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Binary Logic

Binary Logic. Section 1.9. Binary Logic. Binary logic deals with variables that take on discrete values (e.g. 1, 0) and with operations that assume logical meaning (e.g. AND, OR and NOT). Home Alarm Logic. W1, W2, P and D are variables which can take on discrete values.

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Binary Logic

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  1. Binary Logic Section 1.9

  2. Binary Logic • Binary logic deals with variables that take on discrete values (e.g. 1, 0) and with operations that assume logical meaning (e.g. AND, OR and NOT)

  3. Home Alarm Logic W1, W2, P and D are variables which can take on discrete values.

  4. Synthesis of Logic Circuits (Boolean Algebra)

  5. Curriculum Connection

  6. Boolean Algebra

  7. George Boole • An English Mathematician • An inventor of Boolean Logic • Boolean logic=Basis of computer logic • His work was re-discovered by Claude Shannon 70 years after Boole’s death

  8. Associative Law • A+(B+C)=(A+B)+C • (A ∙ B) ∙ C=A ∙(B∙C) • Interpretation: we can group the variables in AND or OR any way we want • Example: • 1+(1+0)=(1+1)+0 • (1∙ 0)0=1(1∙0)

  9. Distributive Law • X ∙(Y+Z)=X ∙ Y+X ∙ Z • (W+X)(Y+Z)=W ∙ Y+X ∙ Y+W ∙ Z+X ∙ Z • In Plain English: An expression can be expanded by multiplying term by term just as in ordinary algebra • Example: • 1 ∙(1+0)=1 ∙ 1+1 ∙ 0

  10. Commutative Laws • X+Y=Y+X • X ∙ Y=Y ∙ X • In Plain English: The order in which we OR or AND two variables are not important • Example • (1+0)=(1+0)

  11. Duality • If the dual of an algebraic expression is desired, we simply • Interchange OR and AND • Interchange 1 and 0 • Example • A+(B+C)=(A+B)+C • (A ∙ B) ∙ C=A ∙(B∙C)

  12. DeMorgan’s Theorem • Basic Operation: • Interchange an OR with an AND • Invert A • Invert B • Example

  13. Logic Gates

  14. Logic Gates • Logic gates are electronic circuits that operate on one or more input signals to produce signals

  15. Hierarchy of Digital Circuits (Packaged Gates)

  16. Curriculum Connection

  17. AND Operation x AND y is equal to z Interpretation: z=1 if and only if x=1 and y=1 A truth table

  18. OR Operation x OR y is equal to z Interpretation: z=1 if x=1 or y=1 This is not binary addition

  19. NOT Operation Not x is equal to x’ Interpretation: x’ is what x is not x’ performs the complement operation

  20. Input-Output Signals for Gates

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