Question

What is a Boolean expression?

Short answer

#tag_title# Basic Boolean Expressions and Evaluation #tag_content# A Boolean expression is a mathematical expression that evaluates to a Boolean value, either True (1) or False (0). The three basic Boolean operations are AND, OR, and NOT. Evaluating a Boolean expression involves plugging in the values of the variables, computing the NOT operations, followed by the AND and OR operations. It is useful in programming and digital circuits to make decisions and control the flow of a program or circuit.

Step-by-step solution

Introduction to Boolean Expressions

A Boolean expression is a mathematical expression that evaluates to a Boolean value, either True (1) or False (0). Boolean expressions are used in programming and digital circuits to make decisions and control the flow of a program or circuit.

Basic Boolean Operations

There are three basic Boolean operations: AND, OR, and NOT. 1. AND: The AND operation (also known as conjunction) compares two Boolean values and returns True only if both inputs are True. In Boolean algebra, AND is often denoted by the symbol \(\cdot\) or simply the inputs side by side (ex: \(AB\)). 2. OR: The OR operation (also known as disjunction) compares two Boolean values and returns True if at least one of the inputs is True. In Boolean algebra, OR is often denoted by the symbol \(+\) (ex: \(A + B\)). 3. NOT: The NOT operation (also known as negation) takes a single Boolean value and returns the opposite value. In Boolean algebra, NOT is often denoted by a bar over the input (ex: \(\overline{A}\)).

Example of a Boolean Expression

Let's consider the Boolean expression: \(F = A\cdot \overline{B} + \overline{A} \cdot B\) We will evaluate this expression for all possible values of \(A\) and \(B\). There are four possible combinations of inputs: \((A, B) = (0, 0), (0, 1), (1, 0), (1, 1)\).

Evaluate Expression for (A, B) = (0, 0)

First, let's plug in the values: \(F = 0\cdot \overline{0} + \overline{0} \cdot 0\). Now, compute the NOT operations: \(F = 0\cdot 1 + 1 \cdot 0\). Then, compute the remaining AND and OR operations: \(F = 0 + 0 = 0\). So, when \(A=0\) and \(B=0\), the expression evaluates to False (0).

Evaluate Expression for (A, B) = (0, 1)

Plug in the values: \(F = 0\cdot \overline{1} + \overline{0} \cdot 1\). Compute the NOT operations: \(F = 0\cdot 0 + 1 \cdot 1\). Then compute the remaining AND and OR operations: \(F = 0 + 1 = 1\). So, when \(A=0\) and \(B=1\), the expression evaluates to True (1).

Evaluate Expression for (A, B) = (1, 0)

Plug in the values: \(F = 1\cdot \overline{0} + \overline{1} \cdot 0\). Compute the NOT operations: \(F = 1\cdot 1 + 0 \cdot 0\). Then compute the remaining AND and OR operations: \(F = 1 + 0 = 1\). So, when \(A=1\) and \(B=0\), the expression evaluates to True (1).

Evaluate Expression for (A, B) = (1, 1)

Plug in the values: \(F = 1\cdot \overline{1} + \overline{1} \cdot 1\). Compute the NOT operations: \(F = 1\cdot 0 + 0 \cdot 1\). Then compute the remaining AND and OR operations: \(F = 0 + 0 = 0\). So, when \(A=1\) and \(B=1\), the expression evaluates to False (0). Now we have the full truth table for the given Boolean expression: | A | B | F | |---|---|---| | 0 | 0 | 0 | | 0 | 1 | 1 | | 1 | 0 | 1 | | 1 | 1 | 0 |