Basic theorems and properties of boolean algebra

fundamental postulates of boolean algebra

1. Boolean Algebra Laws and Theorems Boolean Algebra is a form of mathematical algebra that is used in digital logic in digital electronics. Albebra consists of symbolic representation of a statement (generally mathematical statements). Similarly, there are expressions, equations and functions in Boolean algebra as well.

2. Boolean Algebra Laws and Theorems The main aim of any logic design is to simplify the logic as much as possible so that the final implementation will become easy. In order to simplify the logic, the Boolean equations and expressions representing that logic must be simplified. So, to simplify the Boolean equations and expression, there are some laws and theorems proposed. Using these laws and theorems, it becomes very easy to simplify or reduce the logical complexities of any Boolean expression or function.

3. The article demonstrates some of the most commonly used laws and theorem is Boolean algebra . Basic Laws and Proofs The basic rules and laws of Boolean algebraic system are known as “Laws of Boolean algebra”. Some of the basic laws (rules) of the Boolean algebra are i.Associative law ii. Distributive law iii. Commutative law iv. Absorption law v. Consensus law

4. 1.Associative Law Associate Law of Addition Statement: Associative law of addition states that OR ing more than two variables i.e. mathematical addition operation performed on variables will return the same value irrespective of the grouping of variables in an equation. It involves in swapping of variables in groups. The Associative law using OR operator can be written as A+(B+C) = (A+B)+C  = A + (B + C) = B + (C + A) Eg :(0 + 1) + 0 = 0 + (1 + 0) 1 + 0 = 0 + 1 1 = 1

5. Associate Law of  Multiplication Statement: Associative law of multiplication states that ANDing more than two variables i.e. mathematical multiplication operation performed on variables will return the same value irrespective of the grouping of variables in an equation. The Associative law using AND operator can be written as A * (B * C) = (A * B) * C

6. Distributive law This is the most used and most important law in Boolean algebra, which involves in 2 operators: AND, OR. Statement: The multiplication of two variables and adding the result with a variable will result in same value as multiplication of addition of the variable with individual variables. In other words, ANDing two variables and ORing the result with another variable is equal to AND of ORing of the variable with the two individual variables. Distributive law can be written as A + BC = (A + B)(A + C) similarly, A (B+C) = (A B) + (A C)

7. Commutative law Statement: Commutative law states that the inter-changing of the order of operands in a Boolean equation does not change its result. Using OR operator → A + B = B + A Using AND operator → A * B = B * A This law is also has more priority in Boolean algebra. Eg:1 + 0 = 0 + 1, 1 * 0 = 0 * 1

8. Absorption Law Absorption law involves in linking of a pair of binary operations. A+A=A A.A=A A+AB = A A(A+B) = A A+ĀB = A+B A.(Ā+B) = AB 5th and 6th laws are also called as Redundancy laws.

9. 1.Statement : A + A = A LHS=A+A =(A+A).1 =(A+A).(A+Ā) =A+( A. Ā) =A(RHS) 2.Statement : A . A = A LHS=(A.A)+0 =( A.A) +(A. Ā) =A.( A+Ā) =A +( 1 ) =A(RHS)

10. 3.Statement : A + AB = A Proof: A + AB = A.1 + AB → since A.1 = A =A(1+B) → since 1 + B = 1 = A.1 = A 4.Statement : A (A + B) = A Proof: A (A + B) = A.A + A.B = A+AB → since A . A = A = A (1 + B) = A.1 = A

11. 5.Statement :A + ĀB = A + B Proof: A + ĀB = (A + Ā) (A + B) → since A+BC =( A+B)(A+C) using distributive law = 1 * (A + B) → since A + Ā = 1 =A + B 6.Statement : A * (Ā+B) = AB Proof: A * (Ā + B) = A. Ā + AB = AB → since A Ā = 0

12. Duality Principle in Boolean algebra Statement: Duality principle states that “The Dual of the expression can be achieved by replacing the AND operator with OR operator , along with replacing the binary variables, such as replacing 1 with 0 and replacing 0 with 1 ”. (or) One expression can be obtain from another expression by replacing the every 1 with 0, every 0 with 1, every (+) with (.), every (.) with (+). Any pair of expression satisfying this property is called  dual  expression. This characteristic of Boolean algebra is called the  principle  of  duality . Eg : A + B = 0 , A * B = 1.

13. De Morgan’s Theorem: Statement 1 : “ The compliment of the product of 2 variables is equal to the sum of the compliments of individual variables”. ( A.B)’ = A’ + B’ Statement 2 : “The compliment of the sum of two variables is equal to the product of the compliment of each variable ”. (A + B)’ = A’.B’

14. Truth Tables The De Morgan’s laws are simply explained by using the truth tables. The truth table for De Morgan’s first statement((A.B)’ = A’ + B’) is given below. The truth table for De Morgan’s second statement ((A + B)’ = A’.B’) is given below.