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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This note was uploaded on 01/23/2011 for the course MAS 111 taught by Professor Drchansongheng during the Spring '10 term at Nanyang Technological University.

Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online