Question

What is a Boolean function?

Short answer

Answer: A Boolean function is a mathematical function that takes binary (0 or 1) inputs and returns a binary output. It is based on Boolean algebra and is used to represent and manipulate logical expressions, digital circuits, and data manipulation. The three basic logical operations used in Boolean functions are AND, OR, and NOT. AND returns 1 if both inputs are 1, and 0 otherwise. OR returns 1 if at least one of the inputs is 1, and 0 otherwise. NOT takes a single binary input and inverts it, returning 0 if the input is 1 and 1 if the input is 0.

Step-by-step solution

Introducing Boolean functions

A Boolean function is a mathematical function that takes binary (0 or 1) inputs and returns a binary output. It is based on Boolean algebra, a form of mathematics that deals with the relationships between binary variables and logical operations such as AND, OR, and NOT. Boolean functions are important in many areas, including computer science, logic, and electrical engineering, where they are used to represent and manipulate logical expressions, digital circuits, and data manipulation.

Basic logical operations

There are three basic logical operations that are commonly used in Boolean functions: AND, OR, and NOT. Each of these operations has a specific meaning: 1. AND: The AND operation takes two binary inputs and returns 1 if both inputs are 1, and 0 otherwise. In other words, the output is true (1) only if both input values are true. Symbolically, it can be represented as \(A \cdot B\) or \(A \wedge B\). 2. OR: The OR operation takes two binary inputs and returns 1 if at least one of the inputs is 1, and 0 otherwise. In other words, the output is true (1) if either of the input values is true, or both are true. Symbolically, it can be represented as \(A + B\) or \(A \vee B\). 3. NOT: The NOT operation takes a single binary input and inverts it, returning 0 if the input is 1 and 1 if the input is 0. Essentially, the output is the opposite of the input value. Symbolically, it can be represented as \(\bar{A}\) or \(\neg A\).

Examples of Boolean functions

Now let’s look at some examples of Boolean functions. 1. f(A, B) = AB: This function takes two binary inputs A and B, and returns the AND of the two inputs. Example input-output pairs for this function are: - f(0, 0) = \(0 \cdot 0 = 0\) - f(0, 1) = \(0 \cdot 1 = 0\) - f(1, 0) = \(1 \cdot 0 = 0\) - f(1, 1) = \(1 \cdot 1 = 1\) 2. f(A, B) = A + B: This function takes two binary inputs A and B, and returns the OR of the two inputs. Example input-output pairs for this function are: - f(0, 0) = \(0 + 0 = 0\) - f(0, 1) = \(0 + 1 = 1\) - f(1, 0) = \(1 + 0 = 1\) - f(1, 1) = \(1 + 1 = 1\) 3. f(A) = \(\bar{A}\): This function takes one binary input A, and returns the NOT of the input. Example input-output pairs for this function are: - f(0) = \(\bar{0} = 1\) - f(1) = \(\bar{1} = 0\) You can create more complex Boolean functions by combining the basic logical operations in different ways, such as f(A, B) = \(A \cdot \bar{B} + B \cdot \bar{A}\), which represents the XOR (exclusive-OR) operation.