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Logic gates are the building blocks of digital circuits. These components process one or more input signals to produce an output signal, following the rules of boolean algebra. In essence, logic gates perform basic logical functions that are fundamental to digital components, such as those found in computers and other modern electronic devices.

There are several different types of logic gates, each with its own specific function. Among the most commonly used are the NOT, AND, and OR gates, which are considered the basic gates. In the study of digital logic, showing that a set of gates is 'functionally complete' means that any other gate or logical operation can be constructed using a combination of these gates. A system is functionally complete if it can produce all possible truth tables for logical operations.

The NOT gate, also known as an inverter, is a fundamental component of digital logic circuits. Its operation is simple: it takes a single input and inverts it. If the input is 1 (true), the output is 0 (false), and vice versa. This unary operation is vital in digital systems and plays a crucial role in modifying and controlling signal values.

In the exercise, the NOT gate is part of the set outlined for functional completeness. The reason for its importance is that the NOT gate allows us to switch the logic state, which is particularly useful for creating other types of gates or complex circuits.

The OR gate is a digital logic gate that outputs true (1) if at least one of the inputs is true. In a two-input OR gate, if input A is 1, input B is 1, or both are 1, then the output will be 1. If both inputs are 0, the output is 0. This type of gate is useful when a circuit requires to power on or activate a particular action when multiple conditions are satisfactory.

For maintaining functional completeness, the OR gate is included because it allows the combination of signals in a way that the presence of any input signal will carry through.

An AND gate performs a fundamental logical operation that requires all inputs to be true for the output to be true. In a simple two-input AND gate, both input A and input B must be true (1) for the output to be true (1). If either of the inputs is false (0), the output will also be false (0).

Although the AND gate is not directly included in the set {OR, NOT}, it can still be constructed using a combination of OR and NOT gates. The exercise showcases this construction with the formula: \( A \text{ AND } B = \text{NOT}((\text{NOT} A) \text{ OR } (\text{NOT} B)) \). By allowing the inversion of inputs and then combining them via OR before inverting the result, we can mimic the behavior of an AND gate. This confirms that the set {OR, NOT} is indeed functionally complete, as it can replicate the AND gate's functionality.