What is the minimal form of the Karnaugh map shown below? Assume that $\mathbf{X}$ denotes a don’t care term.

From the given 4-variable Karnaugh map with row variables $cd$ and column variables $ab$:

• The minterms where the output is $1$ are: $m_0$ $(0000)$, $m_2$ $(0010)$, $m_8$ $(1000)$, and $m_9$ $(1001)$.

• The don't care conditions ($X$) are at: $m_1$ $(0001)$, $m_4$ $(0100)$, $m_{10}$ $(1010)$, and $m_{12}$ $(1100)$.

To find the minimal SOP expression:

1. Group 1 (Four Corners): Cells $m_0, m_2, m_8, m_{10}$ (where $m_{10}$ is a don't care) form a group of 4. This gives the term $\bar{b}\bar{d}$.

2. Group 2 (Quad spanning $cd = 00, 01$ and $ab = 00, 10$): Cells $m_0, m_1, m_8, m_9$ (where $m_1$ is a don't care) form a group of 4. This gives the term $\bar{b}\bar{c}$.

Combining these two essential prime implicants covers all $1$s ($m_0, m_2, m_8, m_9$):

Hence, (B) is the correct option.