What Is Boolean Algebra

Boolean algebra is a branch of mathematics that is very simple and close to human reasoning. Boolean algebra derives its name from the peat British mathematician George Boole. Boolean algebra is extensively used in the design of logic circuits. It is also called "twitching algebra.

Boolean expressions represent Boolean functions. The function and operation of a Logic Circuit can be determined by Boolean expressions without implementing the Logic Circuit. Logic circuits can be very large and complex. Such large circuits having many games can be simplified and implemented using fewer gates. Determining a simpler Logic circuit with fewer gates but similar to the original logic circuit can be easily done by evaluating and simplifying Boolean expressions.

Boolean Algebra expressions are written in terms of variables and literals using laws, rules, and theorems of Boolean Algebra. Simplification of Boolean expressions is also based on the Boolean laws, rules, and theorems.

 

Boolean Algebra Definitions

Different definitions of Boolean algebra are as follows:

Variable

A variable is a symbol (usually an uppercase letter) used to represent a logical quantity. A variable can have a 0 or 1 value.

Complement

A complement is the inverse of a variable and is indicated by a prime symbol over the variable. The complement of variable X is X‘. If X = 0 then X: 1 and if X = 1 then X' = 0.

Literal

A Literal is a variable or the complement of a variable.

Boolean Addition

Boolean Addition operation is performed by an OR gate. In Boolean algebra, the expression defining Boolean Addition is a sum term which is the sum of literals.

For example, -, A+B', A‘ +B‘+C

1. A sum term is 1 when anyone literal is a 1
2. A sum term is O when all literal is a 0.

Boolean Multiplication

Boolean Multiplication operation is performed by an AND sale. In Boolean algebra, the expression defining Boolean Multiplication is a product term, which is the product of Metals.

For example,

 A.B, A.B’, A'.B‘.C

1. A product term is 1 when all literal terms are a 1.
2. A product term is 0 when anyone literal is a 0.

 

Laws of Boolean Algebra

The basic laws of Boolean algebra am the name as ordinary algebra and hold true for any number of variables. These laws are:

 

Commutative Law

The commutative law of addition (or two variables is written as:

A+B=B+A

This law states that the order in which the variables are ORed makes no difference.

The commutative law for multiplication for two variables is:

AB=BA

This law states that the order in which the variables are ANDed makes No difference.

 

Associative Law

The associative law of addition for three variables is written as follows:

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

This law states that ORing more than two variables, the result is the same regardless of the grouping of the variables. The figure shows this law as applied In the 2-input OR gate.

The associative law of multiplication, for three variables, is written as follows:

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

This law states that ANDing more than two variables, the result is the same regardless of the grouping of the variables. The figure shows this law as applied to 2—input AND gates.

 

Distributive Law

The distributive law is written for three variables as follows:

This law slates that ORing two or more variables and then ANDing the result with a single variable is equivalent to ANDing the single variable with each of the two or more variables and then using the products.

 

Rules of Boolean Algebra

Rules of Boolean algebra can be proved by replacing the literals with Boolean values 0 and 1. Following are the Boolean algebra rules:

1. A+0= A
2. A+ 1=1
3. A.0=0
4. A.1=A
5. A+A=A
6.  A+A'=1
7. A.A = A
8. A.A'= 0
9. A”=A
10. A+A.B=A
11. A+A’.B=A+B
12. (A+B) . (A+C)= A+B.C

 

Simplification using Boolean algebra

Many times a Boolean expression has to be simplified using laws, rules, and theorems of Boolean algebra. The simplified expression results in fewer variables and a simpler circuit.

Example: Simplify the Boolean expression

AB+ A(B+C)+B(B+C)

Solution

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

=AB+AB+AC+BB+BC       using Distributive law

=AB+AC+B+BC                  BB=B using rule 7

=AB+AC+B                          (B+BC)= B using rule 10

=B+AC                                 (B+AB)= B using rule10

 

DeMorgan’s LAW

DeMorgan‘s theorems are extremely useful in simplifying expression in which a product or sum of variables is inverted. The two theorems are:

 

DeMorgan’s First Theorem

The compliment of a product of variable is equal to the sum of the compliments of the Variables.

DeMorgan’s law stated another way "the complement of two or more variables ANDed is equivalent to the OR of the compliant of the individual variables".

 

DeMorgan’s Second Theorem:

DeMorgan’s second theorem is stated as follows:-

The complement of a sum of variables is equal to the product of the compliment of the variables

DeMorgan’s law may also be stated as:

The compliment of two or more variables ORed is equivalent to the AND of the compliments of the individual variables.

Implications of DeMorgan's Theorem

The formula for expressing this theorem for two variables is:

(x + y) = x * y

 

Standard Form of Boolean Expressions

All Boolean expressions can be converted into and represented in one of the two standard forms:

1. Sum-of-Products form
2. Product—of—Sums form

Sum of Product Form

When two or more product terms are summed by Boolean addition, the result is a Sum-of— Product or SOP expression. For instance,

1. AB + ABC
2. ABC + CDE + B’CD’
3. A'B+A'BC'+AC

The Domain of an SOP expression is the set of variables, both complemented and un-complemented, contained in the expression. An SOP expression may have a single variable term such as X. A SOP expression cannot have a term of more than one variable having an over bar extending over the entire term, such as (AB)’+C.

 

Product of Sums form

When two or more sum terms are multiplied by Boolean multiplication, the result is a Product-of-Sum or POS expression. For instance,

• (A’+B)(A+B'+C)
• (A'+B'+C’)(C+D’+E)(B’+C+D)
• (A+3)(A+B’+QKA'+C)

The Domain of a POS expression is the set of variables, both complemented and un-complemented, contained in the expression. A POB expression can have a single variable term such as X. A POS expression up not have a term of more than one variable having an over bar extending over the entire term such as (A+B)’ ( A+B’+C ).

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