Comparing Infinite Sets
By
Christina Grossman
A Thesis Submitted to the Faculty of the
DEPARTMENT OF MATHEMATICS
In Partial Fulfillment of the Requirements
For the Degree of
MASTER OF MIDDLE SCHOOL MATHEMATICS LEADERSHIP
In the Graduate College
THE UNIVERSITY OF ARIZONA
2010
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In thinking about infinite sets, one might not associate the process of comparing infinite sets because….do they really have a size? Finite sets have a size associated with them; however, how can one size something that is considered infinite? By defining what exactly is an infinite set and looking at the possibility of different sizes of infinite sets when determined by one-to-one and onto functions, the result will reveal the relationship between the infinite sets of N , Z , Q , and R .
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Before exploring infinite sets, think about how numbers are used in everyday life. As children, we are taught how to use numbers very early in our learning. We began our relationship with numbers by counting our toes, fingers, apples, and oranges. These are finite; they can be counted. Counting these items usually began with using numbers 1, 2, 3, … etc.
Everything that we know of the natural world can be counted this way. In turn, this set of numbers that we use is called the set of natural numbers ( N ). Because we use the natural numbers to count objects or other things, they are usually known as the counting numbers.
Finite Sets
We are able to count things as children using the natural numbers and we also learn about how to find the size of groups or in this case sets of things. When looking at a group of people, you count the number of people in that group, thus calculating the size of the group. If wanting to know the size of a set of numbers, you need to be able to count how many numbers, or elements are in the set. From what we know at such a young age, we use the natural numbers as tools to count the elements of something that is finite. The word finite is used to describe a set or a group of elements or objects that can be counted and where the counting stops eventually. In general, sets do not always include numbers. For the sake of familiarity, number sets will be used. For example, if there are eight elements in a set of numbers it might be {1, 2, 3, 4, 5, 6, 7,
8}. Looking further at this set, we can say that 1 is the first element of this set of numbers and 6 is the sixth element of this set of numbers. If there are ten elements in a set of numbers it might be {2, 4, 6, 8, 10, 12, 14, 16, 18, 20} where 14 is the seventh element in this set. Because these particular two sets of numbers have a definite number of elements that we can count, we call these finite sets. With finite sets, the elements can be counted by use of the natural numbers.
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Since there are elements that can be counted, finite sets can now be sized. In mathematics, instead of size, the word cardinality is used. Thus, if a finite set has k elements, the finite set is of cardinality k.
Being able to know the set is finite and its cardinality can be used to compare it to other finite sets. In looking at the two given sets below, there are a few things that can describe the two of them: A = {5, 6, 7, 8} and B = {1, 2, 3, 4}. Set A and B are finite sets. Set A is of cardinality 4. Set B is also of cardinality 4. Look at how each element of the two sets are arranged below.
5 1 6 2 7 3 8 4 Because every element in set B has an element in set A assigned to it, this is called a relation that is onto. Each element in set A does not go to more than one element in set B. This is relation is one-to-one because no two elements of set A go to the same element in B. Just like people are different from one another, so are relations between different sets.
Take the set C = {1, 2, 3} and set D = {3, 6, 9, 12, 15}. Both sets are finite, but their cardinality is different. Set C has a cardinality of 3 as opposed to set D has a cardinality of 5. In this case again, cardinality is used to describe finite sets of numbers. Look at how this relation compares between these two sets.
1 3 6
4
2 9 3 12 15 Each element in set C corresponds to an element in set D and no two elements in set C go to the same element in set D. This relation is called one-to-one. The difference in this case is that not all the elements in set D correspond to an element in set C. This relation is not onto. Another relation between finite sets is depicted in figure 1.
NOT "One-to-One", but onto
Figure 1
http://www.regentsprep.org/Regents/math/algtrig/ATP5/OntoFunctions.htm
In analyzing finite sets, they can be compared to each other by way of cardinality. For example, if two sets have the same cardinality, it is that there is a relation between them that is both one-to-one and onto. The relationship between the status of cardinality and a one to one and onto relation is that for two sets to have the same cardinality there must exist a one-to-one and onto relationship between them. Focusing on cardinality is fairly easy when dealing with finite sets. The interesting aspect comes to play when dealing with varying infinite sets. In
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review, there are key features when comparing finite sets and now infinite sets: (a) the counting elements must be in numeric/alphabetic order; (b) the particular set has to be in one-to-one correspondence with the set being counted; and (c) the cardinality must be the same.
Infinite Sets
A finite set has a first element, second element, until it reaches its kth element. It did not keep going forever on the number of elements in the finite set. In an infinite set, you still have the first element, second element, and so on. However there is no last element because the infinite set will contain a kth element and there will always be a (k+1)th element because there is no limit to the number of elements in an infinite set. When first looking at infinite sets, that was all that I understood about them. In contrast, infinite sets are more than a set of numbers that continue infinitely, they can also either be countable or uncountable. It is possible to figure how a given infinite set is countable or not regardless of whether the number of elements is infinite.
This traces back to the interesting characteristics of the set of natural numbers ( N ).
