Question

Consider the following four variable Boolean function in sum-of-product form \[ F(b_3, b_2, b_1, b_0) = \sum(0, 2, 4, 8, 10, 11, 12), \] where the value of the function is computed by considering \( b_3 b_2 b_1 b_0 \) as a 4-bit binary number, where \( b_3 \) denotes the most significant bit and \( b_0 \) denotes the least significant bit. Note that there are no don’t care terms. Which ONE of the following options is the CORRECT minimized Boolean expression for \( F \)?

• A. \( \overline{b_1}b_0 + \overline{b_2}b_0 + b_1b_2b_3 \)
• B. \( \overline{b_1}b_0 + \overline{b_2}b_0 \)
• C. \( \overline{b_2}b_0 + b_1b_2b_3 \)
• D. \( b_0b_2 + \overline{b_3} \)

Hint:

When simplifying Boolean expressions, try to combine common terms and factor out common variables to minimize the expression.

Answer 1

The Correct Option is A

We are given the Boolean function in sum-of-product form and asked to minimize it.

Step 1: Write the minterms for the given indices
The minterms are generated from the binary representation of the given indices: \[ 0 \rightarrow \overline{b_3} \overline{b_2} \overline{b_1} \overline{b_0}, \quad 2 \rightarrow \overline{b_3} \overline{b_2} b_1 \overline{b_0}, \quad 4 \rightarrow \overline{b_3} b_2 \overline{b_1} \overline{b_0}, \quad 8 \rightarrow b_3 \overline{b_2} \overline{b_1} \overline{b_0}, \quad \text{and so on.} \]
Step 2: Minimize the Boolean expression
By simplifying the sum of minterms, we arrive at the minimized Boolean expression: \[ F(b_3, b_2, b_1, b_0) = \overline{b_3} \overline{b_2} \overline{b_0} + \overline{b_3} b_2 \overline{b_1} \overline{b_0} + b_3 \overline{b_2} b_1. \]
Thus, the correct minimized Boolean expression is \( \boxed{A} \).