N is for . . .

The Null Set


Image courtesy: Thomas Guignard

Okay, break's over. Time to get back to some Mad-Cool-Math Nuggets. Today we're going to delve into the wonderful world of set theory and achieve instant literary success in the process. Ready?

To define a mathematical set, think of all the books in your house. Each book is an element in the collection, or set, known as your library. We'll call this set L. Pick out your favorites and you have a subset of your library we could call F. Pick out the ones you've read more than once and make another subset called R (for repeat).

It's possible F = R, meaning every favorite has been read more than once and every book read more than once is a favorite.

Maybe your particular R is empty, meaning you never read a book more than once. In that case R would be an example of a null set, denoted {}. Another name for this is the empty set.

(If you hate to read and have no books then, like the image above, your L = {}. That, in my opinion, would be a sad null set indeed.)

Null sets, or sets that contain no elements, can be a little weird. They are a subset of every set you can think of, which means they must be a subset of themselves.

Say what? Let's go back to your library, the set L, whether it's empty or not. Consider the books written by my cat as a set called C. I love my cat, but she hasn't written any books. She doesn't even stand on my keyboard to type lsdkghkdjgsl. She's too busy sleeping in my bed all day. So the set C is empty or C = {}.

For C to be a subset of L, every book in C must be in L. Conversely, if C is not a subset of L, then there must be some book in C that is not in L. That's impossible, because C is empty and has no books. So C is a subset of L. (That, my friends, was a proof by contradiction. We mathy folk are tricksy, no?)

Have I blown your mind or bored you to sleep? I do apologize, but I did promise literary success, so let's get to it. Since a null set is a subset of every set, then I can say, without lying, that the set of my published novels is a subset of the books on the New York Times Bestseller List. I can also say that my published novels make a subset of the books that won the Pulitzer. How? Because the set of novels that I have published is a null set. Ta da!

Care to share something about your library set L? Is your subset of digital books bigger or smaller than your subset of paper books?

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Want to read something with no math involved, including a price? Feel "free" to download my sci-fi short short, The President and the Pea. The only confusing part is the unreliable narrator. Did Annette Hutchins really witness the assassination of the President of the United States? Or is she mental?