Applying DeMorgan and removing double negations Consider a Boolean expression containing both double inversions and single inversions. After fully applying DeMorgan’s theorems and canceling double inversions, the result should contain only single inversion marks on single variables. Is this statement valid?

Introduction / Context:
DeMorgan’s theorems are the primary tools for pushing negations inward through sums and products, essential when mapping logic into NAND-only or NOR-only realizations. Understanding what the fully propagated form looks like prevents common sign errors.

Given Data / Assumptions:

• Negations can exist on groups (parenthesized terms) and on single variables.
• We apply DeMorgan repeatedly and remove double inversions through the rule that a double inversion restores the original operand.

Concept / Approach:
DeMorgan’s rules: complement of a sum becomes product of complements, and complement of a product becomes sum of complements. Every time a bar crosses a parenthesis, the operator toggles (+ ↔ ·) and the complement distributes to each literal. Double inversions over any term cancel out.

Step-by-Step Solution:
Start with a complemented group such as (X + Y)̄ or (X · Y)̄.Apply DeMorgan to push the bar to each literal, switching operators.Wherever a literal is already complemented and receives another bar, remove the pair (double negation).Final form places single inversion marks only on variables (literals), not on multi-term groups.

Verification / Alternative check:
Construct truth tables before and after the transformation; they match exactly. Logic network drawings with bubbles (signal inversion marks) also show bubble-pushing results in inversions only on inputs of gates.

Why Other Options Are Wrong:
Limiting validity to two variables or SOP only is unnecessary; the property holds for arbitrary-size expressions and forms as long as DeMorgan and double-negation rules are correctly applied.

Common Pitfalls:
Forgetting to flip the operator when moving the bar, or missing a double-negation cancellation, leading to incorrect literal polarities.

Final Answer:
Valid