55PS12 - SOLUTION SET 12DUE Problem 1#8.6.8 Determine...

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Problem 1 (#8.6.8) . Determine whether the relations represented by the following zero-one matrices are partial orders: a) 1 0 1 1 1 0 0 0 1 b) 1 0 0 0 1 0 1 0 1 c) 1 0 1 0 0 1 1 0 0 0 1 1 1 1 0 1 Solution. By inspection, all three matrices are reflexive and antisymmetric, so the only thing we have to check in each case is transitivity. Relation (a) is not transitive, since the second element relates to the first, and the first to the third, but the second element doesn’t relate to the third. Relation (b) is transitive, and represents the partial order 3 ± 1. Relation (c) also isn’t transitivethe first element relates to the third, and the third to the fourth, but the first doesn’t relate to the fourth. Problem 2 (#8.6.10) . Determine whether the relation with the shown directed graph is a partial order. Solution. The graph shown is almost that of the total ordering a ± c ± d ± b , but it’s missing c ± b . Since it isn’t transitive, it’s not the digraph of a partial ordering. Problem 3 (#8.6.12) . Let ( S,R ) be a poset. Show that ( S,R - 1 ) is also a poset, where R - 1 is the inverse of R . The poset ( S,R - 1 ) is called the dual of ( S,R - 1 ) . Proof. We must show that R - 1 is a partial ordering, given that R is. Since ( a,a ) R , ( a,a ) R - 1 ,so R - 1 is reflexive. Suppose ( a,b ) R - 1 and ( b,a ) R - 1 . Then ( b,a ) R and ( a,b ) R , so a = b . Thus, R - 1 is also antisymmetric. Finally, suppose ( a,b ) R - 1 and ( b,c ) R - 1 . Then ( b,a ) R and ( c,b ) R , so by the transitivity of R, ( c,a ) R . Thus ( a,c ) R - 1 , and R - 1 is also transitive. Thus R - 1 is a partial ordering, and S,R - 1 is a poset, as claimed. Problem 4 (#8.6.28) . Let ( S, ± ) be a poset. We say that an element y S covers an elements x S if x y and there is no element z S such that x z y . The set of pairs ( x,y ) such that y covers x is called the covering relation of ( S, ± ) . What is the covering relation of the partial ordering { ( a,b ) | a divides b } on { 1 , 2 , 3 , 4 , 6 , 12 } ? Solution.
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55PS12 - SOLUTION SET 12DUE Problem 1#8.6.8 Determine...

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