (B\Y = C\Y) implies A\Y = \Y, where al= ,and hence the relation R is transitive. (d) Accordingly, by (a,b,c), the relation R is an equivalence relation on S. (11) Find the matrix MR of the relation R (please enter the matrix row-by-row in the six input fields below, when entering each row, seperate the entries by single spaces; understandably, the i-th row/column of the matrix MR must correspond to the i-th set A, in the list of the elements of S given in Eq. (1.1) above): (I)…
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.
Choose the answer
Let X=\{a,b,c,d,e,f\}X={a,b,c,d,e,f} and SS be the relation on XX defined by S = \{ (a,a), (a,b), (a,c),S={(a,a),(a,b),(a,c), (a,d), (a,e), (a,f),(a,d),(a,e),(a,f), (b,b), (b,d), (b,e),(b,b),(b,d),(b,e), (b,f), (c,c), (c,d),(b,f),(c,c),(c,d), (c,e), (c,f), (d,d),(c,e),(c,f),(d,d), (d,e), (d,f), (e,e), (f,f) \}(d,e),(d,f),(e,e),(f,f)}. Is SS a partial order? If so, draw its Hasse diagram. If not, give an example showing why.
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…
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6) Let A = {4,5,6} and B = {3,6,7}. Define relations U, V, and W from A to B as follows, For all (x, y) E A×B, (x, y) eU x>y (x, y) e V > X-y is an integer. W = {(4,7), (6,3), (5,6)} Indicate whether any of the relations U, V, and W are functions from A to B.

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