ECE 3110: Introduction to Digital Systems

1. ECE 3110: Introduction to Digital Systems Combinational Logic Design Principles

2. Previous… • Variables, expressions, equations • Axioms (A1-A5 pairs) • Theorems (T1-T11 pairs) • Single variable • 2- or 3- variable • Prime, complement, logic multiplication/addition, precedence

3. Axioms (postulates) • A1) X=0 if X‡1 A1’ ) X=1 if X‡0 • A2) if X=0, then X’=1A2’ ) if X=1, then X’=0 • A3) 0 • 0=0 A3’ ) 1+1=1 • A4) 1 • 1=1 A4’ ) 0+0=0 • A5) 0 • 1= 1 • 0 =0 A5’ ) 1+0=0+1=1 Logic multiplication and addition precedence

4. Theorems (Single variable) • Proofs by perfect induction

5. Two- and three- variable Theorems

6. Duality • Swap 0 & 1, AND & OR • Result: Theorems still true • Principle of Duality • Any theorem or identity in switching algebra remains true if 0 and 1 are swapped and • and + are swapped throughout. • Why? • Each axiom (A1-A5) has a dual (A1¢-A5¢)

7. Duality • Counterexample:X + X × Y = X (T9)X × X + Y = X (dual)X + Y = X (T3¢)???????????? X + (X×Y) = X (T9)X× (X + Y) = X (dual)(X× X) + (X× Y) = X (T8)X+ (X× Y) = X (T3¢) parentheses,operator precedence!

8. Dual of a logic expression • If F(X1, X2, X3,… Xn,, +, ‘) is a fully parenthesized logic expression involving variables X1, X2, X3,… Xn and the operators +,, and ‘, then the dual of F, written FD, is the same expression with + and  swapped. • FD(X1, X2, X3,… Xn, +,, ‘)=F(X1, X2, X3,… Xn,, +, ‘)

9. N-variable Theorems • Prove using finite induction • Most important: DeMorgan theorems

10. Finite induction • Step1: Proving the theorem is true for n=2; • Step 2: Proving that if the theorem is true for n=i, then it is also true for n=i+1; • Thus the theorem is true for all finite values of n. • For example: T12

11. Next… • DeMorgan Symbols • Representations of logic functions • Read Chapter 4.2 and take notes • Combinational circuit analysis