GATE CS 2004 Q17 — Boolean Algebra

We can simplify the given Boolean expression $F = x'y' + xy + x'y$ using algebraic simplification or a Karnaugh map.

Method 1: Algebraic Simplification

Rearranging and grouping the terms:

Factor out $x'$ from the first two terms:

Since $y' + y = 1$:

Using the distributive law $A + BC = (A + B)(A + C)$:

Since $x' + x = 1$:

Method 2: Karnaugh Map

The minterms corresponding to the expression $x'y' + xy + x'y$ are:

• $x'y' \implies m_0$
• $x'y \implies m_1$
• $xy \implies m_3$

Plotting these on a 2-variable K-map:

Grouping the 1s:

• The horizontal pair $(m_0, m_1)$ simplifies to $x'$.
• The vertical pair $(m_1, m_3)$ simplifies to $y$.

Thus, the simplified expression is $x' + y$.

Therefore, the correct option is D.