WELL LET’S GO ON NOW TO THE MAIN DISH OF OUR LESSON FOR TODAY…

AND THIS IS ALL ABOUT….

What’s all about this?

› In 1854,George Boole invented two-state algebra, known today as BOOLEAN ALGEBRA.

› Every variable in Boolean Algebra can have only have either of two values: TRUE or FALSE.

› This Algebra had no practical use until Claude Shannon applied it to telephone switching circuits.

› BOOLEAN ALGEBRA is a branch of mathematics that is directly applicable to digital designs.

› It is a set of elements, a set of operators that act on these elements, and a set of axioms or postulates that govern the actions of these operators on these elements.

The Postulates commonly used to define Algebraic Structures are:

A set S is closed with respect to a binary operator * if, application of

the operator on every pair of elements of S results is an element

of S.

A binary operator * on a set S is said to be associative when (x * y) * z = x * (y * z)

For all x ,y ,z , that are elements of the set S .

A binary operator * on a set S is said to be commutative when x * y = y * x ,

For all x, y, that are elements of the set S.

A set S is said to be Identity element with respect a binary operator * on S if there exist an element e in the set S such that x * e = x

For all x that are elements of the set S.

It is said to have an inverse elements when for every x that is an element of the set S, there exists an element x’ that is a member of the set S such that x * x = e.

If * and “ .” Are two binary operators on a set S, * is said to be distributive over “.” when x * (y * z) = (x * y) . (x * z)

For all x, y, z, that are elements of the set S.

BASIC THEOREMS OF BOOLEAN

ALGEBRA

Solution Reason

x + x = ( x + x ) 1 Basic identity (b)

= ( x + x ) ( x + x’) Basic identity (a)

= x + xx’ Distributive (b)

= x + 0 Basic identity (b)

= x Basic identity (a)

Solution Reason

x . x = xx + 0 Basic identity (a)

= xx + xx’ Basic identity (b)

= x (x + x’ ) Distributive (a)

= x . 1 Basic identity (a)

= x Basic identity (b)

Solution Reason

x + 1 = (x+1) . 1 Basic identity (b)

= (x+1) . (x + x’) Basic identity (a)

= x + 1x’ Distributive (b)

= x +x’ Basic identity (b)

= 1 Basic identity (a)

Solution Reason

x . 0 = x.0 + 0 Basic identity (a)

= x.0 + xx’ Basic identity (b)

= x ( 0 + x’ ) Distributive (a)

= x ( x’ ) Basic identity (a)

= 0 Basic identity (b)

From postulate 5, we have x + x’ = 1 and x . X’ = 0, which defines the complement of x. The complement of x’ is x and is also ( x’ )’.

Therefore, we have that ( x’)’ = x.

Solution Reason

x + xy = x.1 + xy Basic identity (a)

= x( 1 + y ) Distributive (a)

= x (y+ 1) Commutative (a)

= x .1 Basic identity (a)

= x Basic identity (b)

Solution Reason

x ( x + y ) = xx + xy Distributive (a)

= x + xy Basic identity (b)

= x1 + xy Basic identity (b)

= x ( 1 + y ) Distributive (a)

= x.1 Basic identity (a)

= x Basic identity (b)

› To complement a variable is to reverse its value.

Thus ,

if x=1, then, x’=0 if x=0, then, x’=1

This operation is equivalent to a logical OR operation. The (+) plus symbol is used to indicate addition or Oring.

0 + 0 = 00 + 1 = 11 + 1 = 1

› Is equivalent to a logical AND operation.

0 . 0 = 00 . 1 = 01 . 1 = 1

The theorems of BOOLEAN ALGEBRA can be shown to hold true by means of a truth table.

If a function has N inputs, there are 2 raise to N possible combinations of these inputs and there will be 2 raise to N entries in the truth table.

Solution:

In this example you have a value of N equal to 2,

Therefore the possible combinations if 2 raise to N

is 4. Let x and y represent the variables.x y x+ y x( x+y )

0 0 0 0

0 1 1 0

1 0 1 1

1 1 1 1

HAKUNAH MATATAH!--

-Rizan ‘2012