Question

What is a Boolean function?

Short answer

Answer: A Boolean function is a mathematical function that takes one or more Boolean inputs (i.e., variables with values of true or false) and produces a single Boolean output. This type of function is crucial in digital logic and computer circuit design. Truth tables are helpful tools for understanding Boolean functions, as they list all possible input combinations and their corresponding outputs. By examining a truth table, we can see how a specific function behaves under various input conditions.

Step-by-step solution

Definition and Basic Concept of a Boolean Function

A Boolean function is a mathematical function that takes one or more Boolean inputs (that is, variables that can have values either true or false) and produces a single Boolean output. These functions are essential in digital logic and computer circuit design.

Examples of Boolean Functions

Now let's go through some examples of Boolean functions: 1. The AND function (denoted as \(\land\)): It takes two Boolean inputs \(A\) and \(B\) and returns true only if both inputs are true, otherwise, it returns false. Mathematically, it can be represented as \(A \land B\). 2. The OR function (denoted as \(\lor\)): It takes two Boolean inputs \(A\) and \(B\) and returns true if either or both of the inputs are true, otherwise, it returns false. Mathematically, it can be represented as \(A \lor B\). 3. The NOT function (denoted as \(\lnot\)): It takes a single Boolean input \(A\) and returns the opposite of its input (i.e., true if the input is false, and false if the input is true). Mathematically, it can be represented as \(\lnot A\).

Truth Tables

A truth table is a table that lists all possible input combinations for a Boolean function and their corresponding outputs. It can help us better understand the behavior of Boolean functions. Let's look at the truth tables for the AND, OR, and NOT functions we just described: 1. AND function: | \(A\) | \(B\) | \(A \land B\) | |-----|-----|------------| | T | T | T | | T | F | F | | F | T | F | | F | F | F | 2. OR function: | \(A\) | \(B\) | \(A \lor B\) | |-----|-----|-----------| | T | T | T | | T | F | T | | F | T | T | | F | F | F | 3. NOT function: | \(A\) | \(\lnot A\) | |-----|---------| | T | F | | F | T | Remember, these are just a few examples of Boolean functions; there are many more out there with various applications!