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: Draw the 4-Variable Karnaugh Map 
The given minterms are: 0, 2, 5, 7, 8, 10, 13, 14, 15. 
Construct a 4-variable K-map by taking (w, x) as row variables and (y, z) as column variables. 
Place 1’s in the cells corresponding to the given minterms. 

Step 2: Make the Largest Possible Groups
From the K-map, we identify the following groups: 

• The minterms (0, 2, 8, 10) form a group of four. This grouping eliminates w and y, giving the term: \[ \overline{x}\,\overline{z} \] 
• The minterms (5, 7, 13, 15) form another group of four. This produces the term: \[ xz \] 
• The pair (10, 14) forms a group of two. This gives the term: \[ wy\overline{z} \] 

Step 3: Write the Simplified Boolean Expression
Combining all the prime implicants obtained from the K-map: \[ f = \overline{x}\,\overline{z} + xz + wy\overline{z} \] 
Step 4: Compare with the Options
The simplified expression matches option (3).