University of Illinois
Fall 2009
ECE 313:
Problem Set 1: Solutions
Sets, Events, Axioms of Probability and Their Consequences
1.
[Subsets of a finite set]
(a) The subsets of Ω =
{
ω
1
, ω
2
, ω
3
, ω
4
}
are
Subsets of size 0:
∅
Subsets of size 1:
{
ω
1
}
,
{
ω
2
}
,
{
ω
3
}
,
{
ω
4
}
Subsets of size 2:
{
ω
1
, ω
2
}
,
{
ω
1
, ω
3
}
,
{
ω
1
, ω
4
}
,
{
ω
2
, ω
3
}
,
{
ω
2
, ω
4
}
,
{
ω
3
, ω
4
}
Subsets of size 3:
{
ω
1
, ω
2
, ω
3
}
,
{
ω
1
, ω
2
, ω
4
}
,
{
ω
1
, ω
3
, ω
4
}
,
{
ω
2
, ω
3
, ω
4
}
Subsets of size 4:
{
ω
1
, ω
2
, ω
3
, ω
4
}
= Ω
There are 16 = 2
4
subsets of the set Ω of 4 elements. 15 = 2
4

1 of them are
nonempty
subsets.
(b) OK, OK, geez, some people are never satisfied . . .
(c) By checking our answers in parts (a) and (b), we see that there is: 1 subset of size 0 and 1 subset
of size 4

0 = 4, 4 subsets of size 1 and 4 subsets of size 4

1 = 3, and lastly 6 subsets of size 2
with 4

2 = 2. For general
n
, whenever we choose a subset of size
k
, there is a unique subset of
size
n

k
that is left out. That is, for every subset of size
k
, there is exactly one corresponding,
complementary subset of size
n

k
, so the total number of subsets of size
k
is the same as the
total number of subsets of size
n

k
.
(d)
i. The vector
{
1
,
1
, . . . ,
1
}
corresponds to Ω.
The vector
{
0
,
0
, . . . ,
0
}
corresponds to
∅
.
The
n
bit vector with each bit flipped with respect to
A
corresponds to
A
c
.
ii. Since
A
∪
B
contains exactly those elements in either
A
or
B
,
z
i
=
x
i
∨
y
i
. Similarly, since
A
∩
B
contains exactly those elements in both
A
and
B
,
w
i
=
x
i
∧
y
i
. In purely arithmetical
terms,
z
i
=
x
i
+
y
i

x
i
y
i
and
w
i
=
x
i
y
i
.