SYEN 3330 Digital SystemsJung H. Kim Chapter 2-3 1 SYEN 3330 Digital Systems Chapter 2 Part 3.

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Presentation transcript:

SYEN 3330 Digital SystemsJung H. Kim Chapter SYEN 3330 Digital Systems Chapter 2 Part 3

SYEN 3330 Digital Systems Chapter Boolean Operator Precedence

SYEN 3330 Digital Systems Chapter Review: Duality Principle

SYEN 3330 Digital Systems Chapter Duality In Proofs

SYEN 3330 Digital Systems Chapter Useful Theorems

SYEN 3330 Digital Systems Chapter Proof of Simplification

SYEN 3330 Digital Systems Chapter Proof of Concensus

SYEN 3330 Digital Systems Chapter Proof of DeMorgan’s Law

SYEN 3330 Digital Systems Chapter Boolean Function Evaluation

SYEN 3330 Digital Systems Chapter Expression Simplification Simplify to contain the smallest number of literals (complemented and uncomplemented variables):

SYEN 3330 Digital Systems Chapter Complementing Functions This generate a lot of terms. You might want to simplify the expression first.

SYEN 3330 Digital Systems Chapter Canonical Forms It is useful to specify Boolean functions of n variables in a manner that is easy to compare. Two such Canonical Forms are in common usage:  Sum of Minterms  Product of Maxterms

SYEN 3330 Digital Systems Chapter Minterms

SYEN 3330 Digital Systems Chapter Maxterms

SYEN 3330 Digital Systems Chapter Maxterms and Minterms The index above is important for describing which variables in the terms are true and which are complemented.

SYEN 3330 Digital Systems Chapter Standard Order

SYEN 3330 Digital Systems Chapter Purpose of the Index The index for the minterm or maxterm, expressed as a binary number, is used to determine whether the variable is shown in the true form or complemented form.

SYEN 3330 Digital Systems Chapter Index Example in Three Variables

SYEN 3330 Digital Systems Chapter Four Variables, Index 0-7

SYEN 3330 Digital Systems Chapter Four Variables, Index 8-15

SYEN 3330 Digital Systems Chapter Minterm and Maxterm Relationship Review: DeMorgan's Theorem (x  y) = (  x +  y) and (x + y) = (  x  y ) Note: For 2 variables: M 2 = (  x + y) and m 2 = (x  y) Thus M 2 is the complement of m 2 and vice-versa. Since DeMorgan's Theorem can be extended to n variables,this holds that for terms of n variables giving: MiMi and m i are complements.

SYEN 3330 Digital Systems Chapter Function Tables for Both Minterms of two variables Maxterms of two variables

SYEN 3330 Digital Systems Chapter Observations

SYEN 3330 Digital Systems Chapter Minterm Function Example

SYEN 3330 Digital Systems Chapter Minterm Function Example F(A, B, C, D, E) = m 2 + m 9 + m 17 + m 23

SYEN 3330 Digital Systems Chapter Maxterm Function Example

SYEN 3330 Digital Systems Chapter Maxterm Function Example

SYEN 3330 Digital Systems Chapter Cannonical Sum of Minterms

SYEN 3330 Digital Systems Chapter Another SOM Example

SYEN 3330 Digital Systems Chapter Shorthand SOM Form Note that we explicitly show the standard variables in order and drop the “m” designators.

SYEN 3330 Digital Systems Chapter Canonical Product of Maxterms

SYEN 3330 Digital Systems Chapter Product of Maxterm Example

SYEN 3330 Digital Systems Chapter Function Complements Or alternately: Then:

SYEN 3330 Digital Systems Chapter Conversion Between Forms

SYEN 3330 Digital Systems Chapter Review of Canonical Forms

SYEN 3330 Digital Systems Chapter Review: Indices

SYEN 3330 Digital Systems Chapter Forms of Terms, Complements

SYEN 3330 Digital Systems Chapter Review: Sum of Minterms Form

SYEN 3330 Digital Systems Chapter Review: Product of Maxterms

SYEN 3330 Digital Systems Chapter Review: Complements, Conversions