The set of natural numbers ( N ) is a set that can be written as {1, 2, 3, 4, 5, 6, 7, 8,
9…n…}. Because this set has an infinite number of elements, it is called an infinite set. By definition, if an infinite set behaves this way, the infinite set is a denumerable set. “Both finite sets and denumerable sets are countable sets because we can count, or enumerate the elements in the set (Plummer, 2009).” By way of counting an infinite set, it does not mean that you can sit there and be able to give the total number of elements in one sitting. It just refers to the fact that in an infinite set, you can choose a first element, then a second element, and so on, and that you can eventually get to every element of the infinite set by counting. What is special about a denumerable set is that is has “…a one-to-one correspondence between the elements in the 6
infinite set and the infinite set of natural numbers ( N ) (Plummer, 2009). “ The infinite set of natural numbers ( N ) is considered a denumerable set, countable. The set of natural numbers
( N ) is used as a reference set for any denumerable set, just as the set {1,2,3} is a reference set for any set of cardinality 3. That is what is so fascinating about set N . In relation to using the infinite set N to describe the cardinality of finite sets, it is also used to describe the cardinality of infinite sets. With finite sets, if two finite sets were of the same cardinality there was a possibility of the relation to be one-to-one and also onto. In the same respect, if the infinite set has a one-to-one and onto correspondence with the set N , then it is considered countable and has the same cardinality as N .
To be countable, the infinite set must have a defined cardinality as discussed with finite sets. One of the questions of researching this area was whether or not there are infinite sets that have different sizes when size is determined by one to one and onto relations. The infinite sets in question are the set of integers ( Z ), the set of rational numbers ( Q ), and the set of real numbers
( R ). To visualize how these infinite sets relate to each other, refer to figure 2.
Irrationals
Rationals
Integers Natural Numbers
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Figure 2: A diagram depicting the set of real numbers.
At first glance, it seems to be that the natural numbers are included in set Z , Q , and R . From what I learned in grade school, if bag T has blue marbles and bag V contains bag T and has some additional red marbles, then it seems quite obvious that bag V has more marbles. This is true of what is known with a quantity that is finite. Transferring that rule to infinite sets seems to be a natural no nonsense thing to do. When looking at figure 2, it seems that the infinite set N is included in Z , Q , and R . Therefore, it would make sense to develop an argument that states Z , Q , and R are bigger in size compared to N . To find this out, the next portion is setting up a relation between N and Z
N and Z
n (1)n N Z is one-to-one and ontoiff : f ()nif n is even and if n is odd . 2 2
n Let fn() if n is even with n as an element of N . Then there is a table of values that can be 2 generated as follows:
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n f ()n
2 1
4 2
6 3
8 4
10 5
… …
With this relation between n and f ()n , there exists a correspondence that is one-to-one and onto. The set of numbers in column n are from the infinite set N . The numbers in column f ()n are from the infinite set of positive integers, Z . This satisfies the condition of establishing a one-to-one and onto correspondence; however, this does not include the infinite set of negative integers Z to include the whole set of integers. There needs to be another defined function that will provide a one-to-one and onto correspondence.
(1)n Let fn() if n is odd when n is an element of N . Then there is a table of values 2 that can be generated as follows:
n f ()n
1 0
3 ‐1
5 ‐2
7 ‐3
9
9 ‐4
… …
With this relation between n and f ()n , there exists a correspondence that is one-to-one and onto. The column containing the values of n is from the infinite set N . The numbers in column f ()n are from the infinite set of negative integers, Z . Therefore including the whole set of integers, Z and Z , that both established the correspondence of one-to-one and onto can then have the following conclusion: If Z has a one-to-one and onto correspondence with N , then Z Formatted: Font: Times New Roman, 12 pt Formatted: Font: Times New Roman, 12 pt and N have the same cardinality. Formatted: Font: Times New Roman, 12 pt Field Code Changed Field Code Changed
Field Code Changed
N and Q
Finding a correspondence between N and Q is not quite as straight and narrow as with the integers, Z . To establish whether or not a one-to-one and onto correspondence exists, a technique that enables to list all the rational numbers needs to be used.
1/1 2/1 3/1 4/1 5/1 6/1 …
½ 2/2 3/2 4/2 5/2 6/2 …
1/3 2/3 3/3 4/3 5/3 6/3 …
¼ 2/4 ¾ 4/4 5/4 6/4 …
1/5 2/5 3/5 4/5 5/5 6/5 …
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1/6 2/6 3/6 4/6 5/6 6/6 …
… … … … … … …
Figure 3: Representation of listing all Q
In figure 3, an array of the positive rational numbers is created. Through this technique a list will be generated. The arrows represent a system that informs how to list the positive rational numbers accordingly. The purpose of this is to simply be able to list the elements of an infinite set. It is not to sit there and list them all, but to demonstrate that it is possible. If the elements of an infinite set can be listed, then figuring out the infinite set’s cardinality is possible.
This is what is described in relation to Z Q and then Z Q .
