We need to find the Boolean expression for the given circuit and find the output values P, Q, R, and S for the given truth table.

1. Problem Description

We need to find the Boolean expression for the given circuit and find the output values P, Q, R, and S for the given truth table.

2. Solution Steps

For Question 13:

First, let's analyze the circuit diagram.

The circuit has two AND gates and one OR gate. The inputs A and B are fed into two AND gates. One AND gate receives the direct inputs A and B. The other AND gate receives the inverted inputs A and B i.e. $\overline{A}$ and $\overline{B}$. The outputs of these two AND gates are then fed into an OR gate.

Therefore, the Boolean expression for the circuit can be written as $(A \cdot B) + (\overline{A} \cdot \overline{B})$. This matches none of the given options. However, according to DeMorgan's law:

$\overline{A+B} = \overline{A} \cdot \overline{B}$

The given options are:

(1) $A \cdot B + \overline{A \cdot B}$

(2) $\overline{A} \cdot B + A \cdot B$

(3) $A \cdot B + \overline{A} \cdot \overline{B}$

(4) $(A \cdot B)(\overline{A})$

The correct expression is $A \cdot B + \overline{A} \cdot \overline{B}$, which matches option (3).

For Question 14:

Let $F = (A \cdot B) + (\overline{A} \cdot \overline{B})$

When A = 0 and B = 0:

$F = (0 \cdot 0) + (\overline{0} \cdot \overline{0}) = 0 + (1 \cdot 1) = 0 + 1 = 1$

So, P =

1.

When A = 0 and B = 1:

$F = (0 \cdot 1) + (\overline{0} \cdot \overline{1}) = 0 + (1 \cdot 0) = 0 + 0 = 0$

So, Q =

0.

When A = 1 and B = 0:

$F = (1 \cdot 0) + (\overline{1} \cdot \overline{0}) = 0 + (0 \cdot 1) = 0 + 0 = 0$

So, R =

0.

When A = 1 and B = 1:

$F = (1 \cdot 1) + (\overline{1} \cdot \overline{1}) = 1 + (0 \cdot 0) = 1 + 0 = 1$

So, S =

1.

Therefore, the output column P, Q, R, S is 1, 0, 0,

1.

3. Final Answer

Question 13: (3) $A \cdot B + \overline{A} \cdot \overline{B}$

Question 14: (2) 1, 0, 0, 1