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For set S = {1, 2, 3, 4}, we define the following relations 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)}, R4 = {(1, 2), (2, 3), (3, 4)}, R, = {(1, 1), (2, 2), (3, 3), (4, 4)}, Re = {(1, 3), (1, 4), (2, 3), (2, 4), (3, 1), (3, 4)}, Determine which of these statements are correct. Check ALL correct answers below. ORĮ is reflexive O R3 is transitive O RĄ is antisymmetric | R3 is symmetric | R2 is reflexive O RĄ is transitive | R3 is reflexive R5 is not reflexive O R6 is symmetric R5 is transitive |R2 is not transitive R1 is not symmetric O RĄ is symetric

For set S = {1, 2, 3, 4}, we define the following relations 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)}, R4 = {(1, 2), (2, 3), (3, 4)}, R, = {(1, 1), (2, 2), (3, 3), (4, 4)}, Re = {(1, 3), (1, 4), (2, 3), (2, 4), (3, 1), (3, 4)}, Determine which of these statements are correct. Check ALL correct answers below. ORĮ is reflexive O R3 is transitive O RĄ is antisymmetric | R3 is symmetric | R2 is reflexive O RĄ is transitive | R3 is reflexive R5 is not reflexive O R6 is symmetric R5 is transitive |R2 is not transitive R1 is not symmetric O RĄ is symetric

Advanced Engineering Mathematics
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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For set S = {1, 2, 3, 4}, we define the following relations 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)}, R4 = {(1, 2), (2, 3), (3, 4)}, R5 = {(1, 1), (2, 2), (3, 3), (4, 4)}, Re = {(1, 3), (1, 4), (2, 3), (2, 4), (3, 1), (3, 4)}, Determine which of these statements are correct. Check ALL correct answers below. ORĮ is reflexive R3 is transitive O RĄ is antisymmetric | R3 is symmetric | R2 is reflexive O RĄ is transitive R3 is reflexive R5 is not reflexive O R6 is symmetric R5 is transitive |R2 is not transitive R1 is not symmetric O RĄ is symetric
Transcribed Image Text:For set S = {1, 2, 3, 4}, we define the following relations 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)}, R4 = {(1, 2), (2, 3), (3, 4)}, R5 = {(1, 1), (2, 2), (3, 3), (4, 4)}, Re = {(1, 3), (1, 4), (2, 3), (2, 4), (3, 1), (3, 4)}, Determine which of these statements are correct. Check ALL correct answers below. ORĮ is reflexive R3 is transitive O RĄ is antisymmetric | R3 is symmetric | R2 is reflexive O RĄ is transitive R3 is reflexive R5 is not reflexive O R6 is symmetric R5 is transitive |R2 is not transitive R1 is not symmetric O RĄ is symetric
In this problem we work out step-by-step the procedure for checking an equivalence relation. Denote by Z the set of all integers. Declare that two integers æ, y are related if x+y is an integer multiple of 7. In symbols: x ~ y + 7 divides x + y We want to check if this is an equivalence relation. That means we need to check if ~ is (1) Reflexive (2) Symmetric (3) Transitive We begin with (1). This means checking to make sure that for all integers æ, we have æ ~ x. Recall the definition of ~ for this problem and we see that this is equivalent to saying that for all integers x, we have that (x+x) is an integer multiple of 7. Is this true? If so, enter Y; if not, enter an integer for which this is false Next, we check (2). This means checking to make sure that for all integers æ, y we have a ~ y # y ~ x. Unwind the definition of ~ as we have done for (1) and we see that x ~ y A x + y = 7m for some integer m y ~ æ → y + x = 7m for some integer m Based on that, is (2) true? If so, enter Y; if not, enter a pair of integers for which this is false. Finally, we check (3). This means checking to make sure that for all integers æ, y, z, if æ ~ y and y ~ z then æ ~ z. Is this true? If so, enter Y; if not, give a triple of integers for which this fails. Finally, based on this calculation, is - an equivalence relation on the set of integers? Enter Y or N.
Transcribed Image Text:In this problem we work out step-by-step the procedure for checking an equivalence relation. Denote by Z the set of all integers. Declare that two integers æ, y are related if x+y is an integer multiple of 7. In symbols: x ~ y + 7 divides x + y We want to check if this is an equivalence relation. That means we need to check if ~ is (1) Reflexive (2) Symmetric (3) Transitive We begin with (1). This means checking to make sure that for all integers æ, we have æ ~ x. Recall the definition of ~ for this problem and we see that this is equivalent to saying that for all integers x, we have that (x+x) is an integer multiple of 7. Is this true? If so, enter Y; if not, enter an integer for which this is false Next, we check (2). This means checking to make sure that for all integers æ, y we have a ~ y # y ~ x. Unwind the definition of ~ as we have done for (1) and we see that x ~ y A x + y = 7m for some integer m y ~ æ → y + x = 7m for some integer m Based on that, is (2) true? If so, enter Y; if not, enter a pair of integers for which this is false. Finally, we check (3). This means checking to make sure that for all integers æ, y, z, if æ ~ y and y ~ z then æ ~ z. Is this true? If so, enter Y; if not, give a triple of integers for which this fails. Finally, based on this calculation, is - an equivalence relation on the set of integers? Enter Y or N.
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