Boolean Algebra Theorems and Laws of Boolean Algebra

Boolean algebra is a different kind of algebra or rather can be said a new kind of algebra which was invented by world famous mathematician George Boole in the year of 1854. He published it in his book “An Investigation of the Laws of Thought”. Later using this technique Claude Shannon introduced a new type of algebra which is termed as Switching Algebra. In digital electronics there are several methods of simplifying the design of logic circuits. This algebra is one of these methods. According to George Boole symbols can be used to represent the structure of logical thoughts. This type of algebra deals with the rules or laws, which are known as laws of Boolean algebra by which the logical operations are carried out.

There are also few theorems of Boolean algebra, that are needed to be noticed carefully because these make calculation fastest and easier. Boolean logic deals with only two variables, 1 and 0 by which all the mathematical operations are to be performed.
Boolean algebra or switching algebra is a system of mathematical logic to perform different mathematical operations in binary system. There only three basis binary operations, AND, OR and NOT by which all simple as well as complex binary mathematical operations are to be done. There are many rules in Boolean algebra by which those mathematical operations are done. In Boolean algebra, the variables are represented by English Capital Letter like A, B, C etc and the value of each variable can be either 1 or 0, nothing else. In Boolean algebra an expression given can also be converted into a logic diagram using different logic gates like AND gate, OR gate and NOT gate, NOR gates, NAND gates, XOR gates, XNOR gates etc.

Some basic logical Boolean operations, AND Operation OR Operation Not Operation Some basic laws for Boolean Algebra A . 0 = 0 where A can be either 0 or 1.
A . 1 = A where A can be either 0 or 1.
A . A = A where A can be either 0 or 1.
A . Ā = 0 where A can be either 0 or 1.
A + 0 = A where A can be either 0 or 1.
A + 1 = 1 where A can be either 0 or 1.
A + Ā = 1
A + A = A
A + B = B + A where A and B can be either 0 or 1.
A . B = B . A where A and B can be either 0 or 1.
The laws of Boolean algebra are also true for more than two variables like,

Cumulative Law for Boolean Algebra

cumulative laws for boolean algebra According to Cumulative Law, the order of OR operations and AND operations conducted on the variables makes no differences.

Associative Laws for Boolean Algebra

This law is for several variables, where the OR operation of the variables result is same though the grouping of the variables. This law is quite same in case of AND operators. associative laws for boolean algebra

Distributive Laws for Boolean Algebra

This law is composed of two operators, AND and OR. distributive laws for boolean algebra Let us show one use of this law to prove the expression Proof:

Redundant Literal Rule

redundant literal rule From truth table,
Inputs Output
Inputs Output
From truth table it is proved that,

Absorption Laws for Boolean Algebra

Proof from truth table,
Inputs Output
Both A and A+A.B column is same. Proof from truth table,
Both A and A.X or A(A+B) column are same. De Morgan’s Therem, Proof from truth table,

Examples of Boolean Algebra

These are another method of simplifying complex Boolean expression. In this method we only use three simple steps.
  1. Complement entire Boolean expression.
  2. Change all ORs to ANDs and all ANDs to ORs.
  3. Now, complement each of the variable and get final expression.
By this method, will be first complemented, i.e..Now, change all (+) to (.) and (.) to (+) i.e.Now, complement each of the variable,This is the final simplified form of Boolean expression, And it is exactly equal to the results which have been come by applying De Morgan Theorem.
Another example, By Second Method, Representation of Boolean function in truth table.
Let us consider a Boolean function, Now let us represent the function in truth table. Thus we have shown some basic laws of Boolean algebra. In the other page we have described De Morgan’s theorems and related laws on it.

Closely Related Articles Digital ElectronicsDe Morgan Theorem and Demorgans LawsTruth Tables for Digital LogicBinary Arithmetic Binary AdditionBinary SubtractionSimplifying Boolean Expression using K MapBinary DivisionExcess 3 Code Addition and SubtractionK Map or Karnaugh MapSwitching Algebra or Boolean AlgebraBinary MultiplicationParallel SubtractorMore Related Articles Binary Adder Half and Full AdderBinary SubstractorSeven Segment DisplayBinary to Gray Code Converter and Grey to Binary Code ConverterBinary to BCD Code ConverterAnalog to Digital ConverterDigital Encoder or Binary EncoderBinary DecoderBasic Digital CounterDigital ComparatorBCD to Seven Segment DecoderParallel AdderParallel Adder or SubtractorMultiplexerDemultiplexer555 Timer and 555 Timer WorkingLook Ahead Carry AdderOR Operation | Logical OR OperationAND Operation | Logical AND OperationLogical OR GateLogical AND GateNOT GateUniversal Gate | NAND and NOR Gate as Universal GateNAND GateDiode and Transistor NAND Gate or DTL NAND Gate and NAND Gate ICsX OR Gate and X NOR GateTransistor Transistor Logic or TTLNOR GateFan out of Logic GatesINHIBIT GateNMOS Logic and PMOS LogicSchmitt GatesLogic Families Significance and Types of Logic FamiliesBinary Number System | Binary to Decimal and Decimal to Binary ConversionBinary to Decimal and Decimal to Binary ConversionBCD or Binary Coded Decimal | BCD Conversion Addition SubtractionBinary to Octal and Octal to Binary ConversionOctal to Decimal and Decimal to Octal ConversionBinary to Hexadecimal and Hex to Binary ConversionHexadecimal to Decimal and Decimal to Hexadecimal ConversionGray Code | Binary to Gray Code and that to Binary ConversionOctal Number SystemDigital Logic Gates2′s Complement1′s ComplementASCII CodeHamming Code2s Complement ArithmeticError Detection and Correction Codes9s complement and 10s complement | SubtractionSome Common Applications of Logic GatesKeyboard EncoderAlphanumeric codes | ASCII code | EBCDIC code | UNICODELatches and Flip FlopsS R Flip Flop S R LatchActive Low S R Latch and Flip FlopGated S R Latches or Clocked S R Flip FlopsD Flip Flop or D LatchJ K Flip FlopMaster Slave Flip FlopRead Only Memory | ROMProgrammable Logic DevicesProgrammable Array LogicApplication of Flip FlopsShift RegistersBuffer Register and Controlled Buffer RegisterData Transfer in Shift RegistersSerial In Serial Out (SISO) Shift RegisterSerial in Parallel Out (SIPO) Shift RegisterParallel in Serial Out (PISO) Shift RegisterParallel in Parallel Out (PIPO) Shift RegisterUniversal Shift RegistersBidirectional Shift RegisterDynamic Shift RegisterApplications of Shift RegistersUninterruptible Power Supply | UPSConversion of Flip FlopsJohnson CounterSequence GeneratorRing CounterNew Articles Ring CounterDischarging a CapacitorCharging a CapacitorElectric PotentialParity Generator