This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: COT3100 Summer2001 Assignment #3 (Solution) Assigned: 06/12, due: 06/26 on lecture. Total 100 pts. 1. Suppose A = {1, 2, 3}, B ={ a , b , c }, R = {(1, a ), (1, b ), (2, b ), (3, c )} is a relation from A to B and S = {( a , b ), ( a , c ), ( b , a ), ( c , c )} is a relation from B to B . Find the following relations: a) (2pts) R S R S = { (1, 4), (1, 5), (1, 6), (2, 4), (3, 6)} b) (2pts) S S S S = {(4, 4), (4, 6), (5, 5), (5, 6), (6, 6)} c) (3pts) R S 1 R S 1 = {(1, 4), (1, 5), (2, 4), (3, 4), (3, 6)} d) (3pts) S R 1 S R 1 = { (4, 1), (4, 2), (4, 3), (5, 1), (6, 3)} 2. Suppose R and S are two relations from A to B , R A B , S A B . Prove or disprove each of the following statements: a) (5pts) If R S , then R 1 S 1 . Proof . Assume R S . To prove R 1 S 1 , take arbitrary ( x , y ) R 1 to show that ( x , y ) S 1 . If ( x , y ) R 1 , then ( y , x ) R in accordance with definition of inverse relation. But R S by assumption, so ( y , x ) S by subset definition. If ( y , x ) S , then ( x , y ) S 1 by the definition of inverse relation. b) (8pts) ( R S ) 1 = R 1 S 1 Proof . We need to prove two subset relations, i) ( R S ) 1 R 1 S 1 and ii) R 1 S 1 ( R S ) 1 . i) Take arbitrary ( x , y ) ( R S ) 1 to prove that ( x , y ) R 1 S 1 . If ( x , y ) ( R S ) 1 , then ( y , x ) R S by the definition of inverse relation. Then there might be two cases: either ( y , x ) R or ( y , x ) S . If ( y , x ) R , then ( x , y ) R 1 by the definition of inverse relation, and ( x , y ) R 1 S 1 , because R 1 R 1 S 1 . If ( y , x ) S , then ( x , y ) S 1 , and ( x , y ) R  1 S 1 , since S 1 R 1 S 1 . ii). Take arbitrary ( x , y ) R 1 S 1 to prove that ( x , y ) ( R S ) 1 . If ( x , y ) R 1 S 1 , then either ( x , y ) R 1 or ( x , y ) S 1 by the definition of the union. In case ( x , y ) R 1 , ( y , x ) R , that implies that ( y , x ) R S . But if ( y , x ) R S , then ( x , y ) ( R S ) 1 . In another case, ( x , y ) S 1 , we may imply, that ( y , x ) S , and then that ( y , x ) R S ....
View
Full
Document
This document was uploaded on 07/14/2011.
 Spring '09

Click to edit the document details