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3 Cardinality last edited February 1, 2016 3 Cardinality While the distinction between finite and infinite sets is relatively easy to grasp, the distinction between di↵erent kinds of infinite sets is not. Cantor introduced a new definition for the “size” of a set which we call cardinality. Cantor showed that not all infinite sets are created equal – his definition allows us to distinguish between countable and uncountable infinite sets. The attempt to understand in- finite sets highlights the role that definitions play in mathematics. 3.1 Definitions Countless times in our everyday lives we attempt to measure the “size” of some- thing. We might want to know how good an athlete is, or how influential a cer- tain artist is. We might want to measure the size of a house, or the “fastness” of a car. To do so, we could record the pass completion rate of a quarterback, the number of top-10 hits of a music artist, the number of bedrooms of a house, or the maximum speed of a car. Such quantities allow us to compare two ob- jects and make statements about which is “better” in some restricted sense. Of course, none of these quantitative descriptions are completely satisfactory as a complete description of size, or quality, or influence. For example, the number of bedrooms of a house is only one way of measuring the size of a house, but perhaps we should consider the sizes of those bedrooms, or the total size of a property, or the total square footage inside it. Similar ambiguities are present when judging cars, artists, or athletes. Mathematicians attempt to make statements as precise as possible, and will rarely use the word ‘size’ without making clear what they mean by it. For example, consider the two shapes in Figure 2 below. Think about which shape Figure 2: Three shapes of di↵erent sizes. is largest. If we are concerned about length, the long, thin rectangular shape is certainly the largest, but if we care about total enclosed area, the circle certainly wins. However, we might care most about total perimeter, in which case the weird shape on the top-right would certainly win out. The way in which we choose to think about size will determine which shape is the “largest”. If do not make clear in advance what we mean by size, we cannot make intelligent statements about it. To make statements precise, mathematicians use definitions,bywhichthey 16 last edited February 1, 2016 make clear how they will use a particular term. We can define the word “size” to mean enclosed area, perimeter, or whatever else we want it to mean. However, until we define the term, judging the correctness of a statement involving the term is not possible. One theme that will arise throughout the semester is that coming up with good definitions is more difficult than we may have expected. All of us know how to use a dictionary, or the internet, to look up the “definition” of a word. But mathematics uses “definitions” in a manner di↵erent from the way we generally think about them. Most of the definitions that come to our mind are extracted from everyday usage. When the Oxford English Dictionary was first published in 1884, it’s primary aim was to document how words are used. The definitions provided were those extracted from actual usage of the words. If such a definition accurately describes the way in which a word is used, then it is correct; otherwise it is not. In mathematics, definitions are stipulated. A mathematician makes clear, or tries to make clear, what they will mean when they use a certain word. They stipulate that the term “rational number” should be understood to mean a number that can be written as the ratio between two whole numbers. They stipulate that the “square root” of a number x be a number y so that when multiplied by itself gives us the initial x. For this reason, a definition can’t be right or wrong, but it can still be good or bad. Let us consider an example of a definition that we might call “bad”. Definition 3. The square root of a number x is another number y such that multiplying it by itself gives us the original number x. At first glance, this definition seems reasonable. Both 4 and -4 would be a square root of 16 by this definition, since when multiplied by themselves the result is 16. However the silly outcome of this bad definition is that 1 would not be a square root of 1, and neither would 0 be a square root of 0. The square root of a number x – as we defined it here – must be a number di↵erent from x. This though sounds silly to us, and should make us realize that the definition we provided above is probably not a very good one. We will see over the semester that the way we define a term can be crucial to understanding its properties. Mathematics is very much about figuring out the “right” definitions, and then using them to prove interesting results. Let’s try making this a bit more concrete. What is an odd number? One idea that might come to mind is this: Definition 4 (First attempt). An odd number is a number that cannot be evenly divided by 2. This definition might sound initially reasonable. By this definition, the num- bers 3, 7, and 119 are all odd, while 4, 8, and 120 are not. However, according to this definition, the numbers 2.5 and 3.14 are both odd, since neither is evenly divisible by 2. Unless we want to think of such numbers as odd, we had better change our definition. One way to do so would be to only modify the above definition slightly: 17 last edited February 1, 2016 Definition 5 (Second attempt). An odd number is an integer that cannot be evenly divided by 2. While this definition is probably a “better” definition, it is still not excellent. To demonstrate why, consider proving the proposition that if n is odd, then so is n2. This proposition sounds reasonable (try a few examples!), but it is unclear how we might go about proving it. Choosing a “good” definition of an odd number, however, allows us to do exactly that. Definition 6 (Final attempt). An odd number is an integer of the form n = 2k +1, where k is an integer. This definition seems quite reasonable, even if it is slightly technical. Ac- cording to this definition 3, 7, and 119 are all odd, whereas 4, 8, 120, 2.5, and 3.14 are not. More importantly, we can use this definition to prove the following: Proposition 1. If n is odd, then so is n2. Proof. Let a be an odd number. Then by definition, we can write this number as a =2k+1, for some integer k. Squaring this number gives us a2 =(2k+1)2 = 4k2 +4k + 1. We can rewrite the last term to give us a2 =2(2k2 +2k) + 1. Since 2k2 +2k is an integer, then the definition of an odd number tells us that 2(2k2 +2k) + 1, which is equal to a2 is an odd number. Oftentimes, a good definition is the most important part of really under- standing an idea. 3.2 The “Size” of a Set In studying sets, we are often interested in saying something about their sizes. What is a good way to define the size of a set? Let us begin with the following suggestion. Definition 7 (First attempt). The size of a set is the sum of its elements; we denote the size of a set A by A . | | At first glance, this appears like a very reasonable definition. Using this definition, we could state that 1, 2, 3 = 6, whereas the size of the set 3, 4 is 7. This definition will allow|{ us to}| prove all sorts of identities, including{ } A A = A A = A , A = A , and A = 0 (can you understand why?).| [ | This| definition\ | | might| | [; also| motivate| | us| to\; think| about some interesting mathematical questions. For example, imagine that we know A and B (the sizes of the sets A and B), what can we say about A B and| A| B |? | However, although this seems like a reasonable| definition,\ | we| might[ | notice that it could only be used when studying sets of elements that can be added together. What would it mean to talk about the size of the set blue, yel- low, green or the set Chicago, Washington, Philadelphia, New York{ ?This problem leads} us to consider{ a di↵erent, more general, definition of size.} 18 last edited February 1, 2016 Definition 8 (Second attempt). The size of a set is its number of elements; we denote the size of a set A by A . | | We can now use this definition of size to discuss not only sets containing numbers, but also sets containing arbitrary elements including colors, cities, and people. We can show that = 0, and note that the empty set is in fact the only such set with size 0. We|; can| also ask questions such as, if we know A and B , what can we know about A B and A B ? | | However,| | soon after we start discussing| [ | this| definition\ | of size, we note that it too has an Achilles’ heel. More specifically, the definition above works very well when discussing finite sets, for which we can assign a number to describe its size.
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