Boolean functions are mathematical functions that operate between binary values (0 and 1). There are four different functions of the form 𝒇∶ 𝑩 → 𝑩 defined over the set B = {0,1}. These functions are used in digital logic circuits, especially in digital gates (such as AND, OR, NOT). In this section, we will discuss these four functions and their properties.

a) Digital “NOT” Function
We mentioned four functions above. Here, we will start with one of these, the digital NOT function. This function inverts the input value. That is:

𝒇(1) = 0
𝒇(0) = 1

Alternatively, this function can be written using the “exclusive-or (XOR)” operator as:

𝒇(𝑥) = 1 ⊕ x

This function simulates the behavior of a NOT gate and is frequently used in digital logic circuits. The function is bijective, meaning each input value corresponds to a unique output value.

b) Constant Functions

There are two constant functions among Boolean functions. These are functions that always produce the same output regardless of the input value. These functions are as follows:

• A function that always produces the value 0, regardless of the input value. Mathematically, we can express this as:
  𝒇(𝒙) = 0 for any 𝒙 ∈ 𝑩.
• A function that always produces the value 1, regardless of the input value. Mathematically, we can express this as:
  𝒇(𝒙) = 1 for any 𝒙 ∈ 𝑩.

These functions are not balanced because their outputs are always constant.

c) Identity Function

This function corresponds exactly to the input value, meaning it is an identity function that outputs the input value without changing it. Such functions are considered balanced because their outputs produce an equal number of 0s and 1s.
Mathematically, this expression is represented as:
𝒇(𝒙) = 𝒙 for any 𝒙 ∈ 𝑩.

1.5.1. Balanced and Constant Functions Concept
In the previous section, we used the terms “balanced” and “constant functions” while examining functions. In this section, we will explain these two concepts.

a. Balanced Functions
A function is called balanced if it produces an equal number of 0s and 1s in terms of its outputs. In other words, if the function outputs 0 and 1 values in equal amounts, the function is considered balanced. For example, the Digital NOT Function and the Identity Function are examples of balanced functions.

b. Constant Functions
Constant functions always produce the same value. That is, regardless of the input, the output is constant. These functions are not balanced. For example, constant functions.