GATE CS 2005 Q18 — Minimization (K-Maps etc.)

To find the simplified switching expression for $f(A, B, C, D) = \Sigma(1, 4, 5, 9, 11, 12)$, we can group the minterms using a 4-variable Karnaugh map (K-map) or algebraic grouping:

The binary representations of the given minterms are:

• $1 = 0001$ ($A'B'C'D$)
• $4 = 0100$ ($A'BC'D'$)
• $5 = 0101$ ($A'BC'D$)
• $9 = 1001$ ($AB'C'D$)
• $11 = 1011$ ($AB'CD$)
• $12 = 1100$ ($ABC'D'$)

We can form three pairs of adjacent minterms to simplify the expression:

1. Group 1: Combine minterms $4$ ($0100$) and $12$ ($1100$): $$A'BC'D' + ABC'D' = BC'D'$$
2. Group 2: Combine minterms $1$ ($0001$) and $5$ ($0101$): $$A'B'C'D + A'BC'D = A'C'D$$
3. Group 3: Combine minterms $9$ ($1001$) and $11$ ($1011$): $$AB'C'D + AB'CD = AB'D$$

Combining these essential prime implicants gives the simplified sum-of-products (SOP) expression: $$f(A, B, C, D) = BC'D' + A'C'D + AB'D$$ This matches option A.