# 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.

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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

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

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