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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:

Two sets having the same cardinality 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:

Two sets which don’t have the same cardinality 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:

Natural numbers visualized like groups of dots. 3 dots for number 3 and so on. 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-one relation |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 |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!

March 4, MMXI — Mathematics, Maths For Normal People, Infinity.