Question

f(w, x, y, z) =\( \Sigma\) (0, 2, 5, 7, 8, 10, 13, 14, 15) 
Find the correct simplified expression. 
 

• A. \(xz + wxy + w\overline{x}\,\overline{z} + \overline{w}xy\overline{z} \)
• B. \(\overline{x}\,\overline{z} + wxy + w\overline{x}\,\overline{z} + \overline{w}xy\overline{z} \)
• C. \(\overline{x}\,\overline{z} + xz + wy\overline{z} \)
• D. \(\overline{x}\,\overline{z} + xz + wxy \)

Hint:

In 4-variable K-map problems, always try to form groups of 8 or 4 first to get maximum simplification. Larger groups reduce the number of literals in the final expression.

Answer 1

The Correct Option is C

Step 1: Construct the 4-variable K-map.
Given minterms: 0, 2, 5, 7, 8, 10, 13, 14, 15.
We place these in a 4-variable Karnaugh map with variables (w, x) as rows and (y, z) as columns.
Step 2: Form largest possible groups.
From the K-map, we observe the following groupings:
• Minterms (0, 2, 8, 10) form a group of 4 → this gives \(\overline{x}\,\overline{z}\)
• Minterms (5, 7, 13, 15) form another group of 4 → this gives \(xz\)
• Minterms (10, 14) and corresponding pair give \(wy\overline{z}\)
Step 3: Write the simplified expression.
Combining all prime implicants obtained from the K-map, we get:
\[ f = \overline{x}\,\overline{z} + xz + wy\overline{z} \] Step 4: Match with the given options.
This expression exactly matches option (3).