MAS111_09_assignment_08 - a,b,c,d,n ∈ Z k ∈ N with a ...

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MAS 111 FOUNDATION OF MATHEMATICS ASSIGNMENT 8 TUTORIAL DATES: 20, 21, 22/10/09 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) } 2. Let R = { (0 , 1) , (0 , 2) , (1 , 1) , (1 , 3) , (2 , 2) , (3 , 0) } . Find R t , the transitive closure of R . 3. Let S = { (0 , 0) , (0 , 3) , (1 , 0) , (1 , 2) , (2 , 0) , (3 , 2) } . Find S t , the transitive closure of S . 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 . (a) List elements of R . (b) Show that R is an equivalence relation. 5. Let A be any nonempty set. Suppose that B A . Define R = { ( E, F ) : E, F A, E B = F B } . Show that R is an equivalence relation on P ( A ), the power set of A . 6. Let
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Unformatted text preview: a,b,c,d,n ∈ Z , k ∈ N with a ≡ b (mod m ) and c ≡ d (mod m ). Then (a) a + c ≡ b + d (mod m ) (b) a-c ≡ b-d (mod m ) 1 (c) ac ≡ bd (mod m ) (d) a k ≡ b k (mod m ) (e) Find the remainder when 6 2009 is divided by 37. 7. Let R be a relation on a set A . (a) Prove that R is transitive if and only if R 2 ⊆ R . (b) Let 1 A = { ( x,x ) : x ∈ A } ( 1 A is called the identity relation on A ). Prove that R is reflexive if and only if 1 A ⊆ R . 8. Suppose that R is a relation on a set A . Is RR-1 the same as 1 A ? Justify your answer. 9. Let R be a relation from A to B and let 1 B be the identity relation on B . Prove that R 1 B = R . 2...
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  • Spring '10
  • DrChanSongHeng
  • Equivalence relation, Binary relation, Transitive relation, relation, Transitive closure, FOUNDATION OF MATHEMATICS

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