Give an example of a topological sort, a maximal chain, and a maximal anti-chain.
Topological Sort: In a directed acyclic graph (DAG), a topological sort is a linear ordering of its vertices in such a way that for every directed edge u -> v, vertex u comes before v in the ordering.
Maximal Chain: A chain in a partially ordered set is a sequence of elements where each element is comparable to the one before it (i.e., for every pair of elements in the chain, one is less than or equal to the other). A maximal chain is a chain that cannot be extended by including any additional element without violating the partial order.
Maximal Antichain: An anti chain in a partially ordered set is a subset of elements where no two elements are comparable (i.e., there is no directed edge between them). A maximal anti chain is an anti chain that cannot have any more elements added to it without violating the partial order.
Let's illustrate these concepts with an example:
Consider a partial order on the set of integers {1, 2, 3, 4, 5, 6} where the relation is defined as follows:
1 <= 2
1 <= 3
2 <= 4
3 <= 4
4 <= 5
5 <= 6
This partial order can be represented as a directed acyclic graph (DAG) where the vertices are the integers, and there are directed edges between them as specified by the relations.
A topological sort of this DAG could be: [1, 2, 3, 4, 5, 6] because it satisfies the condition that for every directed edge u -> v, u comes before v in the ordering.
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