Boolean Expression Calculator

Simplify Boolean expressions, generate truth tables, and minimize logic circuits. Supports AND (&), OR (|), NOT (~), XOR (^), NAND, NOR operations.

Operators: & (AND), | (OR), ~ (NOT), ^ (XOR), ! (NAND), ? (NOR)
Variables: A, B, C, D (single letters) | Parentheses: ( ) for grouping

Simplified Expression

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POS Form

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Expression Info

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Step-by-Step Solution

Enter a Boolean expression and click "Calculate" to see the step-by-step solution.

Boolean Operators

  • & (AND): True only if both inputs are true
  • | (OR): True if at least one input is true
  • ~ (NOT): Inverts the input
  • ^ (XOR): True if inputs are different
  • ! (NAND): NOT AND
  • ? (NOR): NOT OR

Boolean Laws

  • Identity: A & 1 = A, A | 0 = A
  • Null: A & 0 = 0, A | 1 = 1
  • Idempotent: A & A = A, A | A = A
  • Complement: A & ~A = 0, A | ~A = 1
  • De Morgan's: ~(A&B) = ~A|~B, ~(A|B) = ~A&~B

Example Expressions

  • A & B = A AND B
  • A | B = A OR B
  • ~A & B = NOT A AND B
  • (A&B) | (~A&C) = If-Then-Else
  • A ^ B = A XOR B

Frequently Asked Questions

What is Boolean algebra?

Boolean algebra is a branch of algebra that deals with true/false values (1/0) and logical operations like AND, OR, and NOT. It's fundamental to digital circuit design, computer science, and mathematical logic.

How do you simplify Boolean expressions?

Boolean expressions can be simplified using:

  • Boolean algebra laws (identity, null, idempotent, complement, De Morgan's)
  • Karnaugh Maps (K-Maps) for 2-4 variables
  • Quine-McCluskey algorithm for more variables

What is a truth table?

A truth table lists all possible input combinations for a Boolean expression and shows the corresponding output value. It's useful for verifying expression behavior and finding equivalent expressions.

What is the difference between SOP and POS?

SOP (Sum of Products): OR of AND terms (e.g., AB + A'C)
POS (Product of Sums): AND of OR terms (e.g., (A+B)(A'+C))
Our calculator provides both forms for comparison.

How many variables can this calculator handle?

Our Boolean expression calculator supports 2, 3, or 4 variables (A, B, C, D). For more complex expressions, consider using advanced logic minimization software.

What is De Morgan's theorem?

De Morgan's theorem states:
~(A & B) = ~A | ~B
~(A | B) = ~A & ~B
It allows converting between AND/OR operations and is essential for logic simplification.