Question

Which of the following boolean expression is related to De-Morgan theorem ?

• A. $x + xy = x$
• B. $x + (y + z) = (x + y) + z$
• C. $x(y + z) = xy + xz$
• D. $(x + y)' = x'y'$

Hint:

To remember De-Morgan's theorem, use the phrase: "Break the bar, change the sign." When you break the overline (complement), the $+$ becomes $\cdot$ (and vice versa).

Answer 1

The Correct Option is D

De-Morgan's theorems are fundamental rules in Boolean algebra that describe how to distribute a complement (negation) over a logical sum (OR) or a logical product (AND). 1. Defining De-Morgan's Laws: There are two primary statements in De-Morgan's theorem:
• First Law: The complement of a sum is equal to the product of the complements: $(A + B)' = A' \cdot B'$.
• Second Law: The complement of a product is equal to the sum of the complements: $(A \cdot B)' = A' + B'$. 2. Evaluating the Options:
• Option (1): $x + xy = x$ is the Absorption Law.
• Option (2): $x + (y + z) = (x + y) + z$ is the Associative Law.
• Option (3): $x(y + z) = xy + xz$ is the Distributive Law.
• Option (4): $(x + y)' = x'y'$ directly matches the First Law of De-Morgan's theorem. 3. Conclusion: Option (4) is the only expression that represents the relationship defined by De-Morgan's theorem, showing how the negation of an OR operation is equivalent to the AND of individual negations.