Discrete Mathematics Proving Problem: Chain is Distributive Lattice

It is proved that both A and B are elements of chain A, because any two elements in the chain can be compared, that is, a≤b or a≤b. If A ≤ B, the maximum lower bound of A and B is A, and if b≤a, the maximum lower bound of A and B is B and the minimum upper bound is A, so the chain must be a lattice. Enough to prove the law of distribution.

(1) b ≤ a or c ≤ a.

(2) A ≤ B and a ≤ c.

In the first case, a ∨( b∩c)= a =(a∪b)∩(a∪c).

In case (2), a ∨( b∩c)= b∩c =(a∪b)∩(a∪c).

In both cases, the distributive law holds, so A is distributive lattice.