t2keyfa00

t2keyfa00 - COT3100.01, Fall 2000 S. Lang (11/02/2000)...

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COT3100.01, Fall 2000 S. Lang (11/02/2000) Solution Key to Test #2 Nov. 11, 2000 Part I. (36 pts.) True/False questions. (No explanation is needed.) 1. Let A = {1, 2, 3, 4} and R A × A denote a binary relation depicted by the directed graph given in the figure. Answer each of the following True/False questions: 2. Let R A × A and T A × B denote two arbitrary binary relations. Answer each of the following True/false questions: (a) If R is transitive, then R - 1 is transitive. True , because if ( a , b ), ( b , c ) R - 1 , then ( b , a ), ( c , b ) R , thus ( c , a ) R because R is transitive by assumption. Therefore, ( a , c ) R - 1 . (b) The reflexive closure r ( R ) is irreflexive. False , because ( a , a ) r ( R ) for all a A . (c) ( R ο T ) ( R ο R ο T ). False , for example, R = {(1, 2)}, T = {(2, 3)}. Then R ο R = , R ο T = {(1, 3)}, and R ο R ο T = . (d) If R R - 1 , then R is symmetric. True , if ( a , b ) R , then ( a , b ) R - 1 because R R - 1 by assumption. Thus, ( b , a ) R . 3. Let f : A B be a surjection and let g : B C be a surjection, where A , B , C denote finite sets. Answer the following True/false questions: (a) The composed function g ο f : A C is a surjection. True , this is a theorem. (b) | A | | C |. True , because | A | | B | | C |, by the counting principle. 4. Let R A × A denote a binary relation. Answer the following True/false questions: (a) If R defines a function, and the inverse relation R - 1 also defines a function, then R is an injection. True
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t2keyfa00 - COT3100.01, Fall 2000 S. Lang (11/02/2000)...

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