Boolean Algebra Simplifier

How to Use the Boolean Algebra Simplifier

Type a boolean expression using variables (A, B, C, etc.) and operators (AND, OR, NOT, XOR). The tool instantly simplifies the expression using standard boolean algebra identities and generates the truth table. You can use alternative notation: & for AND, | for OR, ! or ~ for NOT, ^ for XOR.

For example, entering A AND (B OR NOT B) simplifies to A because B OR NOT B is always true (complement law), and A AND 1 = A (identity law).

Boolean Algebra Laws

The simplifier applies these fundamental boolean algebra laws:

• Identity: A AND 1 = A, A OR 0 = A
• Annihilation: A AND 0 = 0, A OR 1 = 1
• Idempotent: A AND A = A, A OR A = A
• Complement: A AND NOT A = 0, A OR NOT A = 1
• Double negation: NOT NOT A = A
• XOR: A XOR 0 = A, A XOR A = 0, A XOR 1 = NOT A

Applications

Boolean algebra is the foundation of digital circuit design, programming (conditional logic), database queries (SQL WHERE clauses), search engine queries, and formal logic. Simplifying boolean expressions reduces the number of logic gates needed in hardware circuits, lowers power consumption, and improves execution speed in software.

Frequently Asked Questions

What is boolean algebra?

A branch of algebra where variables are true (1) or false (0), with fundamental operations AND, OR, and NOT.

How does the simplifier work?

It parses the expression and applies algebraic identities like identity, annihilation, idempotent, complement, and double negation laws.

What notation can I use?

AND (&), OR (|), NOT (! or ~), XOR (^), parentheses, postfix NOT (A'), and 0/1 literals.

What is a truth table?

A table listing all possible input combinations and the output for each. For n variables there are 2^n rows.

What is XOR?

Exclusive OR: returns true when exactly one operand is true. Used in cryptography and error detection.