Boolean Algebra Manipulations

Boolean Algebra

Master the mathematical foundation of digital logic

Boolean Laws and Theorems

Boolean algebra is the mathematical foundation for digital logic design. It deals with variables that can have only two values: 0 (false) and 1 (true).

Identity Laws

A + 0 = A
A · 1 = A

Adding 0 or multiplying by 1 doesn't change the value

Null Laws

A + 1 = 1
A · 0 = 0

OR with 1 always gives 1, AND with 0 always gives 0

Idempotent Laws

A + A = A
A · A = A

A variable OR/AND with itself equals itself

Complement Laws

A + A' = 1
A · A' = 0

A variable with its complement gives 1 (OR) or 0 (AND)

De Morgan's Laws

(A + B)' = A' · B'
(A · B)' = A' + B'

Complement of OR equals AND of complements

Distributive Laws

A + (B · C) = (A + B) · (A + C)
A · (B + C) = (A · B) + (A · C)

Distribution of operations

Basic Operations Truth Table

A B A·B A+B A'
0 0 0 0 1
0 1 0 1 1
1 0 0 1 0
1 1 1 1 0

Interactive Boolean Algebra Tools

Boolean Expression Parser & Simplifier

Enter a Boolean expression using variables A, B, C, D and operators +, ·, ', (, )

Original: A·B + A'·C + B·C
Simplified: A·B + A'·C
Simplification Steps:
Step 1: Apply absorption law (A + A·B = A)
Step 2: Factor common terms
Step 3: Final simplified form

Interactive variable Testing

Toggle variables to see how the expression evaluates:

Current Values: A=0, B=0, C=0, D=0
Original Result: 0
Simplified Result: 0

2-variable K-Map

B'
B
A'
0
0
A
0
0

Expression: 0

3-variable K-Map

BC'
BC
B'C
B'C'
A'
0
0
0
0
A
0
0
0
0

Expression: 0

4-variable K-Map

CD'
CD
C'D
C'D'
A'B'
0
0
0
0
A'B
0
0
0
0
AB
0
0
0
0
AB'
0
0
0
0

Expression: 0

Dynamic Truth Table Generator

Enter an expression to generate its truth table:

Minterms & Maxterms

Minterms (where output = 1):
m(1,2,3)

Maxterms (where output = 0):
M(0,4,5,6,7)

Sum of Products (sOP):
A'B'C + A'BC' + A'BC

Product of Sums (POS):
(A+B+C)(A'+B+C)

Boolean Expression Format Converter

Input Expression
Converted Forms
Standard sOP:
A·B + A'·C

Standard POS:
(A+C)·(A'+B)

Canonical sOP:
A·B·C + A·B·C' + A'·B'·C

Canonical POS:
(A+B+C)·(A+B'+C)

Boolean Algebra Quiz

What is the simplified form of A + A·B?
A·B
A
A + B
B
Score: 0 / 0