Consider the Karnaugh map given below. Where X represents “don’t care” and blank represents 0.

Assume for all inputs (a, b, c, d), the respective complements (a̅, b̅, c̅, d̅) are also available. The above logic is implemented using 2-input NOR gates only. The minimum number of gates required is ________.

Concept:

Draw the K- map, convert the K-map into a SOP (sum of product) or POS (product of sum) form. While reducing the K-map in these forms, a don’t care will be needed only when with the use of don’t cares we can reduce the term size.

Diagram: K – Map

From the K-map simplification:

F(a, b, c, d) = a̅.c

F(a, b, c, d) = \(\overline {\left( {a + \bar c} \right)}\)

Diagram:

Therefore, only one NOR gate is needed to implement the minimized function