Prove that for directed graphs, the connected components algorithm finds the set of vertices that can be

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Prove that for directed graphs, the connected components algorithm finds the set of vertices that can be reached from a given initial vertex \(v\). Prove that this set is a closed set (see Example 7), and that if in addition every vertex \(u\) in the set can reach the initial vertex \(v\), then this set is a connected component.

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