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Binary Logic Section 1.9

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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)

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

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Synthesis of Logic Circuits (Boolean Algebra)

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Curriculum Connection

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Boolean Algebra

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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

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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)

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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

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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)

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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)

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DeMorgan’s Theorem

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Logic Gates

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Logic gates are electronic circuits that operate on one or more input signals to produce signals

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Hierarchy of Digital Circuits (Packaged Gates)

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Curriculum Connection

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

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

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

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Input-Output Signals for Gates

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