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

Concept:
• De-Morgan's Theorems are fundamental laws in Boolean algebra used to simplify expressions by relating the complement of a sum/product to the productsum of the complements.
• The first law states: The complement of the sum of two variables is equal to the product of the complements of the variables: \((A + B)' = A' \cdot B'\).
• The second law states: The complement of the product of two variables is equal to the sum of the complements of the variables: \((A \cdot B)' = A' + B'\).

Step 1: Identify the nature of the other options

• Option (A) \(x + xy = x\): This is the Absorption Law.
• Option (B) \(x + (y + z) = (x + y) + z\): This is the Associative Law.
• Option (C) \(x(y + z) = xy + xz\): This is the Distributive Law.

Step 2: Match the target theorem
Option (D) \((x + y)' = x'y'\) is exactly the mathematical formulation of the first De-Morgan's theorem. It shows that an OR gate followed by an Inverter is logically equivalent to an AND gate with inverted inputs (Bubbled AND).