Propositional Logic: Representations of boolean
functions and complete sets of operators
Periklis A. Papakonstantinou
?
University of Toronto
We begin by introducing the concept of a boolean function. This concept has a very intuitive correspondence to
the semantics of a propositional formula. Then, we discuss in what sense we represent boolean functions through
propositional formulas. We conclude by getting into some of the details of the issue of complete sets of operators.
Boolean functions
We will say that a function is
boolean
if it maps true/false tuples to a true/false value. For
several reasons it helps to think of “true” as “1” and of “false” as “0”. For example, digital
circuits perform logical operations on 0 and 1 in exactly the same way we perform logical
operations in propositional Logic. For instance, 1
∧
1 = 1, 0
∧
1 = 0. From now on we blur
the distinction between 0 and “false” and 1 and “true”.
In general, for a function
f
we write
f
:
D
→
R
, to denote that
D
is the domain of
f
and
R
is the range of
f
. Clearly,
f
:
d
→
R
just gives some but not all of the description
of the function. For example, we write that
f
:
R
→
R
for the function
f
which maps
real numbers to real numbers as follows:
f
(
x
) =
x
2
+ 1. Functions that map
R
→
R
have
infinitely many points (the whole set
R
) we must specify in order to describe them. In this
handout we will restrict our attention to some special functions whose full description is
finite (i.e. we can write it down on a piece of paper  as opposed to functions with infinite
domains). Functions in our current interest have domains of constant size. For example, a
function like
f
:
{
0
,
1
}
2
→ {
0
,
1
}
has domain
{
0
,
1
}
2
=
{
(0
,
0)
,
(0
,
1)
,
(1
,
0)
,
(1
,
1)
}
. Thus it
suffices to specify the function on these four values to have its complete description. Here
is the description of such a function
f
:
x
y
f
(
x, y
)
0
0
0
0
1
1
1
0
1
1
1
0
Remark 1.
{
0
,
1
}
2
is the cartesian product
{
0
,
1
} × {
0
,
1
}
. For example, (0
,
1)
∈ {
0
,
1
}
2
,
(1
,
1)
∈ {
0
,
1
}
2
etc. Similarly, for a constant
k
∈
N
we have that
{
0
,
1
}
k
=
{
0
,
1
} × {
0
,
1
} ×
. . .
{
0
,
1
}

{z
}
k
times
.
?
Please submit any typos, remarks, suggestions to
[email protected]
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
2
Periklis Papakonstantinou (ECE 190, Fall 2006)
Remark 2.
The above function
f
corresponds to the addition in the binary system (note
that 1 + 1 = 10 but the value of the function ignores the carry). Also, this function is
sometimes called as
XOR
or as
parity
.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '06
 Carter
 Logic, Boolean Algebra, Boolean function, Logical connective, Propositional calculus

Click to edit the document details