Boolean Algebra Calculator

-- Sample Expressions --
  • -(A+B)=(-A*-B) = De Morgan
  • A+B = A or B
  • A*B = A and B
  • P=>Q = P implies Q
  • (P=>Q)*(Q=>R)=>(P=>R)
  • (A*-B)+(-A*B) = XOR
  • -P+Q = Definition of impliy
  • -(P*(-P+Q))+Q = Modus ponens
Result

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This calculator will help to solve the Boolean Algebra expressions in the simplest way. It automatically applies the rules of algebra to the logic and gives the results instantly. 

What is Boolean Algebra? 

This is a Boolean algebra solver, that allows the user to solve the complex algebraic expressions through applying the rules that are used in algebra over logic. 
This calculator is used for making simplifications in the expressions of logic circuits. It converts the complex expression into a similar expression that has fewer terms. This way, the user would have less combinational logic circuits for implementation. 
It has two common values including true and false while it is represented by 0 and 1. 0 is considered as true and 1 is considered as false. 
There are three common operators to use in the Boolean Algebra which are shown below in the table:

operators used in algebra calculator Function Symbols  input method 
Conjunction AND ^ And 
Disjunction OR v Or
Negation NOT ¬ Not


These are known as Logical operators or Boolean operators. 

Laws of Boolean Algebra: 

All the Boolean simplification calculators work based on specific rules that help to make the Boolean expression easy for logic circuits. Through applying the rules, the function becomes fewer components. 
Here are the simplification rules: 

  • Annulment Law or A + AB = A

This includes the simplification of the expression “A + 1 = 1” and “1A = A”. Through the rules, we get
A + AB 
A (1 + B)             by (A + 1 = 1) 
A (1)                    by (1A = A)
A
In the above steps, we have reduced (B + 1) to 1 with the help of the law “A + 1 = 1”. Here A is not static and can be changed with any of the values. 

  • A + AB = A + B

Applying the Annulment law, we get the following: 
(A + AB) + AB
A + B (A + A)       by factorization 
A + B (1)              by (A + A = 1)
A + B                    by (1A = A)

This way, the factorization, and the annulment law has made it easier for reducing the expressions. 

  • (A+B) (A=C) = A + BC

AA + AC + AB + BC               by multiplying both terms
A + AC + AB + BC                 by (A + AB = A)
A + AB + BC                          by (A + AB = A)
A + BC    

Application of Boolean Algebra

The Boolean algebra can be used on any of the systems where the machine works on 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)

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

 

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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