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 $F$?

To find the minimal sum-of-products (SOP) expression, we plot the given minterms on a 4-variable Karnaugh Map with rows representing $b_3b_2$ and columns representing $b_1b_0$:

The given minterms are $\sum(0, 2, 4, 8, 10, 11, 12)$:

• $m_0 = 0000$

• $m_2 = 0010$

• $m_4 = 0100$

• $m_8 = 1000$

• $m_{10} = 1010$

• $m_{11} = 1011$

• $m_{12} = 1100$

Now, group the 1s into maximal prime implicants:

1. Quad 1 (Column 00): Consists of minterms $\{0, 4, 8, 12\}$. Here, $b_1 = 0$ and $b_0 = 0$, giving the product term:

   $$\bar{b}_1 \bar{b}_0$$

2. Quad 2 (Four corners/edges): Consists of minterms $\{0, 2, 8, 10\}$. Here, $b_2 = 0$ and $b_0 = 0$, giving the product term:

   $$\bar{b}_2 \bar{b}_0$$

3. Pair: The minterm $m_{11}$ ($1011$) is paired with $m_{10}$ ($1010$). Here, $b_3 = 1, b_2 = 0, b_1 = 1$, giving the product term:

   $$b_1 \bar{b}_2 b_3$$

All three terms are Essential Prime Implicants (EPIs) covering all the given minterms without redundancy.

Thus, the minimized Boolean expression is:

This matches Option (A).