LECTURE 5Standard Forms of Boolean Expressions

The Sum-of-Product (SOP) FormTwo standard formsSum-of-products (SOP) formProduct-of-sums (POS) formStandardizationEvaluation, simplification and implementation of boolean expressions become more systematic, easy

SOP FormProduct term : a term that consists of the product of literalsSum-of-products (SOP) : sum of two or more product termsExample :

AB + ABCliteralProduct termSum-of-products

SOP FormSOP can contain single variable term; > A + BC + ABCA single overbar cannot extend over more than one variable, although more than one variable in a term can have an overbar; > ABC but not ABC

Domain of a Boolean ExpressionDomain : set of variables contained in the expression in either complemented or uncomplemented form. Example: AB+ABCDomain: A, B, C

ABC+CDE+BCDDomain: A, B, C, D, E

*AND/OR Implementation of an SOP expressionX=AB+BCD+AC

Y=A + BD + BC + CDABBCDAC

Conversion of a General Expression to SOP FormAny logic expression can be change into SOP form by applying Boolean Algebra techniques Example: Try This:

The Standard SOP FormSOP expressionsABC + ABD + ABCDDomain:

Standard SOP expressionan expression where all the variables in the domain appear in each of its product term.

Examples: ABC + ABC+ ABC XYZ + XYZ Important in: - Constructing truth tables - Karnaugh Map simplification method

Conversion of a SOP expression to Standard SOP expressionA nonstandard SOP expression is converted into standard form using Boolean Algebra Rule 6 ( A + A = 1) Example: A + BStep 1 : Step 1: Multiply each nonstandard product term by 1A+B = A1 + BStep 2: Replace 1 by the sum of a missing variable and its complement = A(B+B) + B = AB + AB + BStep 3: Repeat Steps 1,2 until all product terms contain all variables in the domain

Conversion of a SOP expression to Standard SOP expression =AB + AB + B(A+A) =AB + AB + AB +AB=AB + AB + B1A+B = A1 + B= A(B+B) + B= AB + AB + BExample 2 : ABC + AB + ABCD ABC + AB + ABCD = 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 D A+B= AB + AB + AB

The Products-of-Sum (POS) FormSum term : a term that consists of the sum of literalsProduct-of-sums : multiplication of two or more sum terms Example:

The Products-of-Sum (POS) FormImplementation of the POS expression (A + B)(B + C + D)(A + C)

OR/AND Implementation

Standard POS ExpressionAn expression where all of the variables in the domain appear in each of its sum term.(A+B+D)(A+B+C) is not a standard POS expression.A nonstandard POS expression is converted into standard form using Boolean algebra Rule 8 ( A .A = 0)

Conversion of a POS expression to Standard POS expressionStep 1: Add to each nonstandard product term 0.Step 2: Replace 0 by a product of a missing variable and its complement. ( A .A = 0)Step 3: Apply rule 12: A+BC=(A+B)(A+C)Step 4: Repeat Steps 1,2,3 until each product term contains all variables in the domain.

The Standard POS FormRule 12!

Binary RepresentationSum-of-Products (SOP)An SOP expression is equal to 1 only if one or more of its product terms is equal to 1.ABCD = 1 .0.1.0 = 1.1.1.1 = 1Product-of-Sums (POS)- A POS expression is equal to 0 only if one or more of its sum terms is equal to 0. A+B+C+D = 0 + 1+ 0 + 1 = 0 + 0 + 0 +0

*Converting Standard SOP to Standard POSStep 1: For each product term of the SOP expression, determine the binary number that represent the product term.Step 2: Determine all the binary numbers not obtained in Step 1.Step 3: Write the equivalent sum term for each binary number obtained in Step 2.

- Use similar procedure to convert POS to SOP

Converting Standard SOP to Standard POSExample: Express them in POS form A B C + A B C + A B C + A B C + A BCStep 1: Since there are 3 variables ,there are total of 8 possible combination -> 23 = 8Binary number : 000 + 010 + 011 + 101 + 111( sum term = 1) Step 2 :Binary not obtained: 001, 100 , 110 ( sum term = 0 )Step 3: Equivalent binary expression obtained in step 2: ( A + B+ C )(A + B + C )( A + B + C ) These are the POS expression.

Examine each of the products to determine where the product is equal to a 1. Set the remaining row outputs to 0.

Converting SOP to Truth Table

Opposite process from the SOP expressions. Each sum term results in a 0. Set the remaining row outputs to 1.

Converting POS to Truth Table

Converting from Truth Table to SOP and POS

InputsOutputABCX00000010010001111001101011011111