Boolean Expressions and Their Evaluation: Truth Tables and Logic

Learn about Boolean expressions—expressions that evaluate to true or false. This guide explains how Boolean expressions are constructed using variables, constants, and operators (AND, OR, NOT), and how to evaluate them using truth tables.

Boolean Expressions and Their Evaluation

What is a Boolean Expression?

A Boolean expression is an expression that evaluates to either true or false. It's built using Boolean variables (variables that can only take on the values true or false), Boolean constants (true or false), and Boolean operators (AND, OR, NOT).

Constructing Boolean Expressions

Boolean expressions are defined recursively:

  1. Every element of the Boolean algebra is a Boolean expression.
  2. Every variable name is a Boolean expression.
  3. If a₁ and a₂ are Boolean expressions, then a₁', a₁ ∨ a₂, and a₁ ∧ a₂ are also Boolean expressions (where ' represents NOT, ∨ represents OR, and ∧ represents AND).

Example Boolean Expressions

(An example showing several Boolean expressions over a specific Boolean algebra would be included here.)

Evaluating Boolean Expressions

To evaluate a Boolean expression, you substitute the variables with their assigned values (0 for false, 1 for true) and apply the Boolean operations.

Example: Evaluating a Boolean Expression

Let E(x₁, x₂, x₃) = (x₁ ∨ x₂) ∧ (x₁ ∨ x₂) ∧ (x₂ ∨ x₃). If x₁ = 0, x₂ = 1, x₃ = 0, then:

E(0, 1, 0) = (0 ∨ 1) ∧ (0 ∨ 1) ∧ (1 ∨ 0) = 1 ∧ 1 ∧ 1 = 1

Equivalent Boolean Expressions

Two Boolean expressions are equivalent if they produce the same output (true or false) for every possible combination of input values for the variables.

Example: Equivalent Boolean Expressions

(An example illustrating two equivalent Boolean expressions would be shown here.)

Min-Terms

A min-term is a Boolean expression of n variables where each variable appears exactly once, either in its original form (xi) or complemented (xi').

Conclusion

Boolean expressions are fundamental in logic and computer science, forming the basis of digital circuits and programming logic. Understanding how to construct and evaluate them is essential.