Question

11. Show that the two distributive equalities are equivalent in a lattice. That is, x ∨ (y ∧z) = (x ∨ y) ∧ (x ∨ z) if and only if x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z).

Answer 1

Given that

we have to show that the two distributive equalities are equivalent in a lattice.

x ∨ (y ∧z) = (x ∨ y) ∧ (x ∨ z) if and only if x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z).

let be a finite lattice.

we know that is greatest lower bound and   is least upper bound

. similarly is greatest lower bound of and

is least upper bound of .

Therefore we have

for all  

Thus L is bounded.