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ECE 2110: Introduction to Digital Systems Combinational Logic Design Principles

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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

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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

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4 Theorems (Single variable) Proofs by perfect induction

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5 Two- and three- variable Theorems

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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

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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!

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8 Dual of a logic expression If F(X 1, X 2, X 3,… Xn, , +, ‘) is a fully parenthesized logic expression involving variables X 1, X 2, X 3,… Xn and the operators +, , and ‘, then the dual of F, written F D, is the same expression with + and swapped. F D (X 1, X 2, X 3,… Xn, +, , ‘)=F(X 1, X 2, X 3,… Xn, , +, ‘)

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9 N-variable Theorems Most important: DeMorgan theorems

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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

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

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Dale Roberts Department of Computer and Information Science, School of Science, IUPUI Dale Roberts, Lecturer Computer Science, IUPUI

Dale Roberts Department of Computer and Information Science, School of Science, IUPUI Dale Roberts, Lecturer Computer Science, IUPUI

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