key3fa00 - COT3100.01 Fall 2000 S Lang Solution Key to...

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COT3100.01, Fall 2000 S. Lang Solution Key to Assignment #3 (40 pts.) Due: 10/24 in class First, we define some terms and notations. Definition. Consider a binary relation R A × B . The inverse of R , denoted R –1 , is a binary relation B × A such that R –1 = {( b , a ) | ( a , b ) R }, that is, R –1 contains pairs of elements which have the reverse order as they are in relation R . (In some text R –1 is called the converse of R , denoted R c .) Definition. Let R A × A denote a binary relation. The following relations defined over A are called closures : (a) The reflexive closure of R is r ( R ) = R {( a , a ) | a A }. (b) The symmetric closure of R is s ( R ) = R R –1 . (c) The transitive closure of R is t ( R ) = R R 2 R 3 ... , where R 2 = R R , R 3 = R 2 R , etc., where denotes relation composition. Thus, ( a , b ) t ( R ) ( a , b ) R n , for some n 1 there exist a 1 , a 2 , …, a n A , a n = b , for some n 1, such that ( a , a 1 ), ( a 1 , a 2 ), …, ( a n - 1 , a n ) R , i.e., there exists a direct path of n edges connecting a to b in the digraph for the relation R . It can easily be seen that the names of these closures are justified in that for any binary relation R , r ( R ) is reflexive, s ( R ) is symmetric, and t ( R ) is transitive. In fact, the relation r ( R ) is the smallest relation containing R and is also reflexive; s ( R ) is the smallest relation containing R and is symmetric; t ( R ) is the smallest relation containing R and is transitive. Also, we can define the composition of these closures, e.g., tr ( R ) = t ( r ( R )), rs ( R ) = r ( s ( R )), etc. The following figures show a binary relation R depicted by its digraph representation, and the corresponding closures: Now, answer the following questions neatly and concisely. All proofs need to be justified by using the appropriate definitions, theorems, and logical reasoning.
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