We will divide this study into two parts: the NOT Gate and the NOT Function. We will discuss each of these sequentially below.

a. Classical NOT Gate

The classical NOT gate is a fundamental logic gate in classical computing that performs a simple inversion operation. This gate is expressed through a function defined on a Boolean set B.

The set B is defined as:
B={0,1}

As indicated, the inputs and outputs can only be 0 or 1.

b. Classical NOT (notD) Function

We already know that the NOT gate inverts a given input. Here, we will discuss this in a more mathematical context:
Input: x ∈ B
Output: 1 ⊕ x

The operation ⊕ used here is an XOR operation, which is similar to addition. XOR is an operation that returns 0 or 1 depending on whether the two bits are the same or different. The table below briefly shows the possible outcomes of this gate:

1st Bit	2nd Bit	Result
0	0	0
1	1	0
0	1	1
1	0	1

This operation is bijective, meaning there is a one-to-one correspondence between inputs and outputs, making it a reversible function. In other words, if we apply the NOT gate again to the output produced by the first application, we retrieve the original input. This reversibility is a significant feature and plays an essential role in quantum computing.