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CSE140:
Components and Design Techniques
for Digital Systems
Boolean algebra & Combinational logic
Tajana Simunic Rosing
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Overview
• Reminder:
HW#1 is due today!
• What we covered thus far:
– Number representations
– Switches
–MOS
t
r
a
n
s
i
s
t
o
r
s
– Logic gates
• What is next:
– Boolean algebra
– Combinatorial logic:
• Representations
• Minimization
• Implementations
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Review: CMOS Example
• Remember the rules:
– NMOS connects to GND, PMOS to power supply Vdd
– Duality of NMOS and PMOS
– Rp ~ 3 Rn =>
PMOS in series is much slower than NMOS in series
• Implement Z using CMOS: Z = (A + BC)’
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Another CMOS Example
• Implement F using CMOS:
F=A*(B+C)
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Boolean Algebra and its Relation to Digital Circuits
• An algebraic structure consists of
– a set of elements B
– binary operations { + , • } (OR and AND)
– and a unary operation { ’ } (NOT)
– such that the following axioms hold:
1. the set B contains at least two elements: a, b
2. closure:
a + b
is in B
a • b
is in B
3. commutativity:
a + b = b + a
a • b = b • a
4. associativity: a + (b + c) = (a + b) + c
a • (b • c) = (a • b) • c
5. identity:
a + 0 = aa • 1 = a
6. distributivity: a + (b • c) = (a + b) • (a + c)
a • (b + c) = (a • b) +
(a • c)
7. complementarity:
a + a’ = 1
a • a’ = 0
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Axioms and theorems of Boolean algebra
•
identity
1.
X + 0 = X
1D.
X • 1 = X
•
null
2.
X + 1 = 1
2D.
X • 0 = 0
•
idempotency:
3.
X + X = X
3D.
X • X = X
•
involution:
4.
(X’)’ = X
•
complementarity:
5.
X + X’ = 1
5D.
X • X’ = 0
•
commutativity:
6.
X + Y = Y + X
6D.
X • Y = Y • X
•
associativity:
7.
(X + Y) + Z = X + (Y + Z)
7D.
(X • Y) • Z = X • (Y • Z)
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Axioms and theorems of Boolean algebra (cont’d)
•
distributivity:
8.
X • (Y + Z) = (X • Y) + (X • Z)
8D.
X + (Y • Z) = (X + Y) • (X + Z)
•
uniting:
9.
X • Y + X • Y’ = X
9D.
(X + Y) • (X + Y’) = X
•
absorption:
10. X + X • Y = X
10D.
X • (X + Y) = X
11. (X + Y’) • Y = X • Y
11D. (X • Y’) + Y = X + Y
•
factoring:
12. (X + Y) • (X’ + Z) =
12D. X • Y + X’ • Z =
X • Z + X’ • Y
(X + Z) • (X’ + Y)
•
concensus:
13. (X • Y) + (Y • Z) + (X’ • Z) =
13D. (X + Y) • (Y + Z) • (X’ + Z) =
X • Y + X’ • Z
(X + Y) • (X’ + Z)
•
de Morgan’s:
14. (X + Y + .
..)’ = X’ • Y’ • .
..
14D. (X • Y • .
..)’ = X’ + Y’ +…
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This note was uploaded on 02/14/2008 for the course CSE 140 taught by Professor Rosing during the Fall '06 term at UCSD.
 Fall '06
 Rosing

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