Final answer:
To design an SOP minimum-cost circuit with a K-map, populate a 4-variable K-map with the given minterms and don't cares, group them, and derive the product terms. Identify all prime implicants, with the essential ones covering unique minterms.
Step-by-step explanation:
The student's question involves designing an SOP minimum-cost circuit using a Karnaugh map (K-map) to implement a given boolean function. This function has minterms and doesn't care about conditions denoted by the summation ∑[m(4,5,9,13,15)] and D(0,3,11). An SOP (Sum of Products) circuit can be optimized using a K-map by grouping adjacent ones representing the minterms and don't care.
To solve this, you would begin by creating a 4-variable K-map and populating it with the minterms (4,5,9,13,15) as ones and the don't care (0,3,11) as 'X's. After plotting these points, you would look for the largest possible groups of 1's that can be made which may include don't cares, remembering that groups should contain numbers of cells that are powers of two (1, 2, 4, 8, etc).
Once you have your groups, create the product terms for each group by writing down the variables that remain constant within a group. Your prime implicants are all the product terms you've created. The essential prime implicants are those that cover a minterm that is not covered by any other implicant. These steps will give you the optimized SOP expression.
The Karnaugh map simplifies the process of minimizing boolean functions and is a visual way to find the most efficient SOP expression. Identifying the essential prime implicants guarantees that all necessary minterms of the function are covered while minimizing the number of terms in the final SOP expression.