The digital circuit shown has 3 inputs (x,y, and z). The simplified logical expression for the output (OUT) is:

The correct option is 0.

To find the simplified logical expression for the output (OUT), we can trace the inputs through each logic gate shown in the circuit diagram step-by-step.

Step 1: Determine the output of the NOR gate

The top gate is a 3-input NOR gate with inputs x, y, and z. The boolean expression for the output of this NOR gate is:

$V_1=\overline{x+y+z}$

Using De Morgan's Law, this can be simplified as:

$V_1=\overline{x}·\overline{y}·\overline{z}$

Step 2: Trace the paths through the NOT and AND gates

The input x is connected to the input of a NOT gate. The output of this NOT gate is:

$\overline{x}$

This inverted signal is then fed into one of the inputs of both AND gates below it:

1. The first (upper) AND gate has inputs $\overline{x}$ and y. Its output is:

$A_1=\overline{x}·y$

2. The second (lower) AND gate has inputs $\overline{x}$ and z. Its output is:

$A_2=\overline{x}·z$

Step 3: Determine the output of the OR gate

The outputs of these two AND gates are connected to the inputs of an OR gate. The output of this OR gate is:

$V_2=(\overline{x}·y)+(\overline{x}·z)$

Factoring out $\overline{x}$, we get:

$V_2=\overline{x}·(y+z)$

Step 4: Find the final output (OUT)

The outputs of the NOR gate and the OR gate are the inputs to the final AND gate. Therefore, the final output expression is:

$\text{OUT}=V_1·V_2$

Substituting the expressions we derived:

$\text{OUT}=(\overline{x}·\overline{y}·\overline{z})·[\overline{x}·(y+z)]$

Expanding the terms inside the expression gives:

$\text{OUT}=(\overline{x}·\overline{y}·\overline{z}·\overline{x}·y)+(\overline{x}·\overline{y}·\overline{z}·\overline{x}·z)$

Using the Boolean algebra identities $\overline{x}·\overline{x}=\overline{x}$, $y·\overline{y}=0$, and $z·\overline{z}=0$, we simplify the terms:

1. First term: $\overline{x}·(y·\overline{y})·\overline{z}=\overline{x}·0·\overline{z}=0$

2. Second term: $\overline{x}·\overline{y}·(z·\overline{z})=\overline{x}·\overline{y}·0=0$

Substituting these simplified terms back into the output equation yields:

$\text{OUT}=0+0=0$

Thus, the simplified logical expression for the output of the circuit is 0.