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Lecture No. 9

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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  1. Lecture No. 9 Computer Logic Design Boolean Algebra and Logic Simplification

  2. Veracity of DeMorgan's Theorems • First Theorem • Second Theorem • Alternative Method – use Truth Tables

  3. Application of DeMorgan's Theorems • Apply to any number of variables • Apply to combination of variables

  4. Application of DeMorgan's Theorems • Finding Complement of a Function • Example:

  5. 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 • Helpful to add parenthesis • Complement each literal

  6. 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

  7. Sum-of-Products (SOP) Form • Two or more product terms summed by Boolean addition • Any expression -> SOP using Boolean algebra • Examples: * A + BC

  8. Sum-of-Products (SOP) Form • Conversion to SOP Form:

  9. Implementation of SOP Expression B+AC+AD

  10. 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:

  11. Standard SOP Form • Example: Determine Standard SOP expression SHORTCUT: Introduce all possible combinations of the missing variables AND’ed with the original term

  12. 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:

  13. 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)

  14. Product-of-Sums (POS) Form • Conversion to POS Form:

  15. Implementation of POS expression (A+B)(B+C+D)(A+C)

  16. 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.

  17. Standard POS Form • Example: {Rule 8} {Rule 12} SHORTCUT: Introduce all possible combinations of the missing variables OR’ed with the original term

  18. 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:

  19. 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

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