Example
For n = 2, there are six monotonic Boolean functions and six antichains of subsets of the two-element set {x,y}:
- The function f(x,y) = false that ignores its input values and always returns false corresponds to the empty antichain Ø.
- The logical conjunction f(x,y) = x ∧ y corresponds to the antichain { {x,y} } containing the single set {x,y}.
- The function f(x,y) = x that ignores its second argument and returns the first argument corresponds to the antichain { {x} } containing the single set {x}
- The function f(x,y) = y that ignores its first argument and returns the second argument corresponds to the antichain { {y} } containing the single set {y}
- The logical disjunction f(x,y) = x ∨ y corresponds to the antichain { {x}, {y} } containing the two sets {x} and {y}.
- The function f(x,y) = true that ignores its input values and always returns true corresponds to the antichain {Ø} containing only the empty set.
Read more about this topic: Dedekind Number
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