The listing will first be a mapping from Z Q , the positive integers to the positive rational numbers. Referring back to figure 3, the list will start with 1/1, which is 1 and then follow the arrows. In doing so, the following list M is generated:
List M
1 1/1
2 2/1
3 ½
4 3/1
5 2/2
6 1/3
7 4/1
… …
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Notice how the 5th element is 2/2, a value of 1. When generating the list, all repeated values are deleted and the left column is filled in as shown in the “new” list O.
List O
1 1/1
2 2/1
3 ½
4 3/1
5 1/3
6 4/1
7 3/2
8 2/3
9 ¼
10 5/1
… …
Essentially what is being done is making the list of numbers relatively shorter and shorter because new numbers that are generated will most likely have been repeated and therefore taken out of the list. It now represents a relation between Z Q that is one-to-one and onto. The next step is to develop another array of the negative rational numbers, Q , and relate it to the negative integers, Z . The process of that was used to list all the positive rational numbers is repeated for the listing of all the negative rational numbers in this array and so is the list with no repeated values/elements, list P.
-1/1 -2/1 -3/1 -4/1 -5/1 -6/1 …
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-1/2 -2/2 -3/2 -4/2 -5/2 -6/2 …
-1/3 -2/3 -3/3 -4/3 -5/3 -6/3 …
-1/4 -2/4 -3/4 -4/4 -5/4 -6/4 …
-1/5 -2/5 -3/5 -4/5 -5/5 -6/5 …
-1/6 -2/6 -3/6 -4/6 -5/6 -6/6 …
… … … … … … …
Figure 4: Representation of listing all Q
List P
-1 -1/1
-2 -2/1
-3 -1/2
-4 -3/1
-5 -1/3
-6 -4/1
-7 -3/2
-8 -2/3
-9 -1/4
-10 -5/1
… …
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This list is a representation of the mapping ofZ Q . The last piece is to include 0 to complete the correspondence (Mevlen, Partel, &Wall, pp 61-62). This correspondence now establishes one that is a one-to-one and onto of Z Q , the entire infinite set of integers and the entire infinite set of rational numbers. Because Z has a one-to-one and onto correspondence with N , they have the same cardinality and are considered countable sets. So, if Q has a one-to- one and onto correspondence with Z , they are of the same cardinality and considered countable.
Furthermore, Q has a one-to-one and onto correspondence with N as well. The next inquiry is the nature of the mapping between N and R , the real numbers.
N and R
The infinite set of real numbers, as depicted in figure 2, includes the infinite sets N , Z , and Q . When George Cantor delved into the infinite set of real numbers, R , he began to look at the set between 0 and 1. The thinking behind this, from what I understand, is that the behavior of the real numbers between 0 and 1 would likely be the same behavior from 1 to 2, 2 to 3, etc.
Since the evaluation was performed, on integers, the counting numbers, and fractions, the focus between the set 0 and 1 will be on decimals. A list can be started as follows:
rddd1111213 0. ...
rddd2212223 0. ...
rddd3313233 0......
rdddiiii 0.123 ...
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where dij {0,1,2,3,4,5,6,7,8,9}- (Plummer, 2009)
So, if there is a real number r1 0.34579..., then d11 3, dand12 4, d13 5. The purpose is now to construct a number that is not contained in the list. To do this, pick a number whose first digit is different from d_11, which equals 3, whose second digit is also different from d_22, whose third digit is different from d_33, and so forth. Since the new number differs from every number in the list at one of its digits, it cannot be the same as any number in the list. So the list was never a complete list of all the real numbers m in the first place. Since the list was assumed to be a complete list of the real numbers between 0 and 1, and we’ve just found that it isn’t complete, the only conclusion we can make is that it’s not possible to have a complete list. It will not be of the same cardinality as N . Therefore, there is no correspondence between N and
R that is one-to-one and onto. Therefore R does not belong to the group of N , Z , and Q because it is not of the same cardinality.
Conclusion
In the infinite number system, it all comes to how each infinite set relates to the infinite set of natural numbers, N . If it is the same cardinality as N , then it has a one-to-one and onto correspondence with N . If it does not, then it belongs to another group of infinite sets. In summary, not all infinite sets are of the same size, or cardinality. What the mathematical world did was to open the possibility of sizing infinite sets regardless if the number of elements is infinite. As finite sets can be counted, the infinite sets ofN , Z , and Q can as well.
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Bibliography
Mevlen, Alice. Partel, Barbara, H. Wall, Robert, E. (1990). Mathematical Methods in Linguistics. Chapter 4. 59-62. Plummer, Robert. (2009). Infinite Sets and Countability. CS103, Handout #19. http://www.stanford.edu/class/cs103/handouts/19%20Infinite%20Sets%20and%20Count ability.pdf Trevistan, Luca. (2007). Discrete Mathematics for CS. CS 70, Lecture 27. http://inst.eecs.berkeley.edu/~cs70/sp07/lec/lecture27.pdf Trapa. Peter. (2005). Counting Infinite Sets. www.math.utah.edu/mathcircle/notes/infinity2.pdf
Figure 1: http://www.regentsprep.org/Regents/math/algtrig/ATP5/OntoFunctions.htm
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