SYEN 3330 Digital Systems Chapter 2 Part 3

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

Boolean Operator Precedence SYEN 3330 Digital Systems 2

Review: Duality Principle SYEN 3330 Digital Systems 3

Duality In Proofs SYEN 3330 Digital Systems 4

Useful Theorems • • • SYEN 3330 Digital Systems 5

Proof of Simplification • SYEN 3330 Digital Systems 6

Proof of Concensus • SYEN 3330 Digital Systems 7

Proof of De. Morgan’s Law • SYEN 3330 Digital Systems 8

Boolean Function Evaluation SYEN 3330 Digital Systems 9

Expression Simplification • Simplify to contain the smallest number of literals (complemented and uncomplemented variables): SYEN 3330 Digital Systems 10

Complementing Functions • This generate a lot of terms. You might want to simplify the expression first. SYEN 3330 Digital Systems 11

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 12

Minterms SYEN 3330 Digital Systems 13

Maxterms SYEN 3330 Digital Systems 14

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 15

Standard Order SYEN 3330 Digital Systems 16

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 17

Index Example in Three Variables SYEN 3330 Digital Systems 18

Four Variables, Index 0 -7 SYEN 3330 Digital Systems 19

Four Variables, Index 8 -15 SYEN 3330 Digital Systems 20

Minterm and Maxterm Relationship Review: De. Morgan'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 De. Morgan's Theorem can be extended to n variables, this holds that for terms of n variables giving: Mi and mi are complements. SYEN 3330 Digital Systems 21

Function Tables for Both Minterms of two variables SYEN 3330 Digital Systems Maxterms of two variables 22

Observations SYEN 3330 Digital Systems 23

Minterm Function Example SYEN 3330 Digital Systems 24

Minterm Function Example • F(A, B, C, D, E) = m 2 + m 9 + m 17 + m 23 SYEN 3330 Digital Systems 25

Maxterm Function Example SYEN 3330 Digital Systems 26

Maxterm Function Example • SYEN 3330 Digital Systems 27

Cannonical Sum of Minterms SYEN 3330 Digital Systems 28

Another SOM Example SYEN 3330 Digital Systems 29

Shorthand SOM Form Note that we explicitly show the standard variables in order and drop the “m” designators. SYEN 3330 Digital Systems 30

Canonical Product of Maxterms SYEN 3330 Digital Systems 31

Product of Maxterm Example SYEN 3330 Digital Systems 32

Function Complements Then: Or alternately: SYEN 3330 Digital Systems 33

Conversion Between Forms SYEN 3330 Digital Systems 34

Review of Canonical Forms SYEN 3330 Digital Systems 35

Review: Indices SYEN 3330 Digital Systems 36

Forms of Terms, Complements SYEN 3330 Digital Systems 37

Review: Sum of Minterms Form SYEN 3330 Digital Systems 38

Review: Product of Maxterms SYEN 3330 Digital Systems 39

Review: Complements, Conversions SYEN 3330 Digital Systems 40