Animated Solution for Physics - Semiconductors: The output of the given combination gates represents

The journey to understanding complex logic circuits often begins with breaking them down into their simplest atomic parts. When faced with a combination of gates, our primary goal is to trace the logical signal from the inputs all the way to the final output, simplifying the Boolean expression at each stage.

Imagine you are standing at the input terminals of this logic circuit. We have two distinct signals, $A$ and $B$. The first thing that happens to these signals is that they are fed into two separate NOT gates.

A NOT gate is the simplest of all logic gates; it simply inverts the signal. If the input is high, the output is low, and vice versa. Therefore, the outputs of these first two gates will be the complements of our inputs: $\overline{A}$ and $\overline{B}$.

Now, these inverted signals travel along the wires and enter a NOR gate. A NOR gate is essentially an OR gate followed by a NOT gate. It takes its inputs, performs a logical addition (OR), and then inverts the entire result.

So, when $\overline{A}$ and $\overline{B}$ enter the NOR gate, the intermediate output, let's call it $Y_1$, becomes:

This expression looks a bit intimidating, but this is exactly where De Morgan's Laws come to our rescue! De Morgan's Law states that the complement of a sum is equal to the product of the complements. Applying this to our expression, we split the top bar and change the plus sign to a dot (AND operation):

In Boolean algebra, a double inversion cancels itself out. Flipping a signal twice brings it back to its original state. Thus, $\overline{\overline{A}}$ simplifies to just $A$, and $\overline{\overline{B}}$ simplifies to just $B$. Our intermediate output elegantly reduces to:

We are almost at the finish line! This simplified signal, $A\cdot B$, now enters the final gate in our circuit, which is another NOT gate.

As we know, the NOT gate will invert whatever signal it receives. So, it takes $A\cdot B$ and applies a bar over the entire expression, giving us our final output $Y$:

Take a moment to look at this final Boolean expression. Does it look familiar? Yes! The expression $\overline{A\cdot B}$ is the defining logical operation of a NAND gate.

Through careful step-by-step analysis and the power of De Morgan's laws, we've discovered that this entire combination of four gates is logically equivalent to a single NAND gate. This is a beautiful example of how complex digital circuits can often be optimized and simplified into much more efficient designs.