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MAS111_09_assignment_09 - MAS 111 FOUNDATION OF MATHEMATICS...

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MAS 111 FOUNDATION OF MATHEMATICS ASSIGNMENT 9 HINTS 1. The following two binary relations are defined on the set A = { 0 , 1 , 2 , 3 } . For each relation, determine whether it is reflexive, symmetric, tran- sitive. Give a counterexample in each case in which the relation does not satisfy one of the properties. (a) R 1 = { (1 , 2) , (2 , 1) , (1 , 3) , (3 , 1) } (b) R 2 = { (0 , 0) , (0 , 1) , (0 , 3) , (1 , 1) , (1 , 0) , (2 , 3) , (3 , 3) } Hint : (a) R 1 = { (1 , 2) , (2 , 1) , (1 , 3) , (3 , 1) } is symmetric, but not reflexive (for example, (1 , 1) is not in R 1 ), not transitive (for example, (1 , 2) , (2 , 1) are in R , but (1 , 1) is not in). (b) R 2 = { (0 , 0) , (0 , 1) , (0 , 3) , (1 , 1) , (1 , 0) , (2 , 3) , (3 , 3) } is transitive, but not reflexive ((2 , 2) is not in R ), not symmetric ((2 , 3) is in R , but (3 , 2) is not in). 2. Let R = { (0 , 1) , (0 , 2) , (1 , 1) , (1 , 3) , (2 , 2) , (3 , 0) } . Find R t , the transitive closure of R . Hint : It is easy to use directed graph to find R t . R t = { (0 , 1) , (0 , 2) , (1 , 1) , (1 , 3) , (2 , 2) , (3 , 0) , (0 , 3) , (1 , 0) , (3 , 1) , (1 , 2) , (3 , 2) } . 3. Let S = { (0 , 0) , (0 , 3) , (1 , 0) , (1 , 2) , (2 , 0) , (3 , 2) } . Find S t , the transitive closure of S . Hint : Easy. 4. Let A = { 21 , 23 , 24 , 29 , 30 , 31 , 36 , 37 , 40 , 41 , 42 , 43 , 45 } and let the rela- tion R on A be defined by x R y x - y is divisible by 3 .
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