Solution

The correct answer is Only B and D are correct

Key Points

• A. There exists a Boolean algebra with '5' elements.
  • This statement is false. A Boolean algebra will have 2ⁿ elements, where n is a positive integer (as it consists of all subsets of a set of size n). Therefore, if it has 5 elements, then it is not a Boolean algebra because 5 is not a power of 2.
• B. Every element of Boolean algebra has a unique complement.
  • This statement, however, is true. In Boolean algebra, for every element A, there exists a unique element B such that A ^ B = 0 (the zero element, or "false") and A v B = 1 (the one element, or "true"). Thus, every element does have a unique complement.
• C. If a Lattice 'L' is a Boolean algebra then 'L' is not relatively complemented.
  • This statement is false. A lattice 'L' is called complemented if for every element a there is an element b such that a ∨ b = 1 and a ∧ b = 0. And these elements are called complements of each other. If these complements are unique then the lattice is called relatively complemented. A Boolean algebra is a special kind of lattice where the complement for each element is unique. So, actually, if 'L' is a Boolean algebra, then 'L' is relatively complemented.
• D. The direct product of two Boolean Algebras is also a Boolean algebra.
  • This statement is true. The direct (or Cartesian) product of two Boolean algebras produces a new Boolean algebra. For any two Boolean algebras A and B, their Cartesian product A × B, equipped with componentwise operations, is also a Boolean algebra.

Based on the above analysis, the correct answer is: 2) Only B and D are correct