1
ECE 15A
Fundamentals of Logic Design
Lecture 5
Malgorzata MarekSadowska
Electrical and Computer Engineering Department
UCSB
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Canonical representations of Boolean
functions
Canonical forms are unique representations of Boolean functions.
dnf and cnf are canonical representations of Boolean functions
To convert between dnf and cnf, we interchange the sum and
product symbols and list those numbers which were absent in the
original form.
)
,
,
(
z
y
x
F
(m3,m5,m6,m7)
)
,
,
(
z
y
x
F
(M0,M1,M2,M4)
Example:
3
Boolean functions in standard forms
Sumofproducts (SOP)
Boolean expression that contains AND terms (called
products or implicants) which are ORed together
Example:
F1(x,y,z,w)=xy+z’+x’w’w’
Product of sums (POS)
Boolean expression that contains OR terms (called sum
terms or implicants) which are ANDed together
Example: F2=(x+z’+w)(x’+y+w’)
These normal forms are not unique: the same
functions can be expressed in many different ways
Standard forms usually contain fewer terms and
literals than the corresponding dnfs or cnfs.
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XOR operation
0
1
0
1
1
1
0
A
B
= B
A
(commutative)
(A
B)
C
= A
(B
C)
(associative)
(AB)
(AC)
= A (B
C)
multiplication distributive over
L = (A
B)(A
C)
=
A
(BC)= R
A
B
C
A
B
A
C
L
BC
R
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
0
1
0
1
0
0
0
0
0
1
1
1
1
1
1
1
1
0
0
1
1
1
0
1
1
0
1
1
0
0
0
1
1
1
0
0
1
0
0
1
1
1
1
0
0
0
1
0
A
C
= B
B
C
= A
A
B
C
=
0
If
A
B = C, then
?
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Other logic operations
NOR
f(a,b)
= (a+b)’
NAND
f (a,b) = (ab)’
XNOR (Equivalence)
f (a,b)= ab + a’b’
NOR
0
1
0
1
0
1
0
0
NAND
0
1
0
1
1
1
1
0
XNOR
0
1
0
1
0
1
0
1
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Functionally complete operations
A set of operations is functionally complete
(universal) if and only if every Boolean
function can be expressed entirely by means
of operations from this set.
We have shown that every function can be
expressed as cnf and dnf, so (AND,OR,NOT)
is functionally complete.
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 Spring '08
 M
 Boolean Algebra, Logic gate, Boolean expression

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