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.