Problem 26. Consider the following equivalence relation on the set A = {1, 2, 3, 4, 5, 6):Find the partition generated by RR= {(1, 1), (2, 2), (3, 3), (4,4), (5,5), (6, 6), (1, 2), (1,4), (2, 1), (2, 4), (4, 1), (4, 2), (3,6), (6,3)}.

Answered: Image /qna-images/answer/8e20311a-bed2-4ce4-be4c-d126cd65db53.jpg

Answered: Problem 26. Consider the following…

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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1. We are given the sets S and V: S = {Anne, Theo} V {studies, runs, meditates} = (a) Find the Cartesian Product of S and V: S x V (b) A relation R. is a subset of the Cartesian product. Create such a subset RC (S X V) with three elements.
6. Q6; Relations (a) Define relations S and R on the set A = {a, b, c, d} as follows: {(a, b), (b, c), (c, d), (b, b)}, S = R = {(b, a), (c, c), (a, c), (d, b)}. i. Find So S, So R, R-o So R. Is R-lo SoR= S ? ii. Determine which of S, R, So R, S-1 or R-1 is a function. (b) Let R = {(x, y) E R × R : 1
15. Let S = P({1,2,3,4,5}).. Define an equivalence relation ~ by X-Y iff XO{1,3,5}=Yn{1,3,5}. Give the equivalence class of the set {1,2,3}.
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.
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)…
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.
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.
The difference of two sets, A – B, is defined as the set containing all elements of A except those in B. That is, A - B = AN B'. Find A - B for the pair of sets if U = {1, 2, 3, 4, 5, 6, 7, 8, 9}. (Enter your answer in roster notation. Enter EMPTY or Ø for the empty set.) A = {3, 5, 9} and B = {2, 4, 6, 7, 8, 9} A - B =
Problem 6. Find the number of all relations R on A = {1,2, 3, 4} such that (a) No restrictions; (b) (1, 2) € R; (c) (1, 2) g R; (d) (1,2) eR or (1,4) eR (or both).

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Problem 26. Consider the following equivalence relation on the set A = {1, 2, 3, 4, 5, 6): Find the partition generated by R R= {(1, 1), (2, 2), (3, 3), (4,4), (5,5), (6, 6), (1, 2), (1,4), (2, 1), (2, 4), (4, 1), (4,2), (3,6), (6,3)}.