Prove that, the complement of each element in a Boolean algebra B is unique.
Ans: Proof: Let I and 0 are the unit and zero elements of B correspondingly. Suppose b and c be two complements of an element a ∈ B. After that from the definition, we have
a ∧ b = 0 = a ∧ c and
a ∨ b = I = a ∨ c
We can write b = b ∨ 0 = b ∨ (a ∧ c )
= (b ∨ a) ∧ (b ∨ c) [as lattice is distributive ]
= I ∧ (b ∨ c )
= (b ∨ c )
Likewise, c = c ∨ 0 = c ∨ (a ∧ b )
= (c ∨ a) ∧ (c ∨ b) [as lattice is distributive]
= I ∧ (b ∨ c) [as ∨ is a commutative operation]
= (b ∨ c)
The above two results define that b = c.