does the system of all subsets of a finite set under the operation subset of form a lattice
A partially ordered set is a bounded lattice if and only if every finite set of elements (including the empty set) has a join and a meet. For every element x of a poset it is trivially true (it is a vacuous truth) that ∀ a ∈ ∅ : x ≤ a (\displaystyle \forall a\in \varnothing :x\leq a) and ∀ a ∈ ∅ : a ≤ x (\displaystyle \forall a\in \varnothing :a\leq x) , and therefore every element of a poset is both an upper bound and a lower bound of the empty set. This implies that the join of an empty set is the least element ⋁ ∅ = 0 (\displaystyle \bigvee \varnothing =0) , and the meet of the empty set is the greatest element ⋀ ∅ = 1 (\displaystyle \bigwedge \varnothing =1) . This is consistent with the associativity and commutativity of meet and join: the join of a union of finite sets is equal to the join of the joins of the sets, and dually, the meet of a union of finite sets is equal to the meet of the meets of the sets, i.e., for finite subsets A and B of a poset L,
Lattice (order)