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Math 110: One-to-One Pairings and Cardinality. Name:
Mindscapes from Section 3.2
6. Even odds. Let E stand for the set of all even natural numbers (so E = {2, 4, 6, 8, . . . }) and O stand for the set of all
odd natural numbers (so O = {1, 3, 5, 7, . . . }). Show that the sets E and O have the same cardinality by describing an
explicit one-to-one correspondence between the two sets. Give a formula for this correspondence. What odd integer
is paired with the arbitrary positive even integer e?
E
2
4
6
8
10
12
14
16
18
20
22
24
An even natural number e
Formula: e ↔
O
11. Missing 3. Let T IM be the set of all natural numbers except the number 3 (T IM stands for “three is missing”), so
T IM = {1, 2, 4, 5, 6, 7, 8, 9, . . . }. Show that the set T IM and the set of all natural numbers have the same cardinality
by describing an explicit one-to-one correspondence between the two sets. Your formula may have two parts. One
part for the integers before 3, and another for the rest.
N
1
2
T IM
3
4
5
6
7
8
9
10
11
12
General Formula: n ↔ ?
Describe it below.
Your Two-Part Formula:
1. Show that the Graduation Goals at HWS (describe each in a phrase—look them up on the HWS website) are finite.
(Remember what finite means.)
2. Every student at a certain college is assigned to an academic advisor. Does this imply that there is a one-to-one
correspondence between advisors and students? Explain your answer.
3. Suppose your friend makes the following claim: “The cardinality of N and the cardinality of the set of all possible
infinitely long sequences of 0’s and 1’s are the same!” You respond by saying, “Well that means there is a one-to-one
correspondence between N and these sequences. So you should be able to make a list (first, second, third,. . . ) of
ALL of the sequences.” You friend says, “Yup, and here’s my list.” Your friend’s list starts as follows.
N
Friend’s list of sequences
1
1
0
0
1
1
1
0
...
2
1
1
0
1
1
1
1
...
3
0
0
0
0
0
1
0
...
4
0
0
0
1
0
1
0
...
5
1
0
0
1
0
0
0
...
6
1
0
0
1
1
1
0
...
7
..
.
0
..
.
0
..
.
1
..
.
0
..
.
1
..
.
1
..
.
1
..
.
...
..
.
a) Give the first 7 digits of the sequence that you know is not on the list.
...
b) Carefully describe the process you used to create your ‘missing’ sequence and why that process works.
ZRemember to do the other Mindscapes listed at the website.