Consider the following statements: A. There exists a Boolean algebra with ′5′…

Answer: Only statements B and D are correct.

• Statement A: There exists a Boolean algebra with 5 elements. — False. Any finite Boolean algebra has exactly 2^n elements for some nonnegative integer n (it is isomorphic to the power set algebra of an n-element set). Since 5 is not a power of 2, a Boolean algebra with 5 elements cannot exist.

• Statement B: Every element of a Boolean algebra has a unique complement. — True. Uniqueness proof sketch: if y and z are both complements of x, then y ∨ x = z ∨ x = 1 and y ∧ x = z ∧ x = 0. Then y = y ∧ 1 = y ∧ (x ∨ z) = (y ∧ x) ∨ (y ∧ z) = 0 ∨ (y ∧ z) = y ∧ z. Symmetrically z = y ∧ z, so y = z.

• Statement C: If a lattice L is a Boolean algebra then L is not relatively complemented. — False. Every Boolean algebra is relatively complemented: for any interval [a,b] and any x with a ≤ x ≤ b, a relative complement y (so that x ∨ y = b and x ∧ y = a) can be constructed. One convenient formula is y = (b ∧ x') ∨ a, where x' is the (global) complement of x in the Boolean algebra. Checking gives y ∨ x = b and y ∧ x = a, so relative complements exist in every interval.

• Statement D: The direct product of two Boolean algebras is also a Boolean algebra. — True. If A and B are Boolean algebras, define operations on A × B componentwise (meet, join, complement, 0 and 1). The Boolean axioms hold coordinatewise, so the product is a Boolean algebra. Complements are given componentwise as well.

Conclusion: Only the statements asserting uniqueness of complements and closure under direct product are correct; hence the correct choice is the one that selects those two statements.