Final answer:
Warshall's algorithm is used to find the transitive closure of a relation by updating an adjacency matrix iteratively. For the given relation, apply Warshall's updates based on potential paths through intermediate nodes. A Hasse diagram for divisibility represents the partially ordered set, connecting numbers via edges if they divide without intermediate divisors.
Step-by-step explanation:
Warshall's Algorithm for Transitive Closure
To find the transitive closure of a relation on a set using Warshall's algorithm, we start with the adjacency matrix of the relation and iteratively update it for each node (vertex) in the set. The initial adjacency matrix for the relation {(2,1),(2,3),(3,1),(3,4),(4,1),(4,3)} on the set (1,2,3,4) is:
M0 =
1 2 3 4
1 0 0 0 0
2 1 0 1 0
3 1 0 0 1
4 1 0 1 0
Following Warshall's algorithm, we go through every node k=1 to 4 and update the matrix:
-
- For k=1, no changes are required since M0 already reflects the direct connections to node 1.
- For k=2, we check connections via node 2, which would not induce any additional connections.
- For k=3, we update connections through node 3. Thus, since there is a path from 2 to 3 and from 3 to 1, we add a path from 2 to 1, and similarly for path from 4 to 1 through node 3.
- For k=4, we observe if any new connections are made through node 4, which would not result in any new paths as existing paths cover all connections.
The final matrix represents the transitive closure:
M4 =
1 2 3 4
1 0 0 0 0
2 1 0 1 0
3 1 0 0 1
4 1 0 1 0
Hasse Diagram for Divisibility
To draw the Hasse diagram for divisibility on the set {2,3,6,12,24,36,48,72}, we arrange the numbers in layers representing their power of divisibility. The bottom layer has the smallest number, 2, and we draw edges upwards to numbers it directly divides without intermediate divisors. Thus, 2 would connect to 6, and 6 to 12, and so on. We continue this for all numbers in the set, ensuring that for each pair of numbers, if one divides the other, there is a path in the diagram connecting them.
The Hasse diagram will have edges from smaller to larger numbers indicating direct divisibility. It visualizes the partially ordered set where each element is a divisor of the elements above it, but not of the elements on the same level or below.