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Boolean Algebra and Logic Circuits

Boolean Switching Algebras
A Boolean Switching Algebra is one which deals only with two-valued variables. Boole's general theory covers algebras which deal with variables which can hold n values.

Axioms
Consider a set S = { 0. 1}
Consider two binary operations, + and . , and one unary operation, -- , that act on these elements. [S, ., +, --, 0, 1] is called a switching algebra that satisfies the following axioms S
Closure
If X ÎS and Y ÎS then X.Y ÎS
If X ÎS and Y ÎS then X+Y ÎS

 

Identity
$an identity 0 for + such that X + 0 = X

$an identity 1 for . such that X . 1 = X

Commutative Laws
X + Y = Y + X

X . Y = Y . X

Distributive Laws
X.(Y + Z ) = X.Y + X.Z

X + Y.Z = (X + Y) . (X + Z)

Complement
"X ÎS $a complement X'such that

X + X' = 1
X . X' = 0
The complement X' is unique.

Theorems
A number of theorems may be proved for switching algebras

Idempotent Law
X + X = X

X . X = X

DeMorgan's Law
(X + Y)' = X' . Y', These can be proved by the use of truth tables.

Proof of (X + Y)' = X' . Y'

X Y X + Y ( X+Y)’
0 0 0 1
0 1 1 0
1 0 1 0
1 1 1 0
X Y X’ Y’ X’.Y’
0 0 1 1 1
0 1 1 0 0
1 0 0 1 0
1 1 0 0 0

The two truth tables are identical, and so the two expressions are identical.
(X.Y) = X' + Y', These can be proved by the use of truth tables.
Proof of (X.Y) = X' + Y'

X Y X.Y (X.Y)’
0 0 0 1
0 1 0 1
1 0 0 1
1 1 1 0
X Y X’ Y’ X’+Y’
0 0 1 1 1
0 1 1 0 1
1 0 0 1 1
1 1 0 0 0

Note : DeMorgans Laws are applicable for any number of variables.

Boundedness Law
X + 1 = 1

X . 0 = 0

Absorption Law
X + (X . Y) = X

X . (X + Y ) = X

Elimination Law
X + (X' . Y) = X + Y

X.(X' + Y) = X.Y

Unique Complement theorem
If X + Y = 1 and X.Y = 0 then X = Y'

Involution theorem
X'' = X

0' = 1

Associative Properties
X + (Y + Z) = (X + Y) + Z

X . ( Y . Z ) = ( X . Y ) . Z

Duality Principle
In Boolean algebras the duality Principle can be is obtained by interchanging AND and OR operators and replacing 0's by 1's and 1's by 0's. Compare the identities on the left side with the identities on the right.

Example
X.Y+Z' = (X'+Y').Z

Consensus theorem
X.Y + X'.Z + Y.Z = X.Y + X'.Z

or dual form as below
(X + Y).(X' + Z).(Y + Z) = (X + Y).(X' + Z)
Proof of X.Y + X'.Z + Y.Z = X.Y + X'.Z:

X.Y + X'.Z + Y.Z = X.Y + X'.Z
X.Y + X'.Z + (X+X').Y.Z = X.Y + X'.Z
X.Y.(1+Z) + X'.Z.(1+Y) = X.Y + X'.Z
X.Y + X'.Z = X.Y + X'.Z

(X.Y'+Z).(X+Y).Z = X.Z+Y.Z instead of X.Z+Y'.Z
X.Y'Z+X.Z+Y.Z
(X.Y'+X+Y).Z
(X+Y).Z
X.Z+Y.Z
The term which is left out is called the consensus term.
Given a pair of terms for which a variable appears in one term, and its complement in the other, then the consensus term is formed by ANDing the original terms together, leaving out the selected variable and its complement.
Example :
The consensus of X.Y and X'.Z is Y.Z
The consensus of X.Y.Z and Y'.Z'.W' is (X.Z).(Z.W')

Shannon Expansion Theorem
The Shannon Expansion Theorem is used to expand a Boolean logic function (F) in terms of (or with respect to) a Boolean variable (X), as in the following forms.

F = X . F (X = 1) + X' . F (X = 0)
where F (X = 1) represents the function F evaluated with X set equal to 1; F (X = 0) represents the function F evaluated with X set equal to 0.
Also the following function F can be expanded with respect to X,
F = X' . Y + X . Y . Z' + X' . Y' . Z
= X . (Y . Z') + X' . (Y + Y' . Z)
Thus, the function F can be split into two smaller functions.
F (X = '1') = Y . Z'
This is known as the cofactor of F with respect to X in the previous logic equation. The cofactor of F with respect to X may also be represented as F X (the cofactor of F with respect to X' is F X' ). Using the Shannon Expansion Theorem, a Boolean function may be expanded with respect to any of its variables. For example, if we expand F with respect to Y instead of X,
F = X' . Y + X . Y . Z' + X' . Y' . Z
= Y . (X' + X . Z') + Y' . (X' . Z)
A function may be expanded as many times as the number of variables it contains until the canonical form is reached. The canonical form is a unique representation for any Boolean function that uses only minterms. A minterm is a product term that contains all the variables of F¿such as X . Y' . Z).
Any Boolean function can be implemented using multiplexer blocks by representing it as a series of terms derived using the Shannon Expansion Theorem.

Summary of Laws And Theorms

Identity Dual
Operations with 0 and 1  
X + 0 = X (identity) X.1 = X
X + 1 = 1 (null element) X.0 = 0
Idempotency theorem  
X + X = X X.X = X
Complementarity  
X + X' = 1 X.X' = 0
Involution theorem  
(X')' = X  
Cummutative law  
X + Y = Y + X X.Y = Y X
Associative law  
(X + Y) + Z = X + (Y + Z) = X + Y + Z (XY)Z = X(YZ) = XYZ
Distributive law  
X(Y + Z) = XY + XZ X + (YZ) = (X + Y)(X + Z)
DeMorgan's theorem  
X + Y + Z + ...)' = X'Y'Z'...
or { f ( X1,X2,...,Xn,0,1,+,. ) } =
{ f
( X1',X2',...,Xn',1,0,.,+ ) }
(XYZ...)' = X' + Y' + Z' + ...
Simplification theorems  
XY + XY' = X (uniting) (X + Y)(X + Y') = X
X + XY = X (absorption) X(X + Y) = X
(X + Y')Y = XY (adsorption) XY' + Y = X + Y
Consensus theorem  
XY + X'Z + YZ = XY + X'Z (X + Y)(X' + Z)(Y + Z) = (X + Y)(X' + Z)
Duality  
(X + Y + Z + ...)D = XYZ...
or {f(X1,X2,...,Xn,0,1,+,.)}D
= f(X1,X2,...,Xn,1,0,.,+)
(XYZ ...)D = X + Y + Z + ...
Shannon Expansion Theorem  
f(X1,...,Xk,...Xn) Xk * f(X1,..., 1 ,...Xn) + Xk' * f(X1,..., 0 ,...Xn)
f(X1,...,Xk,...Xn) [Xk + f(X1,..., 0 ,...Xn)] * [Xk' + f(X1,..., 1 ,...Xn)]