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SYEN 3330 Digital Systems. Chapter 2 Part 3. Boolean Operator Precedence. Review: Duality Principle. Duality In Proofs. Useful Theorems. Proof of Simplification. Proof of Concensus. Proof of DeMorgan’s Law. Boolean Function Evaluation. Expression Simplification.
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SYEN 3330Digital Systems Chapter 2 Part 3 SYEN 3330 Digital Systems
Boolean Operator Precedence SYEN 3330 Digital Systems
Review: Duality Principle SYEN 3330 Digital Systems
Duality In Proofs SYEN 3330 Digital Systems
Useful Theorems SYEN 3330 Digital Systems
Proof of Simplification SYEN 3330 Digital Systems
Proof of Concensus SYEN 3330 Digital Systems
Proof of DeMorgan’s Law SYEN 3330 Digital Systems
Boolean Function Evaluation SYEN 3330 Digital Systems
Expression Simplification • Simplify to contain the smallest number of literals (complemented and uncomplemented variables): SYEN 3330 Digital Systems
Complementing Functions • This generate a lot of terms. You might want to simplify the expression first. SYEN 3330 Digital Systems
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
Minterms SYEN 3330 Digital Systems
Maxterms SYEN 3330 Digital Systems
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
Standard Order SYEN 3330 Digital Systems
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
Index Example in Three Variables SYEN 3330 Digital Systems
Four Variables, Index 0-7 SYEN 3330 Digital Systems
Four Variables, Index 8-15 SYEN 3330 Digital Systems
Review: DeMorgan's Theorem · ` ` ` · ` (x y) = ( x + y) and (x + y) = ( x y ) Note: For 2 variables: ` · ` M = ( x + y) and m = (x y ) 2 2 Thus M is the complement of m and vice - versa. 2 2 Since DeMorgan's Theorem can be extended to n variables, this holds that for terms of variables n giving : Mi and mi are complements. Minterm and Maxterm Relationship SYEN 3330 Digital Systems
Function Tables for Both Minterms of two variables Maxterms of two variables SYEN 3330 Digital Systems
Observations SYEN 3330 Digital Systems
Minterm Function Example SYEN 3330 Digital Systems
Minterm Function Example • F(A, B, C, D, E) = m2 + m9 + m17 + m23 SYEN 3330 Digital Systems
Maxterm Function Example SYEN 3330 Digital Systems
Maxterm Function Example SYEN 3330 Digital Systems
Cannonical Sum of Minterms SYEN 3330 Digital Systems
Another SOM Example SYEN 3330 Digital Systems
Shorthand SOM Form Note that we explicitly show the standard variables in order and drop the “m” designators. SYEN 3330 Digital Systems
Canonical Product of Maxterms SYEN 3330 Digital Systems
Product of Maxterm Example SYEN 3330 Digital Systems
Function Complements Then: Or alternately: SYEN 3330 Digital Systems
Conversion Between Forms SYEN 3330 Digital Systems
Review of Canonical Forms SYEN 3330 Digital Systems
Review: Indices SYEN 3330 Digital Systems
Forms of Terms, Complements SYEN 3330 Digital Systems
Review: Sum of Minterms Form SYEN 3330 Digital Systems
Review: Product of Maxterms SYEN 3330 Digital Systems
Review: Complements, Conversions SYEN 3330 Digital Systems
