Boolean Algebra: Simplification using Karnaugh Map and Standard Forms, Lecture notes of Organizational Development

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SIMPLIFICATION OF B OOLEAN

ALGEBRA

Presented By: Ms. Poonam Anand

S IMPLIFICATION USING B OOLEAN A LGEBRA

 A simplified Boolean expression uses the fewest

gates possible to implement a given expression.

A B C AB+A(B+C)+B(B+C)

S IMPLIFICATION USING B OOLEAN A LGEBRA

 Try these:

AB AC AB C

ABC ABC ABC ABC ABC

AB C BD AB C

[ ( + ) + ]

8

S TANDARD F ORMS OF B OOLEAN E XPRESSIONS

 All Boolean expressions, regardless of their form,

can be converted into either of two standard

forms:

 The sum-of-products (SOP) form

 The product-of-sums (POS) form

 Standardization makes the evaluation,

simplification, and implementation of Boolean

expressions much more systematic and easier.

THE SUM- OF-PRODUCTS (SOP) F ORM  An SOP expression  when two or more product terms are summed by Boolean addition.

 Examples:

 Also:

AB ABC AC

ABC CDE BC D

AB ABC

A + ABC+BC D  In an SOP form, a single overbar cannot extend over more than one variable; however, more than one variable in a term can have an overbar:

 example: is OK!

 But not :

AB C ABC

I MPLEMENTATION OF AN SOP

 AND/OR implementation  NAND/NAND

implementation

X=AB+BCD+AC

A B B C D A C X A B B C D A C X

THE STANDARD SOP FORM

 A standard SOP expression is one in which all the variables in the domain appear in each product term in the expression.

 Example:

 Standard SOP expressions are important in:

 Constructing truth tables

 The Karnaugh map simplification method

AB CD + ABCD + ABC D

CONVERTING PRODUCT TERMS TO STANDARD SOP  Step 1: Multiply each nonstandard product term by a term made up of the sum of a missing variable and its complement. This results in two product terms.  As you know, you can multiply anything by 1 without changing its value.  Step 2: Repeat step 1 until all resulting product term contains all variables in the domain in either complemented or uncomplemented form. In converting a product term to standard form, the number of product terms is doubled for each missing variable.

BINARY REPRESENTATION OF A STANDARD PRODUCT TERM  A standard product term is equal to 1 for only one combination of variable values.

 Example: is equal to 1 when A=1, B=0,

C=1, and D=0 as shown below

 And this term is 0 for all other combinations of

values for the variables.

ABCD = 1 • 0 • 1 • 0 = 1 • 1 • 1 • 1 = 1

ABC D

PRODUCT - OF -SUMS (POS)

I MPLEMENTATION OF A POS

 OR/AND implementation

X=(A+B)(B+C+D)(A+C)

A B B C D A C X

THE STANDARD POS FORM

 A standard POS expression is one in which all the variables in the domain appear in each sum term in the expression.

 Example:

 Standard POS expressions are important in:

 Constructing truth tables

 The Karnaugh map simplification method

( A +B +C +D)(A+B +C+D)(A+B+C +D )

CONVERTING A SUM TERM TO STANDARD POS ( EXAMPLE)  Convert the following Boolean expression into standard POS form: ( A +B+C)(B+C+D)(A+B+C +D ) ( )( )( )( )( ) ( )( )( ) ( )( ) ( )( ) A B C D A B C D A B C D A B C D A B C D A B C B C D A B C D B C D B C D AA A B C D A B C D A B C A B C DD A B C D A B C D

• • • • • • • • • • + + + + + + + + + + + + = + + = + + + = + + + + + + + + = + + + = + + + + + +

BINARY REPRESENTATION OF A STANDARD SUM TERM  A standard sum term is equal to 0 for only one combination of variable values.

 Example: is equal to 0 when A=0, B=1,

C=0, and D=1 as shown below

 And this term is 1 for all other combinations of

values for the variables.

A+ B +C+D = 0 + 1 + 0 + 1 = 0 + 0 + 0 + 0 = 0

A +B +C+ D