1
ECE 15A
Fundamentals of Logic Design
Lecture 4
Malgorzata Marek-Sadowska
Electrical and Computer Engineering Department
UCSB
2
Today : Boolean functions
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:
F1(a,b,c,d,x)=(a+b)c’ + (a+b’x)d
F2(a,b,c,d)= ab’c+a’d
⋅
3
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
)
(
i
i
i
x
f
∏
=
∀
i
i
i
i
x
x
x
f
i
'
)
(
4
Theorem 1a
In a Boolean algebra, every function which contains no
constants can be represented in a disjunctive normal
form.
Proof. Suppose that f(x1,x2,…,xn) is not in dnf.
If it contains expression (g+h)’ or (gh)’ for some functions
g,h, then they can be written as g’h’ and g’+h’. This process
may be continued until each (’) applies to a single variable xi.
Next, we can apply distributive law of ( ) over (+) and express
f as a polynomial.
Each term tj with a missing variable xi can be expressed as
tj(xi+xi’). Using the laws of Boolean algebra, we can eliminate
duplicate terms.
⋅
5
Example
f(a,b,c)=(ab’+ac)’+b’
=(ab’)’(ac)’+b’=(a’+b)(a’+c’)+b’=
a’+a’b+a’c’+bc’+b’=
a’(b+b’)(c+c’)+
a’b(c+c’)
+
a’(b+b’)c’
+(a+a’)bc’+(a+
a’)b’(c+c’)=
a’bc+a’bc’+a’b’c+a’b’c’+abc’+
a’bc’
+ab’c+ab’c’+
a’b’c
+a’b’c’
= a’bc+a’bc’+a’b’c+a’b’c’+abc’+ab’c+ab’c’
6
Disjunctive normal form
Unique representation of a function in a
given number variables
Example:
f(a,b) = a’b
f(a,b,c) = a’b = a’b(c’+c) = a’bc+a’bc’
We will assume that disjunctive normal form
refers to a dnf which contains the fewest
possible number of variables.