1. **Stating the problem:** We are given two Boolean expressions: $$F1 = XYZ + X'Y'Z + XY'Z' + XYZ$$ $$F2 = X'YZ + X(Y'Z + YZ')$$ and a truth table with input values for $X$, $Y$, and $Z$: | X | Y | Z | |---|---|---| | 0 | 0 | 1 | | 0 | 1 | 0 | | 1 | 0 | 0 | We need to find the values of $F1$ and $F2$ for each input row and identify which option matches these values. 2. **Recall Boolean algebra rules:** - $X'$ means NOT $X$. - $+$ means OR. - Multiplication (e.g., $XY$) means AND. - Evaluate each term for given inputs, then OR the terms. 3. **Evaluate $F1$ for each input:** - For $X=0, Y=0, Z=1$: - $XYZ = 0\cdot0\cdot1=0$ - $X'Y'Z = 1\cdot1\cdot1=1$ - $XY'Z' = 0\cdot1\cdot0=0$ - $XYZ$ (repeated) = 0 - $F1 = 0 + 1 + 0 + 0 = 1$ - For $X=0, Y=1, Z=0$: - $XYZ = 0\cdot1\cdot0=0$ - $X'Y'Z = 1\cdot0\cdot0=0$ - $XY'Z' = 0\cdot0\cdot1=0$ - $XYZ$ (repeated) = 0 - $F1 = 0 + 0 + 0 + 0 = 0$ - For $X=1, Y=0, Z=0$: - $XYZ = 1\cdot0\cdot0=0$ - $X'Y'Z = 0\cdot1\cdot0=0$ - $XY'Z' = 1\cdot1\cdot1=1$ - $XYZ$ (repeated) = 0 - $F1 = 0 + 0 + 1 + 0 = 1$ 4. **Evaluate $F2$ for each input:** - For $X=0, Y=0, Z=1$: - $X'YZ = 1\cdot0\cdot1=0$ - $Y'Z = 1\cdot1=1$ - $YZ' = 0\cdot0=0$ - $X(Y'Z + YZ') = 0\cdot(1+0) = 0$ - $F2 = 0 + 0 = 0$ - For $X=0, Y=1, Z=0$: - $X'YZ = 1\cdot1\cdot0=0$ - $Y'Z = 0\cdot0=0$ - $YZ' = 1\cdot1=1$ - $X(Y'Z + YZ') = 0\cdot(0+1) = 0$ - $F2 = 0 + 0 = 0$ - For $X=1, Y=0, Z=0$: - $X'YZ = 0\cdot0\cdot0=0$ - $Y'Z = 1\cdot0=0$ - $YZ' = 0\cdot0=0$ - $X(Y'Z + YZ') = 1\cdot(0+0) = 0$ - $F2 = 0 + 0 = 0$ 5. **Summary of results:** | X | Y | Z | F1 | F2 | |---|---|---|----|----| | 0 | 0 | 1 | 1 | 0 | | 0 | 1 | 0 | 0 | 0 | | 1 | 0 | 0 | 1 | 0 | 6. **Compare with options:** The matching option is: | F1 | F2 | |----|----| | 1 | 0 | | 0 | 0 | | 1 | 0 | which corresponds to the second option. **Final answer:** The correct option is the second one.