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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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