Boolean Algebra Calculator & Laws
Evaluates and simplifies boolean algebraic expressions with reference laws for digital logic design.
Complete Boolean Algebra Axioms, Laws & Theorems
¬(A ∧ B) = ¬A ∨ ¬B
¬(A ∨ B) = ¬A ∧ ¬B
"Break the line, change the sign." Essential for converting between NAND and NOR logic.A ∧ (A ∨ B) = A
A ∨ (A ∧ B) = A
Simplifies redundant terms in Karnaugh map minimization.A ∧ (B ∨ C) = (A ∧ B) ∨ (A ∧ C)
A ∨ (B ∧ C) = (A ∨ B) ∧ (A ∨ C)
Notice that OR distributes over AND in boolean algebra, unlike regular arithmetic!A ∧ A = A, A ∨ A = A
¬(¬A) = A (Double Negation)
Repeatedly applying the same signal causes zero changes to state.A ∧ 1 = A, A ∧ 0 = 0 (Null)
A ∨ 0 = A, A ∨ 1 = 1 (Null)
0 is the identity for OR; 1 is the identity for AND.A ∧ ¬A = 0 (Contradiction)
A ∨ ¬A = 1 (Tautology)
A signal and its inverse cannot be simultaneously 1 (Law of Non-Contradiction).What is Boolean Algebra?
Boolean algebra is a branch of algebra in which variables have truth values of True (1) or False (0), manipulated through logical conjunction (AND), disjunction (OR), and negation (NOT).
Formula & Step-by-Step Calculation
Duality and negation in boolean algebra.
Worked Step-by-Step Examples
Simplify: A ∨ (A ∧ B)
Common Real-World & Academic Use Cases
- ✓ Minimizing gate count in digital circuit fabrication
- ✓ Optimizing SQL query WHERE clause conditions
- ✓ Compiler conditional branch pruning
How to Use the Boolean Algebra Calculator & Laws
Review Axioms
Inspect identity, idempotent, and absorption rules.
Apply Transformations
Simplify digital logic expressions.
Frequently Asked Questions
Q: What is Karnaugh mapping (K-map)?
A graphical method used to simplify boolean expressions up to 4–6 variables without complex algebraic steps.