In the Karnaugh map shown below, X denotes a don't care term. What is the…

Solution: derive the minimal expression from the Karnaugh map by forming largest possible groups (including don't-cares).

Step 1: K-map layout and useful observation

• Rows correspond to cd in order 00, 01, 11, 10; columns correspond to ab in order 00, 01, 11, 10.

• The rows with d = 0 are the top and bottom rows (cd = 00 and cd = 10). Those rows contain the 1s and some don't-cares that let us form larger groups.

Step 2: Form groups using don't-cares to maximize grouping

• Group 1: Make a 2x2 block that covers the two rows with d = 0 and the two columns where b = 0 (ab = 00 and ab = 10). This block yields the product term b'd' (b = 0 and d = 0).

• Group 2: Make a 2x2 block that covers the two rows with d = 0 and the two columns where a = 0 (ab = 00 and ab = 01). This block yields the product term a'd' (a = 0 and d = 0).

Step 3: Combine the product terms

The minimal sum-of-products is b'd' + a'd'.

You can factor the result as d'(a' + b'), but the minimal SOP form is b'd' + a'd'.

Why this is minimal: each term has two literals and the two terms are necessary to cover all 1-cells (they cannot be combined into a single fewer-literal term without leaving some required 1s uncovered).