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Consider a relation on the set .
1. Let us state the steps of the Warshall's algorithm:
3. For all such that and for all let
4. If then stop and else go to step 2.
Let us find transitive closure of the relation using Warshall's algorithm:
It follows that
2. Let us write the matrix representation of It follows from the previous item that
3. Let us check whether the relation of is an equivalence relation.
Since but we conclude that the relation is not symmetric, and hence it is not an equivalence relation.
Let us check whether the relation of is a partial order.
Since the relation is reflexive. Taking into account that and imply for any , we conclude that this relation is antisymmetric. Since is the transitive closure, it is transitive.
We conclude that is a partial order.