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ECE 15A
Fundamentals of Logic Design
Lecture 4
Malgorzata MarekSadowska
Electrical and Computer Engineering Department
UCSB
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Last time :
We have proved all
fundamental laws stated for sets.
Theorem 1
: Every statement or algebraic identity deducible from P1
P4 remains valid if the operations (+) and ( ), and the identity
elements 0 and 1 are interchanged throughout.
Theorem 2:
For every element a in Boolean algebra B: a+a =a
and aa
= a.
Theorem 3
: For each element a in Boolean algebra B,
a + 1 = 1
and
a 0 = 0.
Theorem 4
. For each pair of elements a, b in a Boolean algebra B,
a +
ab = a
and
a(a + b) = a.
Theorem 5
. In every Boolean algebra B each of the binary operations
(+) and ()is associative. That is, for every a, b and c in B:
a + (b + c) = (a + b) + c
and
a(bc) = (ab)c
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Last time: cont.
Theorem 6.
The element a’ associated with element a
in a Boolean algebra is unique.
Theorem 7.
For every a in a Boolean algebra B,
(a’)’=a.
Theorem 8
. In any Boolean algebra, 0’ = 1 and 1’ = 0.
Theorem 9.
For every a and b in Boolean algebra B,
(ab)’ = a’ + b’
and
(a+b)’ = a’b’ [De Morgan’s laws]
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Example: 2valued Boolean algebra
Boolean algebra on B={0,1} and operators {+,
,’}
From theorem 2 [a+a=a, aa=a]:
0+0 = 0
1+1 = 1
0
0 = 0
1
1 = 1
From Theorem 3
[a+1=1, a0=0]:
0+1=1
0
0 = 0
1+1 = 1
1
0 = 0
From Theorem 8:
0’ = 1
1’ = 0
+
0
1
0
0
1
1
1
1
0
0
0
0
1
1
0
1
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Today: Boolean functions
B={0,1}
B
is a set of all tuples (b1, b2,…
bk), such that
bi is either 0 or 1 for all 1<=i<=k.
B
may be viewed as the set of all kbit strings.
Example: k = 3. B
= {(0,0,0), (0,0,1), (0,1,0),
(0,1,1),(1,0,0),(1,0,1),(1,1,0),(1,1,1)}.
B
has a size of 2
.
k
k
3
k
k
6
Boolean function
Boolean function
f: B
> B
, k>0, m>0.
kbit input is mapped to an mbit output
k
m
B
B
k
m
mapping
Domain
Codomain
An element in a domain is assigned one
element in the codomain.
m=1: singleoutput function.
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Boolean functions – algebraic form
Constant – a symbol which represents a specified element of
Boolean algebra
Variable: a symbol a,b,c,…etc. representing unspecified
elements
Literal  a variable with specified polarity: a,a’,b, b’,…etc.
Boolean function
: any expression which represents the
combination of finite set of symbols, each representing a
constant or variable, by the operations of (+), ( ), or (‘).
Examples of singleoutput functions:
¾
F1(a,b,c,d,x)=(a+b)c’ + (a+b’x)d
¾
F2(a,b,c,d)= ab’c+a’d
⋅
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Disjunctive normal form
Definition. A Boolean function is in disjunctive
normal form in n variables x1,x2,…,xn, for n>0, if
The function is a sum of terms
where
No two terms are identical
0 and 1 are in disjunctive normal form in n variables for any
n>0.
Example: f(a,b,c,d) = ab’cd+a’bc’d+abcd’
The terms of DNF are called minterms
)
(
1
i
n
i
i
x
f
∏
=
⎩
⎨
⎧
=
∀
i
i
i
i
x
x
x
f
i
'
)
(
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Example
f(a,b,c,d) = ab’cd+a’bc’d+abcd’
f(0,0,1,0)= 0 1
1 0 + 1 0 0 0 + 0 0 1 1 =
0+0+0=0
f(1,0,1,1)= 1
1
1
1 + 0 0 0 1 + 1 0 1 0 = 1+0+0=1
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 Winter '08
 M
 Boolean Algebra, Boolean function, Conjunctive normal form, Normal Form

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