To obtain the Boolean function $\text{F(X, Y)}=\text{X}\overline{\text{Y}}+\overline{\text{X}}$, the inputs PQRS in the figure should be

The correct answer is 1110.

Step-by-step Explanation:

1. Analyze the Multiplexer from the Image:
The given image displays a 4:1 Multiplexer with:
- Four data inputs: $P$ (connected to input line 0), $Q$ (connected to input line 1), $R$ (connected to input line 2), and $S$ (connected to input line 3).
- Two select lines: $S_1$ as the Most Significant Bit (MSB) connected to variable $X$, and $S_0$ as the Least Significant Bit (LSB) connected to variable $Y$.
- Output function: $F(X,Y)$.

2. Write the General Equation for the 4:1 Multiplexer Output:
The output of a 4:1 multiplexer is determined by the selection inputs as follows:
$F(X,Y)=\overline{X}\overline{Y}\cdot P+\overline{X}Y\cdot Q+X\overline{Y}\cdot R+XY\cdot S$

3. Simplify the Target Boolean Function:
We are given the target output Boolean function:
$F(X,Y)=X\overline{Y}+\overline{X}$
Using the Boolean distributive law, we can simplify this target expression:
$F(X,Y)=(\overline{X}+X)(\overline{X}+\overline{Y})$
Since $\overline{X}+X=1$, this simplifies to:
$F(X,Y)=\overline{X}+\overline{Y}$

4. Substitute the Values from the Option 1110:
Let us test the inputs $PQRS=1110$ (which means $P=1$, $Q=1$, $R=1$, and $S=0$):
$F(X,Y)=\overline{X}\overline{Y}\left(1\right)+\overline{X}Y\left(1\right)+X\overline{Y}\left(1\right)+XY\left(0\right)$
This simplifies to:
$F(X,Y)=\overline{X}\overline{Y}+\overline{X}Y+X\overline{Y}$
Factor out the common terms:
$F(X,Y)=\overline{X}(\overline{Y}+Y)+X\overline{Y}$
Since $\overline{Y}+Y=1$, this becomes:
$F(X,Y)=\overline{X}+X\overline{Y}$
Applying the distributive law again:
$F(X,Y)=(\overline{X}+X)(\overline{X}+\overline{Y})=\overline{X}+\overline{Y}$
This perfectly matches the simplified target Boolean function. Thus, the inputs $PQRS$ must be 1110.