course Mth 151
6/14 12
Question: `q001. Note that there are 8 questions in this assignment.
The set { 1, 2, 3, ... } consists of the numbers 1, 2, 3, etc.. The etc. has no end. This set consists of the familiar counting numbers, which most of us have long known to be unending. This is one example of an infinite set. Another is the set of even positive numbers { 2, 4, 6, ... }. This set is also infinite. There is an obvious one-to-one correspondence between these sets. This correspondence could be written as [ 1 <--> 2, 2 <--> 4, 3 <--> 6, ... ], where the ... indicates as before that the pattern should be clear and that it continues forever.
Give a one-to-one correspondence between the sets { 1, 2, 3, ... } and the set { 1, 3, 5, ... } of odd numbers.
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
A one to one correspondence means that one number from one set will be paired with one number from the second set so the answer would be {1<->1, 2<->3, 3<->5…}
3
Okay
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Question: `q002. Writing [ 1 <--> 1, 2 <--> 3, 3 <--> 5, ... ] for the correspondence between { 1, 2, 3, ... } and { 1, 3, 5, ... } isn't bad, but the pattern here might be a bit less clear to the reader than the correspondence [ 1 <--> 2, 2 <--> 4, 3 <--> 6, ... ] given for { 1, 2, 3, ... } and { 2, 4, 6, ... }. That is because in the latter case it is clear that we are simply doubling the numbers in the first set to get the numbers in the second.
It might not be quite as clear exactly what the rule is in the correspondence [ 1 <--> 1, 2 <--> 3, 3 <--> 5, ... ], except that we know we are pairing the numbers in the two sets in order. Without explicitly stating the rule in a form as clear as the doubling rule, we can't be quite as sure that our rule really works.
How might we state the rule for the correspondence [ 1 <--> 1, 2 <--> 3, 3 <--> 5, ... ] as clearly as the 'double-the-first-number' rule for [ 1 <--> 2, 2 <--> 4, 3 <--> 6, ... ]?
Your solution:
The way I would state this is that you are adding the next counting number, for instance 1+0=1 2+1=3 3+2=5 the next in the sequence would be 4+3= 7 you are just adding the next number in 0,1,2,3,4…. to the first number and the answer Is the second number.
2
Should make it more clear by saying double the first number and subtract one
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Question: `q003. The 'double-the-number' rule for the correspondence [ 1 <--> 2, 2 <--> 4, 3 <--> 6, ... ] could be made even clearer.
First we let n stand for the nth number of the set {1, 2, 3, ... }, like 10 stands for the 10th number, 187 stands for the 187th number, so whatever it is and long as n is a counting number, n stands for the nth counting number.
Then we note that the correspondence always associates n with 2n, so the correspondence could be written0
[ 1 <--> 2, 2 <--> 4, 3 <--> 6, ... , n <--> 2n, ... ].
This tells us that whatever counting number n we choose, we will associate it with its double 2n. Since we know that any even number is a double of the counting number, of the form 2n, this rule also tells us what each even number is associated with. So we can argue very specifically that this rule is indeed a 1-to-1 correspondence.
In terms of n, how would we write the rule for the correspondence [ 1 <--> 1, 2 <--> 3, 3 <--> 5, ... ]?
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Your solution:
[1<->1, 2<->3, 3<->5, n<->2n-1…]
3
Okay
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Question: `q004. Write an unambiguous rule involving n for the correspondence between { 1, 2, 3, ... } and { 5, 10, 15, ... }.
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Your solution:
The rule would be to multiply the first number by 5 so you would state n<->5n
3
Ok
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Question: `q005. Write an unambiguous rule involving n for the correspondence between { 1, 2, 3, ... } and { 7, 12, 17, ... }.
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Your solution:
The first thing I notice is that the numbers in the second set go up by 5 but you can not simply you n<->5n because that is 5,10,15 so with that in mind I noticed that 5+2=7, 10+2=12, 15+2=12 so that would mean you add 2 to 5n in so your rule is n<->5n+2
3
ok
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