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Introduction to Combinational Logic Design: Axioms, Theorems, and Duality Principles

This course provides an in-depth analysis of combinational logic design principles, covering essential axioms (A1-A5) and theorems (T1-T11) important for understanding digital systems. Key topics include single, two, and three-variable theorems, proof techniques such as perfect induction, and the principle of duality. Students will learn how to apply DeMorgan's theorems and recognize representations of logic functions. An essential resource for anyone seeking expertise in digital logic design and combinational circuits.

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Introduction to Combinational Logic Design: Axioms, Theorems, and Duality Principles

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  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

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