My Cardinal Is Bigger Than Yours (Maths For Normal People I)
Are you a non-mathematician? Do you want to know a little about interesting mathematical phenomena? I am starting a series of articles on selected mathematical topics. I am going to explain them very simply and comprehensibly. No prior knowledge of higher mathematics will be needed. Today, we are going to talk about sets and their sizes. In the next article, we will use this knowledge to understand the mightiest beast itself – infinity.
Thinking about infinity requires radical changes in the way you normally think. Many results are counter-intuitive, unbelievable, and difficult to imagine. For this reason, I am going to explain things slowly and in small chunks. A mathematician could summarize my texts in a few short symbolic statements, but I am hoping to write for non-mathematicians here :-)
Before we can behold the splendors and miseries of the infinite, we must study one of the most fundamental concept of mathematics – a set. It is set theory which breeds the true infinities.
Let’s Not Be Sad! It’s a Set, He Said
Set theory forms the foundation of all modern mathematics. Even though it appears very simple at first glance, it has surprising depths and is extremely potent. It would take a long series of articles to discuss it thoroughly, but precise descriptions don’t concern us now.
We can intuitively understand the term set as an arbitrary collection of objects. Sets are usually denoted by capital letters and their members are enclosed in curly brackets. For instance, let us regard the set of all positive even numbers smaller than 9:
<$$ A = \{2, 4, 6, 8\} $$>Or the set of English vowels:
<$$ B = \{a, e, i, o, u, y\} $$>Or the set of things that are now on my desk:
<$$ C = \{\text{laptop}, \text{mouse}, \text{bottle of water}, \text{pen}, \text{pencil}\} $$>What about the set of unicorns grazing on the London banks of Thames? That is an empty set. (As far as we know. The unicorns could be invisible, after all.) We denote empty sets by a special symbol:
<$$ D = \emptyset $$>Show Me How Big Is Your Set
We can ask ourselves how many members does a set have. This is called the cardinality of a set. I denote it by vertical bars. The previous example sets have the following cardinalities:
<$$ |A| = 4, \quad |B| = 6, \quad |C| = 5, \quad |D| = 0 $$>Any number that can be a cardinality of some set is called a cardinal. Every natural number is a cardinal. <$\frac{1}{2}$>, for instance, is not a cardinal, since you can’t have a set with only half of an item.
For the usual finite sets you can think of, cardinality can be found out simply by counting the members. However, such a definition is not good enough in mathematics. There are a few reasons. Like: “It doesn’t work.”
How could counting “not work”? It breaks down when we consider infinitely large sets. If you’re not Chuck Norris, you can’t just count to infinity.
Feeding Leopards
Let’s forget for a moment about the actual size of a set and compare two sets instead. Say somebody gives you a pile of left hand gloves and a pile of right hand gloves. How do you know that you can pair them? No counting of the piles!
You would obviously start pulling out left and right hand gloves and putting them together. If there are gloves left, you know that the piles weren’t of the same size.
The mathematical definition says exactly this. Two sets have the same cardinality if, and only if, all their members can be put into a one-to-one correspondence (mathematicians say that there exists a bijection between the sets). This means that to every member from the first set we can assign one member from the second set.
We can illustrate the one-to-one correspondence graphically by drawing lines between the sets’ members:
Every leopard from set L can be assigned to a sheep from set S. Well, rather the sheep will be “assigned” to the leopard. Therefore, |L| = |S|. That’s why are the leopards looking so happy. Note: no animals were harmed during the writing of this article. Except for that delicious chicken I had for lunch.
Note that by “drawing lines” we can find out that two sets have equal sizes without actually counting them. This is crucial.
What happens if two sets aren’t equally big? It won’t be possible to find a one-to-one correspondence, or draw lines between the members:
Not every squirrel from set Q can get its own acorn from set R. Therefore, |Q| ≠ |R|.
Six Equals Six By Any Other Word
We are now just one step away from defining cardinality correctly. Well, rather than a step it’s a big intellectual leap. We must imagine that natural numbers are sets. Sounds crazy? Maybe, but wait. You can visualize a number like a bunch (= a set) of dots:
Numbers as sets. If you want the gory details, refer to my previous article, What Are Natural Numbers. The math symbolics there might be a little tough, though, and it’s not really necessary.
Let’s say you believe me that numbers can be represented by sets. Thanks. Now we can finally define cardinality: a set has cardinality <$N$> if it can be put into a one-to-one correspondence with <$N$>, where <$N$> is a number represented by a set. An example will make this clear. Say we want to prove that the cardinality of our set <$B$> of English vowels is six:
|B| = 6 expressed by one-to-one relation.
We can likewise prove it is not five or seven or any number other than six:
|B| ≠ 5 and |B| ≠ 7 expressed graphically. It is not possible to construct one-to-one relations here.
I know what you’re thinking now. Why do we define cardinality in this complicated way? It’s just the count of members, for heaven’s sake! It’s pointless to go through all this trouble with one-to-one relations and drawing numbers as sets. Right?
Yes and no. For finite sets, this definition is an overkill. No denying that. But once we start playing with infinite sets, the power of the definition will appear as clear as a day. It is a core principle of mathematics to generalize as much as we can, and here we have a general definition that works for all kinds of sets, be they finite, infinite, horribly infinite or whatever. It’s a nice little definition.
What’s Next?
Next week, we are going talk about the most basic infinite set – the set of all natural numbers. We will start exploring the mind-boggling properties of infinity and today’s cardinality definition will come in handy. Prepare to be stunned.
In the meantime, please give me feedback on this article. Was it interesting? Did it tell you anything new? Was it too easy? Too difficult? Too long? Too short? Not enough cute animals? I welcome all constructive comments!
2 comments Add your own…
I have spent 40 plus years making my living as a professional photographer, graphic designer, artist etc., with a strong subset of PC computer building with point of sale software which I rarely do that type of work anymore unless it is a favor to a friend, I am a copy writer with a measure of success, a traveler to far places near and far, allowing my curious nature to drink in the subtles of nature, untold chance interactions with the most unusual human beings, often they reshaped my mind, broadening my understanding of my fellow sojorners, whether they turned a bit this way or that as I related my world views to them or my world view molded a bit toward their world view, maybe the word "world might be better stated to be as small as the lightest, gentelest of possible vibrations or waves as you might choose to envision them;. I am a husband, a father to orphans ( my favorite, my daughter now 22 who my wife and I adopted from a hellhole in Kathmandu, Nepal at the age of 10, plenty old enough to have already suffered a large meassure of man's indignity to man, or rather man's cruelty to children, I cannot even begin to digest such puzzles. Not so much that these puzzles are unsolvable, but that the reward for going there in my mind would be too painful to bear, so I will leave the reason, the reasoning to those of sterner hearts and minds. Sunita, being my 3rd rescued older child, just had her first child at 22, this tallies the grandchildren to the stagering number of 10, all not a direct lineage of my flesh, but they are all from the lineage if my heart and of my dear wife;s heart to an even greater degree. I would be willing to send a photo here and there, of children and the beautiful place we are so forutunate to have now lived here for decades.
I was the our presidents personal Photographer for about 15 years, always being in the mix of all the dignataies, je
tting their sort of short storieskl as I would also write up information about then for
farious publications
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