ECE 3110: Introduction to Digital Systems

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ECE 3110 Introduction to Digital Systems

• Combinational Logic Design Principles

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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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Axioms (postulates)

• A1) X0 if X1 A1 ) X1 if X0
• A2) if X0, then X1 A2 ) if X1, then X0
• A3) 0 00 A3 ) 111
• A4) 1 11 A4 ) 000
• A5) 0 1 1 0 0 A5 ) 10011

Logic multiplication and addition precedence
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Theorems (Single variable)

• Proofs by perfect induction

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Two- and three- variable Theorems
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N-variable Theorems

• Prove using finite induction
• Most important DeMorgan theorems

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Finite induction

• Step1 Proving the theorem is true for n2
• Step 2 Proving that if the theorem is true for
  ni, then it is also true for ni1
• Thus the theorem is true for all finite values of
  n.
• For example T12

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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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Duality

• CounterexampleX 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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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,?,
  , )

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Next

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