Minimum sum of product expression for f (w, x, y, z) shown in Karnaugh-map…

1. Analyze the K-Map Content First, we identify the location of 1s (Minterms) and Ds (Don't Cares) based on the provided image. We assume blank cells represent 0.

• Variables: Inputs w,x,y,z.

  • Rows (y,z): 00,01,11,10

  • Columns (w,x): 00,01,11,10

• Minterms (1s):

  • Row 00, Col 01 (m4​): 1

  • Row 00, Col 10 (m8​): 1

  • Row 01, Col 01 (m5​): 1

  • Row 11, Col 00 (m3​): 1

• Don't Cares (D):

  • m0​ (00,00), m13​ (01,11), m7​ (11,01), m15​ (11,11), m2​ (10,00), m10​ (10,10).

2. Form Groups (Prime Implicants) We try to cover all '1's using the largest possible groups of 1s and Ds.

• Group 1: Center Quad (Covers m5​)

  • Cells: m5​(1),m7​(D),m13​(D),m15​(D).

  • This forms a 2×2 block in the center.

  • Logic: x=1 (cols 01, 11) and z=1 (rows 01, 11).

  • Term: xz

• Group 2: Four Corners (Covers m8​)

  • Cells: m0​(D),m8​(1),m2​(D),m10​(D).

  • This uses the four corners of the map.

  • Logic: x=0 (cols 00, 10) and z=0 (rows 00, 10).

  • Term: x′z′

• Remaining 1s to cover: m4​ and m3​.

  • For m4​ (0100): Best grouped with m5​ (0101) to form term w′xy′. (Or with m0​ to form w′y′z′).

  • For m3​ (0011): Best grouped with m7​ (0111) to form term w′yz. (Or with m2​ to form w′x′y).

3. Evaluate Options The minimal expression must include terms to cover all 1s. Our derived expression starts with:

f(w,x,y,z)=xz+x′z′+…

Let's check the given options:

• (A) xz+y′z: This covers the center (xz), but y′z (Row 1) would include minterms that are '0' in the map (like m1​and m9​). Also, it fails to cover m3​,m4​,m8​.

• (B) xz′+zx′: This represents an Exclusive-OR (x⊕z), which does not match our groups (x′z′ and xz).

• (C) x′y+zx′: This fails to cover the specific 1s identified (e.g., m4​).

Since none of the expressions match the derived minimal groups, the correct choice is "None of these".

Correct Option: (D)