8. Let A = {p, q, r, s, t, u, v), and let R be the following equivalence relation on A:R = {(p, p), (q, q), (r, r), (s, s), (t, t), (u, u), (v, v), (p, q), (q, p), (q, t), (t, q), (r, v), (v, r), (p, t), (t, p) }.Write out each equivalence class as a set of elements. Then abbreviate each equivalence class by choosingsome representative of that class and bracketing it (this bracketed element "represents" the whole class).

Equivalence Relation: "A relation E on a set A is called an equivalence relation if E…

Answered: 8. Let A = {p, q, r, s, t, u, v}, and…

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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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 A × B| (x, y) ERor (x, y) E S} RO S= {(x, y) E A x B | (x, y) ERand (x, y) E S}. Let A = {-4, 4, 7, 9} and B = {4, 7}, and define relations R and S from A to B as follows. For every (x, y) E Ax B, x R y x| = [y] and x Sy x - y is even. Which ordered pairs are in A x B, R, S, R U S, and RN S? (Use set-roster notation to enter your answers.) Ax B = R = S = RUS = ROS =
. Let R1, R2, R3 be relations on { a, b, c, d, e }. Find the transitive closures, given their definitions below. i) R1 = { (a, c),(b, d),(c, a),(d, b),(e, d) } ii) R2 = { (b, c),(b, e),(c, e),(d, a),(e, b),(e, c) } iii) R3 = { (a, b),(a, c),(a, e),(b, a),(b, c),(c, a), (c, b),(d, a),(e, d) }
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.
3. Let R be a relation defined on the set A = N × N where (a, b) R (c, d) if að = cd. Show R is an equivalence relation. Then, list the elements of the equivalence class [(9, 2)].
{(x,a), (x,b), (x,c), (y,a), (y,b),(y,c)} is the Cartesian product AXB of two sets given as A = {a,b,c} B = {x,y} A = {x,y} B = {1,2,3} A = {x,y} B = {a,b,c} A = {x,a} B = {y,b.c}
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.
2. Let U = {1, 2, 3} and A = U × U. In each case, show that = is an equivalence on A and find the quotient set A. (a) (a, b) = (a₁, b₁) if a + b = a₁ + b₁. (b) (a,b) = (a₁, b₁) if ab = a₁b₁. (c) (a, b) = (a₁, b₁) if a = a₁. (d) (a, b) = (a₁, b₁) if a - b = a₁-b₁.

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