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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. Synthesis of Logic Circuits (Boolean Algebra) Curriculum Connection Boolean Algebra 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 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) 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 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) 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) DeMorgan’s Theorem • 𝐴+𝐵 =𝐴∙𝐵 • 𝐴∙𝐵 =𝐴+𝐵 • Basic Operation: – Interchange an OR with an AND – Invert A – Invert B • Example –1+0=1∙0 Logic Gates Logic Gates • Logic gates are electronic circuits that operate on one or more input signals to produce signals Hierarchy of Digital Circuits (Packaged Gates) Curriculum Connection AND Operation x AND y is equal to z Interpretation: z=1 if and only if x=1 and y=1 A truth table OR Operation x OR y is equal to z Interpretation: z=1 if x=1 or y=1 This is not binary addition NOT Operation Not x is equal to x’ Interpretation: x’ is what x is not x’ performs the complement operation Input-Output Signals for Gates