# Boolean Algebra Calculator

Tips to use this calculator:
• Select text and press '!' to Not the selection
• Press '+' for an 'or' gate. Eg; A+B
• Press '!' to insert a 'not' gate
• Side by side characters represents an 'and' gate. Eg; AB+CA
• -- Sample Expressions --
• -(A+B)=(-A*-B) = De Morgan
• A+B = A or B
• A*B = A and B
• (A*-B)+(-A*B) = XOR
• -P+Q = Definition of impliy
• -(P*(-P+Q))+Q = Modus ponens

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.

1. Enter the Expression.
2. 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.

## What is Boolean Algebra?

Mathematics has different branches e.g algebra, geometry e.t.c. These branches are further divided into sub-branches. Boolean algebra is one such sub-branch 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;

1. A + B = B + A

2. A.B = B.A

Associative law:

This law states;

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

2. A(B.C) = (A.B)C

Distributive law:

Using this law, we know;

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

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

Identity law:

By identity law:

1. A + 0 = A

2. A . 1 = A

Annulment law:

Here;

1. A . 0 = 0

2. A + 1 = 1

Idempotent law:

By this law:

1. A + A = A

2. 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 real-time 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:

NameAlgebraic function
ANDF = A.B
ORF = A+B
NOTF = A
NANDF = (A.B)
NORF = (A+B)

## Table of Boolean Algebra

ABCA+BA+C(A+B)(A+C)BCA+BC
00000000
00101000
01010000
01111111
10011101
10111101
11011101
11111111

## Truth Table for Binary Logical Operations

Here is a truth table for all binary logical operations:

pq F NOR   ¬p  ¬q  XOR NAND AND  XNOR qpORT
TTFFFFFFFFTTTTTTTT
TFFFFFTTTTFFFFTTTT
FTFFTTFFTTFFTTFFTT
FFFTFTFTFTFTFTFTFT
Com
Assoc
NegTORpqXNORANDNANDXOR¬q¬pNORF
DualTNAND¬p¬qXNORNORORXORqpANDF
L id  F   F TTT,FT  F
R id    F F TT  T,FTF

## 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:

1. Boolean algebra explained | source by Wikipedia
2. Boolean laws – theorems | Goerge Boole (1854)-Tutorialspoint.Com
3. The Mathematics of Boolean Algebra (Stanford Encyclopedia of Philosophy) | Plato.Stanford.Edu
4. Boolean Algebra -- from Wolfram MathWorld

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