Boolean Algebra Calculator

Author: Henrick Yau
- June 1, 2025
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Calculators

Evaluate boolean expressions, create truth tables, and simplify logical statements. Supports AND (&), OR (|), NOT (!), XOR (⊕), NAND, NOR operations with variables A-Z and constants 0/1.

Expression Input

Use A-Z for variables, &(AND), |(OR), !(NOT), ⊕(XOR), parentheses for grouping
Expression Builder

Variable Values

Analysis Options

Specify variables for truth table (auto-detected if empty)

Boolean Algebra Fundamentals

Boolean algebra is a branch of mathematics dealing with operations on logical values. It forms the foundation of digital logic design and computer science, using variables that can only take values of true (1) or false (0).

Basic Operations
  • AND (&, ∧): Returns 1 only when both inputs are 1
  • OR (|, ∨): Returns 1 when at least one input is 1
  • NOT (!, ¬): Returns the opposite value of the input
  • XOR (⊕): Returns 1 when inputs are different
  • NAND (↑): NOT AND - opposite of AND operation
  • NOR (↓): NOT OR - opposite of OR operation
Boolean Laws
  • Identity Laws: A + 0 = A, A · 1 = A
  • Null Laws: A + 1 = 1, A · 0 = 0
  • Idempotent Laws: A + A = A, A · A = A
  • Complement Laws: A + A' = 1, A · A' = 0
  • Commutative Laws: A + B = B + A, A · B = B · A
  • Associative Laws: (A + B) + C = A + (B + C)
  • Distributive Laws: A · (B + C) = A · B + A · C
  • De Morgan's Laws: (A + B)' = A' · B', (A · B)' = A' + B'
Normal Forms
  • DNF (Disjunctive Normal Form): Sum of products - OR of AND terms
  • CNF (Conjunctive Normal Form): Product of sums - AND of OR terms
  • Canonical Forms: Include all variables in each term
  • Minimal Forms: Simplified expressions with fewest terms
Applications
  • Digital Logic Design: Logic gates and circuit design
  • Computer Programming: Conditional statements and logic
  • Database Queries: SQL WHERE clauses and filtering
  • Set Theory: Operations on sets and relations
  • Artificial Intelligence: Logic programming and reasoning

Operator Reference

Operation Symbols Description Example
AND &, ∧, AND, && Logical conjunction A & B
OR |, ∨, OR, || Logical disjunction A | B
NOT !, ¬, NOT, ~ Logical negation !A
XOR ⊕, XOR, ^ Exclusive OR A ⊕ B
NAND ↑, NAND NOT AND A ↑ B
NOR ↓, NOR NOT OR A ↓ B

Disclaimer: This calculator provides boolean algebra evaluations and simplifications based on standard logical operations. Results are computed using established boolean algebra rules and may not represent the only possible simplified form. Always verify critical logic designs with appropriate tools and testing.

Common Boolean Logic Formulas:

  • AND: \( A \land B = A \cdot B \)
  • OR: \( A \lor B = A + B \)
  • NOT: \( \lnot A \)
  • XOR: \( A \oplus B \)
  • NAND: \( \lnot (A \cdot B) \)
  • NOR: \( \lnot (A + B) \)

What This Boolean Algebra Calculator Does

The Boolean Algebra Calculator is a simple yet powerful tool that helps you work with logical expressions. Whether you're a student learning about digital circuits or a professional dealing with logic gates, this calculator lets you evaluate boolean expressions, generate truth tables, and simplify logic formulas in seconds.

What You Can Do With This Tool

Here’s what this calculator helps you achieve:

  • Evaluate logic expressions instantly with support for AND, OR, NOT, XOR, NAND, and NOR operations
  • Auto-generate truth tables for up to 4 variables
  • Convert logic statements to Disjunctive Normal Form (DNF) and Conjunctive Normal Form (CNF)
  • Simplify expressions using common boolean laws
  • Visualize Karnaugh Maps for minimization

Entering and Evaluating Logic Expressions

Follow these steps to use the Boolean Algebra Calculator effectively:

  1. Enter a logic expression in the input field (e.g., (A & B) | !C)
  2. Select the format you’re comfortable with: symbolic (&, |, !), text (AND, OR, NOT), programming style (&&, ||, !), or mathematical symbols (∧, ∨, ¬)
  3. Use the on-screen buttons to build your expression quickly
  4. Assign values (0 or 1) to the variables if you want to evaluate the expression for specific inputs
  5. Click Evaluate Expression to see results, simplification steps, and truth table
  6. Enable extra options to view DNF, CNF, or Karnaugh Map if needed

Benefits for Homework and Logic Design

This tool saves time and reduces error when analyzing boolean logic. Whether you're checking your homework or designing logic gates, this tool ensures:

  • Accurate evaluation of logical conditions
  • Clear visualization through truth tables and Karnaugh Maps
  • Step-by-step simplification using standard laws
  • Quick generation of DNF and CNF forms, useful in logic design and programming

Related Math and Logic Calculators

If you're interested in logic or math calculators, you might also benefit from:

Questions About Boolean Expressions

What kind of expressions can I input?

You can use variables A-Z and logical operators such as AND (&), OR (|), NOT (!), XOR (⊕), NAND (↑), and NOR (↓). You can also use parentheses to group parts of your expression.

How many variables can I use?

The calculator supports up to 4 variables for truth tables and Karnaugh maps, though expressions with more variables can still be evaluated.

Can I simplify any logic expression?

Yes. The tool applies standard boolean algebra laws like De Morgan’s laws, identity, null, and idempotent rules to simplify your expression.

Is this useful for programming or circuit design?

Absolutely. This calculator is valuable for programming conditions, logic gate design, and studying digital electronics or computer science principles.

Simplifying Logical Statements and Truth Tables

The Boolean Algebra Calculator is a fast and reliable solution for evaluating logical statements and simplifying boolean expressions. With features like truth tables, Karnaugh maps, and step-by-step simplification, it’s a practical logic analysis tool suitable for learners and professionals alike. Combine it with tools like a percent error calculator or a scientific calculator to enhance your understanding across math and logic topics.