Everybody Loves My Baby

Target Audience: Everyone

I'm not entirely clear on the terms of the nondisclosure agreements I signed, so let's just say that this problem may or may not resemble something which I may or may not have seen on a test that may or may not be something you'd have to take to get into most math grad schools. It ran something like this:

Suppose you have a relation Y on a set S which obeys the following axioms:

1) There exists x in S such that, for each y in S, y Y x.
2) There exists z in S such that, if x Y w, then w = z.

What can we say about x, y, and z?

(If you're seeing a capital Y for the relation instead of a heart symbol, you should probably install the Webdings font. Not that it matters here, so long as you don't confuse the upper- and lowercase y's.)

Now if you have my taste in music, you might have recognized this situation as the set-up of everyone's second-favorite Spencer Williams tune (after, of course, Basin Street Blues) -- namely, "Everybody Loves My Baby." I'll reproduce the chorus below:

Everybody loves my baby
But my baby don't love nobody but me
Nobody but me
Yes
Everybody wants my baby
But my baby don't want nobody but me
That's plain to see!

To clarify, in the role of "My Baby" will be played by x this evening, while z will be covering the part of "Me." Now we're going to see a prime example of being just clever enough to get yourself killed. Recognizing the premise of the song, I visualized precisely the sort of scene that the listener was intended to -- a guy with a jealous girlfriend who gets a little too much attention from other girls. But deep underneath this cute, seemingly innocuous show tune there's a hidden semantic error with some surprising consequences.

The problem is that we're fooling ourselves with the definition of "everybody." In particular, unless you're a strict postmodernist who eats, sleeps, and breathes gender deconstruction, you probably thought of "everybody" as "all the girls," or perhaps "all the guys," if you didn't realize that it was a female vocal. But "everybody" in this case really is "everybody." Which means... well, I'm sensing that I'm getting a little touchy-feely here, so let's state this as a mathematical proof:

By (1), x is in S. [My baby is real! Really! You just can't meet her because she lives in Canada.]

Now (1) says that y Y x for each y in S. [Like I said, everybody loves my baby.]

In particular, let's take y = x. [Wait a minute here...]

We have x Y x. [Narcissist!]

From [2], this forces x = z. [Uh oh.]

In other words, my baby and I are one and the same. I'll leave you to draw whatever sort of conclusions you want about the nature of loyalty, depending on your relative level of cynicism.
In any case, I may or may not have figured this out right after leaving the hypothetical test center, causing me to be torn between appreciation for the test writer who came up with this question and the dissapointment of losing some points on the hypothetical test.

In completely incidental, unrelated news, I'm probably going to be retaking my GRE Subject Test in Math this November at USC. If you're preparing for the test, I'll let you in on what I know about the prep materials: there are basically two books, one which I have linked below and the other which I have deliberately omitted. As a matter of fact, neither book is all that good; they are both riddled with errors in the very places where they shouldn't be -- those strange trigonometric identities which you only use once or twice in the course of your life. This book, however, is the lesser of the two evils in my opinion. Another thing to keep in mind is that the book has a lot of stuff in it that will never be on the test. Stick to the basics and you'll do fine; even when I had a really bad day taking mine I got a high enough score for a lot of grad schools.

Finally, if you're intrigued by the song which inspired this post, you can check out the "Love Me or Leave Me" original cast recording, also linked below. And if you're gonna be taking the GRE with me at USC, send me an email or something.