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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.