GATE CS 2006 Q38 — Minimization (K-Maps etc.)

To prevent a static-1 hazard (where the output momentarily drops to 0 when a single input changes between two adjacent 1-minterms), every pair of adjacent minterms must be covered by at least one common product term (cube) in the Boolean expression.

Step 1: Represent the minterms in binary

The given function is $f(w,x,y,z) = \sum(5, 7, 11, 12, 13, 15)$. Let's write their 4-bit binary representations:

• $5 = 0101_2$
• $7 = 0111_2$
• $11 = 1011_2$
• $12 = 1100_2$
• $13 = 1101_2$
• $15 = 1111_2$

Step 2: Identify adjacent pairs and their covering cubes

Two minterms are adjacent if they differ by exactly one bit. Let's find all adjacent pairs and the corresponding simplified product term that covers them:

1. $(5, 7)$: $0101$ and $0111$ differ in $y$ $\implies$ covered by $\overline{w}xz$
2. $(5, 13)$: $0101$ and $1101$ differ in $w$ $\implies$ covered by $x\overline{y}z$
3. $(7, 15)$: $0111$ and $1111$ differ in $w$ $\implies$ covered by $xyz$
4. $(11, 15)$: $1011$ and $1111$ differ in $x$ $\implies$ covered by $wyz$
5. $(12, 13)$: $1100$ and $1101$ differ in $z$ $\implies$ covered by $wx\overline{y}$
6. $(13, 15)$: $1101$ and $1111$ differ in $y$ $\implies$ covered by $wxz$

Step 3: Determine the required cover

To ensure hazard-free transitions between all adjacent 1-minterms, the cover must include all 6 of these product terms:

Comparing this with the given options, Option D contains exactly these 6 product terms.