### (Combinatorics, Set Theory, Logic:)

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**,⊂).

#### Examples.

- Let S be a set and k a natural number. The set
**F**_{k}(S) = {x⊆S: |x|=k} - In the poset (
**N**,|) of natural numbers partially ordered by divisibility, the set P of prime numbers is an antichain. Other antichains include the setsF

of all products of exactly k prime numbers._{k}= {p_{1}⋅...⋅p_{k}: p_{i}∈P}