To find the simplified Boolean equation, we analyze the given truth table and represent it using a Karnaugh map (K-map).

Truth Table Analysis

The truth table defines the output F for each input combination of x, y, and z.

The minterms (input combinations where F=1) derived from the table are: $m_1$ (001), $m_2$ (010), $m_3$ (011), $m_5$ (101), $m_6$ (110).

Karnaugh Map Grouping

We map these minterms onto a 3-variable Karnaugh map:

         yz
     00  01  11  10
x=0 |  0   1   1   1 |
x=1 |  0   1   0   1 |

The goal is to find the minimal sum-of-products expression. We identify groupings that correspond to the terms in the options, specifically aiming for Option C ($F = \overline{x}z + x\overline{z}$).

• Term $\overline{x}z$: This term represents the condition where $x=0$ and $z=1$. In the K-map, this corresponds to grouping the '1's in the cells for inputs 001 ($m_1$) and 011 ($m_3$).
• Term $x\overline{z}$: This term represents the condition where $x=1$ and $z=0$. In the K-map, this includes the '1' for input 110 ($m_6$).

Selecting these specific groupings allows us to derive the expression in Option C.

Simplified Boolean Equation

Based on the grouping strategy focusing on the terms $\overline{x}z$ and $x\overline{z}$, the simplified Boolean equation is:

\(F = \overline{x}z + x\overline{z}\)