GATE CS 2011 Q14 — Minimization (K-Maps etc.)

We are given the Boolean product-of-sums (POS) expression:

We can simplify this step-by-step using Boolean algebra identities.

Step 1: Simplify the first two terms

Let us group the first two terms:

Using the distributive law identity, $(X + Y)(X + \bar{Y}) = X$, where $X = P + \bar{Q}$ and $Y = \bar{R}$:

Step 2: Combine with the third term

Now, substitute this back into the original expression:

Using the distributive law $X + YZ = (X + Y)(X + Z)$ in reverse, where $X = P$, $Y = \bar{Q}$, and $Z = Q + \bar{R}$:

Step 3: Expand and simplify

Distribute $\bar{Q}$ over $(Q + \bar{R})$:

Since $\bar{Q}Q = 0$:

Thus, the simplified Sum of Products (SOP) form is $(P + \bar{Q}\cdot\bar{R})$, which corresponds to option B.