The boolean algebra calculator is an expression simplifier for simplifying algebraic expressions. It is used for finding the truth table and the nature of the expression.
How to use the boolean calculator?
Follow the 2 steps guide to find the truth table using the boolean algebra solver.
 Enter the Expression.
 Click "Parse"
Take help from sample expressions in the input box or have a look at the boolean functions in the content to understand the mathematical operations used in expressions.
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What is Boolean Algebra?
Mathematics has different branches e.g algebra, geometry e.t.c. These branches are further divided into subbranches. Boolean algebra is one such subbranch of algebra.
It has two binary values including true and false that are represented by 0 and 1. Where 1 is considered as true and 0 is considered as false.
Boolean expressions are simplified to build easy logic circuits.
Laws of Boolean Algebra
Boolean algebra has a set of laws or rules that make the Boolean expression easy for logic circuits. Through applying the laws, the function becomes easy to solve.
Here are the simplification rules:
Commutative law:
According to this law;

A + B = B + A

A.B = B.A
Associative law:
This law states;

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

A(B.C) = (A.B)C
Distributive law:
Using this law, we know;

A . ( B + C ) = ( A . B ) + ( A . C )

A + ( B . C ) = (A + B ) . (A + C )
Identity law:
By identity law:

A + 0 = A

A . 1 = A
Annulment law:
Here;

A . 0 = 0

A + 1 = 1
Idempotent law:
By this law:

A + A = A

A . A = A
There are some other rules but these six are the most basic ones.
Application of Boolean Algebra
Boolean algebra can be used on any of the systems where the machine works in two states. For example, the machines that have the option of “On” or “Off”.
Here are some of the realtime applications in our daily life that are using the concept of Boolean algebra:

Coffee/Tea machine

Rocket Launcher

Elevator for two floors

Car (Starting and turning off the engine)
Boolean Expression and Functions
Here is a table with Boolean functions and expressions:
Name  Algebraic function 

AND  F = A.B 
OR  F = A+B 
NOT  F = A 
NAND  F = (A.B) 
NOR  F = (A+B) 
Table of Boolean Algebra
A  B  C  A+B  A+C  (A+B)(A+C)  BC  A+BC 

0  0  0  0  0  0  0  0 
0  0  1  0  1  0  0  0 
0  1  0  1  0  0  0  0 
0  1  1  1  1  1  1  1 
1  0  0  1  1  1  0  1 
1  0  1  1  1  1  0  1 
1  1  0  1  1  1  0  1 
1  1  1  1  1  1  1  1 
Truth Table for Binary Logical Operations
Here is a truth table for all binary logical operations:
p  q  F  NOR  ↚  ¬p  ↛  ¬q  XOR  NAND  AND  XNOR  q  →  p  ←  OR  T 

T  T  F  F  F  F  F  F  F  F  T  T  T  T  T  T  T  T 
T  F  F  F  F  F  T  T  T  T  F  F  F  F  T  T  T  T 
F  T  F  F  T  T  F  F  T  T  F  F  T  T  F  F  T  T 
F  F  F  T  F  T  F  T  F  T  F  T  F  T  F  T  F  T 
Com  ✓  ✓  ✓  ✓  ✓  ✓  ✓  ✓  
Assoc  ✓  ✓  ✓  ✓  ✓  ✓  ✓  ✓  
Adj  F  NOR  ↛  ¬q  ↚  ¬p  XOR  NAND  AND  XNOR  p  ←  q  →  OR  T  
Neg  T  OR  ←  p  →  q  XNOR  AND  NAND  XOR  ¬q  ↛  ¬p  ↚  NOR  F  
Dual  T  NAND  →  ¬p  ←  ¬q  XNOR  NOR  OR  XOR  q  ↚  p  ↛  AND  F  
L id  F  F  T  T  T,F  T  F  
R id  F  F  T  T  T,F  T  F 
Boolean Algebra Laws
Use the following rules and laws of boolean algebra to evaluate the boolean expressions:
AND Form  OR Form  
Commutative Law  A.B=B. A  A + B = B + A 
Associate Law  (A. B) . C = A. (B C)  (A + B) + C = A + (B + C) 
Distributive Law  (A+B)+ C = (A+C). (B+C)  (A + B) C = (A. C) + (B C) 
Identity Law  A. 1 A  A + 0 = A 
Zero and One Law  A. 0 = 0  A+ 1 = 1 
Inverse Law  A.A' = 0  A+ A' = 1 
Idempotent Law  A.A= A  A+A = A 
Absorption Law  A(A+B) = A  A + A.B = A A+ A'B = A+B 
DeMorgan's Law  (A. B)'=(A)'+ (B)'  (A+B)' = (A)' . (B)' 
Double Complement Law  x = x 
References:
 Boolean algebra explained  source by Wikipedia
 Boolean laws – theorems  Goerge Boole (1854)Tutorialspoint.Com
 The Mathematics of Boolean Algebra (Stanford Encyclopedia of Philosophy)  Plato.Stanford.Edu.
 Boolean Algebra  from Wolfram MathWorld.

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