2.Let A = {1,2, 3}, B = {4, 5, 6}, C = {7,8, 9}. Consider the relations R from A toB, 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.

Answered: Image /qna-images/answer/df7439b2-c4ca-4021-9284-fb257174dd8d.jpg

Answered: Let A = {1,2,3}, B = {4, 5, 6}, C =…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.CM: Cumulative Review

Problem 4CM: Use a software program or a graphing utility to write v as a linear combination of u1, u2, u3, u4,...

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Use a software program or a graphing utility to write v as a linear combination of u1, u2, u3, u4, u5 and u6. Then verify your solution. v=(10,30,13,14,7,27) u1=(1,2,3,4,1,2) u2=(1,2,1,1,2,1) u3=(0,2,1,2,1,1) u4=(1,0,3,4,1,2) u5=(1,2,1,1,2,3) u6=(3,2,1,2,3,0)
Determine whether the set S={1,x2,2+x2} spans P2.
Let T be a linear transformation from R2 into R2 such that T(4,2)=(2,2) and T(3,3)=(3,3). Find T(7,2).
Consider the relations R = {(1,5), (2, 2), (3, 4), (5, 2)}, S = {(2,4), (3, 4), (3, 1), (5, 5)}, T = {(1,4), (3, 5), (4, 1)} on the set A = {1,2,3, 4, 5}. Find (a) SoR = (b) RoT = (c) ToS = (d) Ro R= (e) SoS = (() ТоТ%—
3. Let A = {-1, –3, – 5}, B = {a, B, 7}, C = {x, y, z}. Consider the relations R from A to B and S from B to C respectively. R = {(-1,8), (–3, a), (–3, 7)} and S = {(a, y), (B, x), (7, y), (y, 2)} Find the following relations (a) Find the composition Ro S. (b) Find the matrices MR, Ms and MRos of the respective relations R, S and Ro S. (c) Compare M Ros to the product MRMS.
3. Let R1 = {(1, 2), (2, 3), (3, 4)} and R2 = {(1, 1), (1, 2), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3), (3, 4)} be relations from {1, 2, 3} to {1, 2, 3, 4). Find a) R1 U R2 b) R1 n R2. c) R1 - R2 d) R2 - R1.
3. Let A = {1,2, 3} and B = {1,2, 3,4}. The relations R1 = {(1,1), (2,2), (3,3)} and R2 = {(1,1), (1,2), (1,3), (1,4)} can be combined using set operations. Find the following: a. R1 U R2 b. Rị n R2 c. R1 \ R2 d. R2 \ R1
Let P={1.2,3,4,12,36} and the relation set R={(1,1) ,(2,2),(3,3),(4,4),(12,12),(36,36),(1,2),(1,3),(1,4),(1,12),(1,36),(2,4)(2,12),(2,36),(3,12),(3,26),(4,12),(4,36)}. Show that (P,R) is a poser. Represent the relation using the diagraph and then draw the Hasse diagram.
Define relations R₁, ..., R₁ on {1, 2, 3, 4} by R₁ = {(2, 2), (2, 3), (2, 4), (3, 2), (3, 3), (3, 4)}, R₂ = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (4,4)}, R3 = {(2, 4), (4, 2)}, R₁ = {(1, 2), (2, 3), (3, 4)}, R5 = {(1, 1), (2, 2), (3, 3), (4,4)}, R6 = {(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. R5 is not reflexive OB. R4 is antisymmetric c. R6 is symmetric D. R₂ is not transitive E. R₂ is reflexive F. R4 is symmetric G. R3 is reflexive H. R5 is transitive OI. R₁ is not symmetric J. R₁ is reflexive K. R4 is transitive OL. R3 is symmetric M. R3 is transitive
Let A = {1, 2, 3, 4} and B = {a, b, c, d, e}. How many different relations are there from A to B. 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…
12. Consider the following relations on {1, 2, 3, 4}. R1 = {(2,2), (2,31.2,4).(3,2).(3,3).(3,4)} R2 = {(1,1L(1,2),(2,1).(2,2).(3,3).(4,4)} R3 = {2,4)l4,2)} R4 = {(1,21,(2,3).(3,4)} Rs = {(1,1L(2,2).(3,3).(4,4)} %3D a) Which of these relations are reflexive? Justify your answers. b) Which of these relations are symmetric? Justify your answers. c) Which of these relations are antisymmetric? Justify your answer. d) Which of these relations are transitive? Justify your answers.

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