MATH 74 HOMEWORK 4 SOLUTIONS
1.
How many elements does each of the following sets have?
(a)
{
1
,
2
,
3
,
4
}
Four elements: 1, 2, 3, and 4
(b)
{{
1
,
2
}
,
{
3
,
4
}}
Two elements:
{
1
,
2
}
and
{
3
,
4
}
(c)
{
1
,
{
2
,
3
}
,
{
4
,
{
5
,
6
}}}
Three elements: 1,
{
2
,
3
}
,
{
4
,
{
5
,
6
}}
(e)
{
n
2
:
n
∈
N
and 1
≤
n
≤
9
}
Nine elements: 1, 4, 9, 16, 25, 36, 49, 64, and 81
(f)
{
n
2

n
:
n
∈
N
and 1
≤
n
≤
9
}
Nine elements: 0, 2, 6, 12, 20, 30, 42, 56, and 72
(g)
{
n
2

10
n
+ 9 :
n
∈
N
and 1
≤
n
≤
9
}
Five elements: 0,

7,

12,

15, and

16
Remark.
The point of this exercise was twofold:
•
The fact that the elements of a set
A
can themselves be sets does not enter
into a count of the elements of
A
•
The fact that if
f
is some function, the set
{
f
(
n
) : 1
≤
n
≤
9
}
does not
necessarily have nine elements. It will have fewer than nine elements if
f
assigns the same value to diﬀerent numbers between
1
and
9
.
This is an important point. The mere statement
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 Fall '07
 COURTNEY
 Sets, #, Empty set, other applications

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