Question

For the Boolean function: \[ F(A, B, C, D) = \Sigma m(0, 2, 5, 7, 8, 10, 12, 13, 14, 15), \] the essential prime implicants are:

• A. \(BD, \, \overline{B}\overline{D}\)
• B. \(BD, \, AB\)
• C. \(AB, \, \overline{B}\overline{D}\)
• D. \(BD, \, \overline{B}\overline{D}, \, AB\)

Hint:

To simplify Boolean functions, use a Karnaugh Map to group adjacent minterms. Look for the largest possible groups to minimize the expression.

Answer 1

The Correct Option is A

Step 1: List the minterms.
The given minterms are \(0, 2, 5, 7, 8, 10, 12, 13, 14, 15\). These represent the binary indices where the function evaluates to 1. Step 2: Plot the Karnaugh Map (K-map).
Place the minterms onto a 4-variable K-map. Group the adjacent cells with ones to form larger groups, ensuring minimal grouping for simplification. Step 3: Identify the essential prime implicants.
From the grouping: \[ {The essential prime implicants are: } BD \, {and } \overline{B}\overline{D}. \] Final Answer: \[ \boxed{{(1) } BD, \, \overline{B}\overline{D}} \]