ECEN 303: Assignment 1
Problems:
1. Following the argument presented in the notes, prove that
parenleftBigg
intersectiondisplay
α
∈
A
S
α
parenrightBigg
c
=
uniondisplay
α
∈
A
S
c
α
.
Suppose that
x
belongs to
(intersectiontext
α
∈
I
S
α
)
c
. That is, there exists an
α
′
∈
I
such that
x
is not an
element of
S
α
′
. This implies that
x
belongs to
S
c
α
′
, and therefore
x
∈
uniontext
α
∈
I
S
c
α
. Thus, we
have shown that
(intersectiontext
α
∈
I
S
α
)
c
⊂
uniontext
α
∈
I
S
c
α
. The converse is obtained by reversing the argument.
Suppose that
x
belongs to
uniontext
α
∈
I
S
c
α
. Then, there exists an
α
′
∈
I
such that
x
∈
S
c
α
′
. This
implies that
x /
∈
S
α
′
and, as such,
x /
∈
(intersectiontext
α
∈
I
S
α
)
. Alternatively, we have
x
∈
(intersectiontext
α
∈
I
S
α
)
c
.
Putting these two inclusions together, we get the desired result.
2. John, Jim, Jay, and Jack have formed a band consisting of 4 instruments. If each of the boys
can play all 4 instruments, how many different arrangements are possible? What if John and
Jim can play all 4 instruments, but Jay and Jack can each play only piano and drums?
There are 4! different arrangements when each of the boys can play all 4 instruments. On
there other hand, there are only 4 possible arrangements in the second scenario.
3. For years, telephone area codes in the United States and Canada consisted of a sequence of
three digits. The first digit was an integer between 2 and 9; the second digit was either 0 or
1; the third digit was any integer between 1 and 9. How many area codes were possible? How
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 Fall '07
 Chamberlain
 Long integer, Sα, possible arrangements, α∈I Sα

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