Let X = {1, 2, 3, 4, 5, 6). Which of the following could be an equivalence class of an equivalence relation on X?O {(1, 2), (3, 4), (5, 6))O (1,3,5)O {(1, 2), (3, 4), (5, 6))O (12)(34) (56)

Answered: Let X = {1, 2, 3, 4, 5, 6}. Which of…

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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13. Which of these relations on {0, 1, 2, 3} are equivalence relations? Determine the properties of an equivalence re- lation that the others lack. a. {(0, 0), (0, 2), (2, 0), (2, 2), (2, 3), (3, 2), (3, 3)} b. {(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 2), (3, 3)}
What is the composite of the relations R and S, where R is the relation from {1, 2, 3} to {1, 2, 3, 4} with R = {(1, 1), (1, 4), (2, 3), (3, 1), (3, 4)} and S is the relation from {1, 2, 3, 4} to {0, 1, 2} with S = {(1, 0), (2, 0), (3, 1), (3, 2), (4, 1)}?
(a) Show that the relation defined on Z by m~n if 4m+n is divisible by 5 (b) is an equivalence relation. Show that the relation defined on Z by m~n if 5m +n is divisible by 4 is not an equivalence relation.
Discrete Maths
Please just answer parts iv. and v.
Question 2 List all possible equivalence relations on the set {1,2,3,4} without repetition up to isomorphism. Justify your answer.
) 7. Let A = {1,2,3,4}×{1,2,3,4}. Define an equivalence relation - by (x1,x2) ~ (X3,x4) iff x1x2 = xx4. List the elements of the equivalence class [(2,2)].
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.
Define relations R 1 , … , R 6 on { 1 , 2 , 3 , 4 } by R 1 = { ( 2 , 2 ) , ( 2 , 3 ) , ( 2 , 4 ) , ( 3 , 2 ) , ( 3 , 3 ) , ( 3 , 4 ) } , R 2 = { ( 1 , 1 ) , ( 1 , 2 ) , ( 2 , 1 ) , ( 2 , 2 ) , ( 3 , 3 ) , ( 4 , 4 ) } , R 3 = { ( 2 , 4 ) , ( 4 , 2 ) } , R 4 = { ( 1 , 2 ) , ( 2 , 3 ) , ( 3 , 4 ) } , R 5 = { ( 1 , 1 ) , ( 2 , 2 ) , ( 3 , 3 ) , ( 4 , 4 ) } , R 6 = { ( 1 , 3 ) , ( 1 , 4 ) , ( 2 , 3 ) , ( 2 , 4 ) , ( 3 , 1 ) , ( 3 , 4 ) } , Which of the following statements are correct? Check ALL correct answers below. A. R 2 is not transitive B. R 4 is antisymmetric C. R 5 is transitive D. R 6 is symmetric E. R 3 is transitive F. R 3 is reflexive G. R 3 is symmetric H. R 2 is reflexive I. R 4 is transitive J. R 5 is not reflexive K. R 4 is symmetric L. R 1 is not symmetric M. R 1 is reflexive
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
Let R be an equivalence relation on A = {0, 1,5, 6, 8, 9} and R = {(6, 1), (1,6), (6, 6), (1, 1), (0, 0), (5, 5), (5, 8), (5,9), (8,5), (8, 8), (8, 9), (9, 5), (9, 8), (9,9)}. Show the partition of A defined by the equivalence classes of R.

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Let X = {1, 2, 3, 4, 5, 6}. Which of the following could be an equivalence class of an equivalence relation on X? O {(1, 2), (3, 4), (5, 6)) O (1,3,5) O [1, 2), (3, 4), (5, 6}} O (12)(34)(56)