Define relations \( R_1, \ldots, R_6 \) 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. \( R_5 \) is not reflexive- □ B. \( R_4 \) is antisymmetric- □ C. \( R_6 \) is symmetric- □ D. \( R_2 \) is not transitive- □ E. \( R_2 \) is reflexive- □ F. \( R_4 \) is symmetric- □ G. \( R_3 \) is reflexive- □ H. \( R_5 \) is transitive- □ I. \( R_1 \) is not symmetric- □ J. \( R_1 \) is reflexive- □ K. \( R_4 \) is transitive- □ L. \( R_3 \) is symmetric- □ M. \( R_3 \) is transitive

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Answered: Define relations R₁, ..., R₁ on {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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Let X = {a, b, c, d, e,f} and S be the relation on X defined by S = || {(а, а), (а, b), (а, с), (а, d), (а, е), (а, , (b, b), (b, d), (b, е), (b,), (с, с), (с, d), (c, e), (c,f), (d,d), (d, e), (d,f), (e, e), (f,f)}. Is S a partial order? If so, draw its Hasse diagram. If not, give an example showing why.
Consider the relation R = {(1, 2), (1, 4), (2, 3), (3, 1), (4, 2)} on the set {1, 2, 3, 4}. What is the transitive closure of R? O {(1, 2), (1, 4), (2, 3), (3, 1), (4, 2), (1,3), (2,1). (3, 2), (3, 4), (4,3)} O {(1.3), (2,1), (3, 2), (3, 4), (4,3), (1, 1), (2, 2). (2, 4), (3, 3), (4, 1), (4, 4)} O {(1, 2), (1, 4), (2, 3), (3,1), (4,2), (1,3), (2,1), (3, 2), (3, 4). (4, 3), (1, 1). (2, 2), (2, 4), (3. 3). (4, 1). (4, 4)} O {(1,3), (2,1). (3, 2), (3, 4), (4,3), (1, 1), (2, 2), (2, 4), (3, 3), (4, 1), (4, 4)}
There are 6 towns {A, B, C, D, E, F} with the following one way street relation {(A, B), (A, C), (B, D), (C, B), (D, C), (D, F), (E, C), (E, F)}. Show all pairs of distinct towns that are reachable from one another in both directions.
3. Draw the Hasse diagram of the relation R on A, where A = (1, 2, 3, 4}, andR = {(1, 1), (1, 2), (2, 2), (2, 4), (1, 3), (3, 3), (3, 4), (1, 4), (4, 4)}
If R = {(1, 2), (1, 4), (2, 3), (3, 1), (4, 2)}, what is the reflexive closure of R? O {(1, 2), (2, 1). (1, 4), (4, 1). (2, 3), (3, 2), (3, 1). (1, 3), (4, 2), (2, 4)} O {(1, 1), (1, 2), (1, 4), (2, 2), (2, 3), (3, 1), (3, 3), (4, 2), (4, 4)} O {(1, 2), (1, 4), (2, 3). (3, 1), (4, 2)} O {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1). (3. 2). (3, 3), (3, 4), (4, 1), (4, 2). (4, 3), (4, 4)}
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.
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:
6. If A = {1, 2, 3} and B = {2, 3, 4}, and R is the relation xRy when x + y is divisible by 3, then R = O {1, 2), (2, 4)} all of these O {(1, 2), (2, 4), (3, 3)} O {1, 2), (3, 3)}
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.
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.
If R = {(1, 2), (1, 4), (2, 3). (3, 1), (4, 2)}, what is the symmetric closure of R? O {(1, 2), (2, 1), (1, 4), (4, 1), (2, 3), (3, 2), (3, 1), (1, 3), (4, 2), (2, 4)} O {(1, 1), (1, 2), (1, 4), (2, 2), (2, 3), (3, 1). (3, 3), (4, 2), (4, 4)} O {(1, 2), (1, 4), (2, 3), (3, 1), (4, 2)} O {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1). (3, 2). (3, 3), (3, 4), (4. 1), (4, 2), (4, 3), (4, 4)}

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