19. Transitive Closure of Relations

Try these first

1. 1.

   Define transitive closure in your own words.

   Hint

   Think about how elements connect through others.

2. 2.

   What does a Boolean matrix represent?

   Hint

   Recall the binary nature of relationships.

3. 3.

   The transitive closure R* of a relation R includes:

   • Only direct connections
   • Only indirect connections
   • Both direct and indirect connections

   Hint

   Recall the definitions of transitivity and reachability.

4. 4.

   True or False: You need to compute all powers of R to find R*.

   • True
   • False

   Hint

   Consider the distinct nodes and path lengths.

5. 5.

   Design a directed graph with 5 nodes and 10 edges; calculate its transitive closure using the connectivity relation.

   Hint

   Start with direct connections, then look for paths that connect indirectly.

6. 6.

   Given a Boolean matrix representing R, develop an algorithm to compute its R* with minimal computational effort and justify your approach.

   Hint

   Consider calculating powers of the matrix and their significance.