In the combinatorics of sets, especially finite sets,
an *antichain* is a family **F** of sets such that no
member contains any other member:

∀x,y∈**F**. ¬(x⊂y)

More generally, we can define antichains whenever (P,<) is
a *poset* (a partially ordered set). A subset F⊆P is an *antichain* if no two elements of F can
be compared:

∀x,y∈F. ¬(x<y)

The definition for sets is a special case of the definition for
posets (when the family **F** is a set): take the poset
(**F**,⊂).

- Let S be a set and k a natural number. The set
**F**_{k}(S) = {x⊆S: |x|=k}

of all subsets of S of size k is an antichain.
- In the poset (
**N**,|) of natural numbers partially ordered
by divisibility, the set P of prime numbers is an antichain.
Other antichains include the sets
F_{k} =
{p_{1}⋅...⋅p_{k}: p_{i}∈P}

of all products of exactly k prime numbers.