Discrete Mathematical Structures by Pearson

8-24 Discrete Mathematical Structures

THEOREM 8.7 In a lattice L with universal lower bound 0 and universal upper bound 1, show

that 0 is the unique complement of 1 and 1 is the unique complement of 0.

Proof: Since 0 and 1 are lower and upper bounds

0 ∨ 1 = 1 and 0 ∧ 1 = 0

and hence 0 and 1 are complements to each other. To show that 1 is a unique complement of 0,

let us assume that a is another complement of 0.

Then,

0 ∨ a = a 0 ∧ a = 0

As 0 ∨ 1 = 1 ∴ a = 1

Thus, the complement of 0 is unique.

Similarly, it can be proved that 0 is the unique complement of 1.

8.20 DISTRIBUTIVE LATTICES

A lattice L is said to be distributive if for any elements ...