Binary Logic/Boolean Algebra

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
• 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
• Example
– (1+0)=(1+0)
• 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
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
z=1 if and only if x=1 and y=1
A truth table
OR Operation
x OR y is equal to z
z=1 if x=1 or y=1
This is not binary
NOT Operation
Not x is equal to x’
x’ is what x is not
x’ performs the complement operation
Input-Output Signals for Gates