1. Program modules can import other modules. [You have seen this al-ready in the Haskell course.] They can also depend indirectly on mod-ules via some chain of importation, so that for instance M depends on M ± if M imports M ± and M ± imports M ± . The relation ‘depends’ is the transitive closure of the relation ‘imports’. 2. Two people are related if one is the parent of the other, if they are mar-ried, or if there is a chain of such relationships joining them directly. We can model this by a universal set People , with three relations Mar-ried , Parent and Relative . The relation Relative is deFned using the transitive closure: Relative = (( Parent ∪ Parent-1 ) ∪ Married ) + D EFINITION 3.16 (T RANSITIVE C LOSURE ) Let R be a binary relation on A . The transitive closure of R , written t ( R ) or R + , is ± n ≥ 1 R n : that is, R ∪ R 2 ∪ R 3 ∪ R 4 ∪ . . . The transitive closure of a binary relation always exists. To cast more light on R +
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