8. Let A = {1, 2, 3, 4, 5).(a) Determine, with justification, if the following relation is reflexive.R== {(1, 1), (1, 2), (3, 1), (2, 2), (5, 4), (4, 4), (5, 5), (3, 2), (3, 4), (2, 5), (2, 1)}(b) Determine, with justification, if the following relation is symmetric.R = {(1, 1), (2, 2), (2, 3), (3, 2), (4, 5), (5, 1), (3, 4), (5, 4), (1, 5), (4,4)}(c) Determine, with justification, if the following relation is antisymmetric.R = {(2, 2), (3, 1), (1, 1), (3, 3), (5, 4), (5, 5), (2, 4), (4, 3), (4,5)}(d) Determine, with justification, if the following relation is transitive.R = {(1, 2), (2, 3), (1, 3), (4, 4), (3, 4), (3, 5), (1, 4), (2,4)}

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Answered: 8. Let A = {1, 2, 3, 4, 5}. (a)…

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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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
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.
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.
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
1. Let A = {a, b, c} and B = {r,s, t, u}. Furthermore, let R = {(a, s), (a, t)}, (b, t) be a relation from A to B. Determine dom R ww and ran R.
17. Let R and S be relations on A= {1, 2, 3, 4} defined by R={(a, b) :b=5-a} and S = {(a, b): a
Let G = {-23, –7, -3, –1,1,3, 5, 7, 14, 27, 42}. Decide if the following statements are true/false. Give a support to your answer. (a) Vr E G : if I 7. (d) 3r, y E G : (r² - y² = 0) ^ (x > 10) ^ (y > 10).
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.
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…
For the set A = {2, 3, 4, 5, 6, 7, 8), relation R defined by R = {(x,y) = AxA | x divides y} Answer the following: What is the Domain and Range of Relation R? Domain = {2,3,4] and Range = {4,6,8) Domain A and Range A Domain A and Range = A' (or A-Compliment) Domain A and Range = A Draw Arrow, Matrix and DiGraph diagrams for the Relation R. 3 diagrams total. legibly, and upload a clear photo.
9.16. Let A = {a, b, c, d}. How many relations defined on A are reflexive, symmetric and transitive and contain the ordered pairs (a, b) , (b, c) , (c, d)?
Let A = {1, 2, 3, 4} and B = {a, b, c, d, e}. How many different relations are there from A to B. Why?

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