Boolean Logic

Symbolic Logic
Boolean algebra derives its name from the mathematician George Boole. Symbolic Logic uses values, variables and operations :

• True is represented by the value 1.
• False is represented by the value 0.

Variables are represented by letters and can have one of two values, either 0 or 1. Operations are functions of one or more variables.

• AND is represented by X.Y
• OR is represented by X + Y
• NOT is represented by X' . Throughout this tutorial the X' form will be used and sometime !X will be used.

These basic operations can be combined to give expressions.
Example :

• X
• X.Y
• W.X.Y + Z

Precedence
As with any other branch of mathematics, these operators have an order of precedence. NOT operations have the highest precedence, followed by AND operations, followed by OR operations. Brackets can be used as with other forms of algebra. e.g.
X.Y + Z and X.(Y + Z) are not the same function.

Function Definitions
The logic operations given previously are defined as follows :

Define f(X,Y) to be some function of the variables X and Y.

f(X,Y) = X.Y

• 1 if X = 1 and Y = 1
• 0 Otherwise

f(X,Y) = X + Y

• 1 if X = 1 or Y = 1
• 0 Otherwise

f(X) = X'

• 1 if X = 0
• 0 Otherwise

Truth Tables
Truth tables are a means of representing the results of a logic function using a table. They are constructed by defining all possible combinations of the inputs to a function, and then calculating the output for each combination in turn. For the three functions we have just defined, the truth tables are as follows.

AND

X	Y	F(X,Y)
0	0	0
0	1	0
1	0	0
1	1	1

OR

X	Y	F(X,Y)
0	0	0
0	1	1
1	0	1
1	1	1

NOT

X	F(X)
0	1
1	0

Truth tables may contain as many input variables as desired
F(X,Y,Z) = X.Y + Z

X	Y	Z	F(X,Y,Z)
0	0	0	0
0	0	1	1
0	1	0	0
0	1	1	1
1	0	0	0
1	0	1	1
1	1	0	1
1	1	1	1

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 S and Y S then X.Y S
If X S and Y S then X+Y S

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 S a complement X'such that

X + X' = 1

X . X' = 0

The complement X' is unique.

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)

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')

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)]