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Antichains of a power set

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Would someone be able to explain to me how we would find antichains and chains of a powerset?

For example, if we had the power set of 4 - P([4]) then how would we derive the antichains, and symmetric chains?

What about for the power set of [5]? (where [5] is the set of numbers {1,2,3,4,5}.
Is there are a general formula to follow or simply a trial and error method?

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Solution Summary

The expert examines antichain as a power set.

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There is no formula--calculating the number of antichains is a difficult problem--but there is an algorithm. The method for generating maximal antichains is a construction: Start with a single subset and expand it to a collection of sets, none of which are comparable. Repeat with each subset.

Let's build an ...

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