Boolean Algebra Notes.docx

Logical Statement  A statement which can be definitely Either True or False is known as Logical Statement  A statement like “Man is Tall” is a Logical statement  Logical statement is represented by Variable For example Example 1: Let a Variable X represents one Logical Statement So X=”The Man is Tall” Let another Variable Y represents another Logical Statement So Y=”The Man is Wise” Compound Statement  Compound statement can be formed by combining both the Logical statements like Following statement is Compound statement The Man is Tall and The Man is Wise So Compound statement can be formed by combining both the Logical statements  So Compound Statement can be written in symbolic form as X AND Y for Example 1 above where X=”The Man is Tall” may be definitely either True or False Logical Statement similarly

Y=”The Man is Wise” may be definitely either True or False Logical Statement Therefore Compound Statement X AND Y will be true only if both X and Y are True Truth Table  A Truth Table is formed to show all possibilities of Truth ability or Falsehood of Logical Statement X as well as Logical Statement Y and Truth ability or Falsehood of Compound Statement symbolically represented by X AND Y X FALSE FALSE TRUE TRUE

AND

Y FALSE TRUE FALSE TRUE

X AND Y FALSE FALSE FALSE TRUE

Logical AND Operation  A “.” Is used to represent AND operator so X AND Y will be represented as X.Y also The Truth and False value for a Boolean variable is represented by Truth=1 and False=0, so if X is True it is represented by X=1 and X=0 when value of X is False  So Above Truth Table can also be written using representation discussed above as X 0 0 1 1

.

Y 0 1 0 1

X.Y 0 0 0 1

 Logical AND operation give output true only if both the input Logical statement’s are True  It’s a Binary Operator because (.) or Logical AND Operator is working on more Than 1 operand’s

Compound statement can be formed by combining both the Logical statements using Logical AND Operation where a “.” Operator is used to indicate Logical AND Operator Compound statement can also be formed by combining both the Logical statements using two more operations Logical OR Operation where a “+” Operator is used to indicate Logical OR Operator and Logical Not Operation where a “-” Operator is used to indicate Logical Not Operator Logical OR Operation  A “+” Is used to represent OR operator so X OR Y will be represented as X+Y also The Truth and False value for a Boolean variable is represented by Truth=1 and False=0, so if X is True it is represented by X=1 and X=0 when value of X is False  So Above Truth Table can also be written using representation discussed above as X 0 0 1 1

+

Y 0 1 0 1

X+Y 0 1 1 1

 Logical OR operation give output true if any of the input Logical statement is True  It’s a Binary Operator because (+) or Logical OROperator is working on more Than 1 operand’s Logical NOT Operation

 A “-” Is used to represent NOT operator so NOT X will be represented as X also The Truth and False value for a Boolean variable is represented by Truth=1 and False=0, so if X is True it is represented by X=1 and X=0 when value of X is False  So Above Truth Table can also be written using representation discussed above as X 0 1

X

1 0  Logical NOT operation give output true if input Logical statement is False and False if input Logical statement is True and  It’s a Unary Operator because (-) or Logical NOT Operator is working on only 1 operand  Compound statement can be represented by Logical Expression Consisting of Variables (representing any Logical Statement) as well as operations defined on it(Logical AND(.),OR(+) and NOT(-)

Boolean Algebra  Algebra of Logic(Which deals with Logical variables to represent Logical statements)  It is one of the tool to analyze and design Logical circuits  Logical circuits are used to solve Logical expressions Definition Boolean Algebra:  Boolean Algebra may be defined as the part of mathematics which deals with some variable’s(Logical Variable) which can take value from Set S of elements with only two values allowed{0,1} and three operators two binary((Logical

OR)+,Logical AND(.)) , one Unary((Logical NOT(-)) and some basic Postulates which are always assumed to be true.These Postulates are known as Fundamental Laws of Boolean Algebra  Other Basic Theorems can be derived or proved from these fundamental laws  Fundamental laws are used to simplify Logical expressions  So Logical expression formed by Logical variable’s and operators can be simplified by applying Fundamental laws and since logic circuits are physical realization of any Logical expressions so such circuit would be cheaper and more reliable Postulate 1 of Boolean Algebra

 A Boolean variable X has two possible values 0 and 1.The values are mutually exclusive If X=0 THEN X ≠ 1 If X=1 THEN X ≠ 0 Postulate 2 of Boolean Algebra

 NOT Operator (-) to perform Not Operation is defined as:

0 = 1 and 1 = 0



NOT Operator (-) is used to find Compliment of a single Variable

So if X=0 then X =1 if X=1 then X =0

Postulate 3 of Boolean Algebra

 Logical Multiplication is performed by Logical AND Operator (.) Logical AND Operation can be is defined as: 0.0=0 0.1=0 1.0=0 1.1=1

Postulate 4 of Boolean Algebra

 Logical Addition is performed by Logical OR Operator ( +) Logical OR Operation can be is defined as: 0+0=0 0+1=1 1+0=1 1+1=1 Postulate 5 of Boolean Algebra



Identity Element  There exist an Identity element 0 for Logical OR operation and 1 for Logical AND operation such that logical variable gives output as itself for the Logical operations

(a) x + 0 = x (b) x . 1 = x

Postulate 6 of Boolean Algebra

Commutative Law

(a) x + y = y + x (b) x . y = y . x Postulate 7 of Boolean Algebra

Associative Law

(a) x + (y + z) = (x + y) + z (b) x . (y . z) = (x . y) . z Postulate 8 of Boolean Algebra

Distributive Law

(a) x . (y + z) = (x . y) + (x . z) (b) x + (y × z) = (x + y) . (x + z)

Postulate 9 of Boolean Algebra



Inverse Element



There exist an Inverse element for a variable for a Logical OR Operation which gives 1 when Logically ORED with the variable And for a Logical AND operation which gives 0 when Logically ANDED with the variable

(a) x + X = 1 (b) x . X = 0 Postulate 10 of Boolean Algebra Closure Property

For all x,y∈ B where B is Set of values {0,1} X+Y ∈ B

and

X.Y ∈ B

Proving a Theorem By using Postulates

Theorem: x+1=1 Proof: L.H.S. =x+1 = (x +1).1 by postulate 5(b) = (x +1).(x+ x ) by postulate 9(a) = x + (1. x ) by postulate 8(b) = x + x by postulate 5(b) = x by postulate 9(a) = R.H.S.

Proving a Theorem By Perfect Induction

=x+x = (x +x).1 by postulate 5(b) = (x + x) . (x+ x ) by postulate 9(a) = x + (x . x ) by postulate 8(b)

= x + 0 by postulate 9(b) = x by postulate 5(a) = R.H.S.

Proving a Theorem By Principle Of Duality

= x.x = x.x+0 by postulate 5(a) = x.x + x. x by postulate 9(b) = x. (x + . x ) by postulate 8(a) = x . 1 by postulate 9(a) = x by postulate 5(b)

Compliment of Boolean Function

Compliment of Boolean Function(Example)