boolean algebra presfgdgdfdgentations.pptx

Boolean algebra stuffs

1. BOOLEAN ALGEBRA AND ITS POSTULATES AND THEOREMS

2. What is boolean Algebra ?

3. Boolean means a result that can only have one of two possible values: true or false. Boolean algebra is branch of algebra which deals with binary values and represented by 0 and 1.

4. • The main aim of Boolean algebra 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. Why we use Boolean algebra ?

5. Postulates of Boolean algebra • If X ≠ then X = 1 and If X ≠ 1 then X= 0. • OR relation ( Logical Addition)- 0 + = 1 + = 1 + 1 = 1 1 + 1 = 1

6. • AND Operation (Logical Multiplication) – 0.0 = 0.1 = 1.0 = 1.1 = 1 • NOT Operation ( Complement) – A + A’ = 1 A . A’ = 0

7. Basic Laws of Boolean Algebra Some of the basic laws of the Boolean algebra are • I. Associative law. • iv Indempotence Law

• ii. Distributive law. • v Complementarity Law

• iii. Commutative law • vi Absorption Law

8. Associative Law - :Associative law of addition states that .: the order of an operation doesn‘t matter when the variables have the same priority. • when any three real numbers are added or multiplied, then the grouping (or association) of the numbers does not affect the result. • X + (Y+Z) = (X + Y) +Z. X + (Y+Z) = (X + Y) +Z • X(YZ) = (XY)Z

9. Distributive Law - Distributive Law states that the multiplication of two variables and adding the result with a variable will result in the same value as multiplication of addition of the variable with individual variables. Distributive Law can be written as - X(Y+Z) = XY + XZ • X(Y+Z) = XY + XZ • X + YZ = (X+Y)(X+Z)

10. Commutative Law - Commutative Law states that the interchanging of the order of operands in a Boolean equation does not change its resu l t. Commutative law can be written as – • X+Y = Y+X • X. Y = Y. X

11. Indempotent Law - Indempotent law states that a variable would remain unchanged when AND operation and OR Operation are performed with itself. Idempotent law can be written as – X + X = X. X. X = X

12. Complementarity Law - Complementarity Law states that in case a complement is added to any variable, then it would give one, whereas when we multiply this variable with its own complement, then it would result in ‘0‘ . Complementarity law can be written as - • X+ X' = 1 • X. X' = 0.

13. Absorption Law - Absorption law states that – If you have a term A and you perform OR operation on it with the result of A AND B , the result is just A .
If you have a term A and you perform AND Operation on it with the result of A OR B , the result is just A . It can be written as - • X + XY = X • X(X+Y) = X

14. Principle of Duality - Principle of Duality states that – Changing each OR sign (+) to an AND sign (.) Changing each AND sign (.) t o an OR sign (+) Replacing each by 1 and each 1 by . For example :- Take postulate 2 related to logical addition – a) 0+0=0. (b) 0+1=1. (c)1+0=1. (d)1+1=1 Now using principle of duality change ( +) sign into(. ) And (. )Sign into (+) And change 1 into 0 and 0 into 1. it give ( i ) 1.1=1. (ii) 1.0=0. iii)0.1=0. (iv)0.0=0 It is same as the postulate 3 related to multiplication.

15. Demorgan’s Theorem - Demorgan’s First Theorem – • It states that the Compliment of addition of Two terms is equal to the multiplication of complement of each term . (X+Y)’ = X’.Y’ Demorgan’s Second Theorem – • It states that the complement of the product of all the terms is equal to the sum of the complement of each term. ( X. Y)’ = X’+ Y’

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