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 byClaude Shannon 70 years afterBoole’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

• Basic Operation:– Interchange an OR with an AND– Invert A– Invert B

• Example

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 Operationx AND y is equal to z

Interpretation:z=1 if and only if x=1 and y=1

A truth table

OR Operationx OR y is equal to z

Interpretation:z=1 if x=1 or y=1

This is not binaryaddition

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