Lecture No. 9

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# Lecture No. 9 - PowerPoint PPT Presentation

Lecture No. 9. Computer Logic Design Boolean Algebra and Logic Simplification. Veracity of DeMorgan's Theorems. First Theorem Second Theorem Alternative Method – use Truth Tables. Application of DeMorgan's Theorems. Apply to any number of variables Apply to combination of variables.

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## Lecture No. 9

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Lecture No. 9

Computer Logic Design

Boolean Algebra and Logic Simplification

Veracity of DeMorgan's Theorems
• First Theorem
• Second Theorem
• Alternative Method – use Truth Tables
Application of DeMorgan's Theorems
• Apply to any number of variables
• Apply to combination of variables
Application of DeMorgan's Theorems
• Finding Complement of a Function
• Example:
Application of DeMorgan's Theorems

Shortcut for finding Complement of a Function

• Take dual of the function
• swap 1’s and 0’s
• Swap AND and OR gates
• Complement each literal
Types of Boolean Expressions
• Define Domain of an expression
• set of all variables (complemented or otherwise)
• Boolean expressions may be expressed as:
• Sum-of-Products (SOP) Form
• Product-of-Sums (POS) Form
• Each form may contain single variable terms
• May contain complemented and un-complemented terms
• A SOP and POS expression can’t have a term of more than one variable having an over bar extending over the entire term
Sum-of-Products (SOP) Form
• Two or more product terms summed by Boolean addition
• Any expression -> SOP using Boolean algebra
• Examples:

* A + BC

Sum-of-Products (SOP) Form
• Conversion to SOP Form:
Standard SOP Form & Minterms
• SOP expressions containing all Variables in the Domain in each term are in Standard Form.
• Standard product terms are also called Minterms.
• Any non-standard SOP expression may be converted to Standard form by applying Boolean Algebra Rule 6 to it.
• Example:
Standard SOP Form
• Example: Determine Standard SOP expression

SHORTCUT: Introduce all possible combinations of the missing variables AND’ed with the original term

Characteristics of a Minterm
• Minterm is a standard product term in which all variables appear exactly once (complemented or uncomplemented)
• Represents exactly one combination of the binary variables in a truth table for which the function produces a “1” output. That is the binary representation or value.
• Has value of 1 for that combination and 0 for all others
• For n variables, there are 2n distinct minterms
• Example:
Product-of-Sums (POS) Form
• Two or more sum terms multiplied by Boolean multiplication
• Any expression -> POS using Boolean algebra
• Examples:

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

Product-of-Sums (POS) Form
• Conversion to POS Form:
Standard POS Form & Maxterms
• POS expressions containing all Variables in the Domain in each term are in Standard Form.
• Standard sum terms are also called Maxterms. A Maxterm is a NOT Minterm.
• Any non-standard POS expression may be converted to Standard form by applying Boolean Algebra Rule 8 and Rule 12A+BC=(A+B)(A+C) to it.
Standard POS Form
• Example:

{Rule 8}

{Rule 12}

SHORTCUT: Introduce all possible combinations of the missing variables OR’ed with the original term

Characteristics of a Maxterm
• Maxterm is a standard sum term in which all variables appear exactly once (complemented or uncomplemented)
• Represents exactly one combination of the binary variables in a truth table for which the function produces a “0” output. That is the binary representation or value.
• Has value of 0 for that combination and 1 for all others
• For n variables, there are 2n distinct maxterms
• Example:
Why Standard SOP and POS Forms?
• Direct mapping of Standard Form expressions and Truth Table entries.
• Alternate Mapping methods for simplification of expressions
• Minimal Circuit implementation by switching between Standard SOP or POS
• PLD based function implementation