GATE CS 1996 Q2.24 — Minimization (K-Maps etc.)

To find the equivalent Boolean expression in Product-of-Sums (POS) form, we focus on grouping the $0$s in the Karnaugh map.

Step 1: Identify the groups of $0$s

From the K-map, the $0$s are located in the cells where:

• $B = 0$ and $D = 0$
• $B = 1$ and $D = 1$

Since these groups are independent of variables $A$ and $C$, they form two larger subcubes (quads) of size 4.

Step 2: Write the Sum Terms

• For the group where $B = 0$ and $D = 0$, the corresponding sum term is: $$(B + D)$$
• For the group where $B = 1$ and $D = 1$, the corresponding sum term is: $$(\overline{B} + \overline{D})$$

Step 3: Combine into POS Form

Multiplying the sum terms together gives the simplified POS expression:

Verification

Expanding this POS expression yields the Sum-of-Products (SOP) form:

This represents the XOR operation $B \oplus D$, which matches the given K-map distribution.