The algebra of algebra textbooks is the branch of mathematics where you manipulate numbers by exploiting various rules. For example, if we are given the equation x
-1=0, we can exploit the following rule to find out what x
inverse of subtraction
We add 1 to both sides and find out that x
=1. Here are some more rules for the algebra of numbers, but by no means all:
There are lots of algebras. Many of them are algebras over different sets of numbers and many of them are not. The set of values that an algebra operates on is called the sort. For common high school algebra the sort is the complex numbers.
There is also an algebra over the sort of sets. The objects are sets and the operations are union, intersection, and complement of sets. The rules look similar to the rules for the algebra of complex numbers. In fact, if we use the symbol "+" to stand for union and "*" to stand for intersection, the rules of commutativity and associativity look identical for sets as for numbers. If we also let "0" stand for the empty set and "1" stand for the universal set, then the identity rules looks the same too. The arithmetic rule of distribution also looks the same:
distribution of *
however, the algebra of sets adds the complementary rule
distribution of +
This rule doesn't hold for numbers, only sets. If we let "-a
" stand for the set complement of a
, then the rule of double negation holds for both systems:
) = a
The minus sign represents an inverse operation in both algebras. An
inverse operation is one that can be combined with another operation to
produce the identity element for that operation:
Obviously these inverse rules apply only for sets, not numbers.
Another algebra is the algebra of functions. Let f
be a function, a mapping that takes a number and returns a number. The value of f
is written f
). Let g
be a function also. Then f
") is another function. Since f
is a function, we can find the value of x
and we write it (f
). It's defined as the value of f
at the value of g
, or (f
)). The operation of function composition forms an algebra over the sort of functions with the following rules (I'm leaving out some important restrictions here. Exercise: what restrictions?):
The symbol "1" represents the identity function --the function such that for all x
. Since function composition is not commutative it is necessary to spell out both the left and right identity rules. Some algebras have an identity element that only works on the left or only on the right.
There is also an algebra of strings with concatenation. A string is just a sequence of letters like "abc
" or "xyz
". Concatenation (represented by the "+" sign joins two strings together: "abc
" + "xyz
". The operation of string concatenation forms an algebra over the sort of strings. I'll leave the rules as an exercise for the reader.
CORRECTION: I'm sure many of my readers noticed that I made some errors in the section on the algebra of sets. First, traditionally an inverse for a binary operation is another binary operation. That is, the inverse for a+b
, not just -b
. I was being too terse and failed to distinguish between an inverse operation and an operation that produces an inverse of an element.
Second, it should have been clear from my definition of "inverse" and my rule inverse
, that set complement is not a true inverse. If a
are inverses with respect to an operation +, then a+(-a)=0
, where 0 is the identity element for +. With sets, a+(-a)=1
, which is not the identity element for +, it is the identity for *.
For fifteen years, I have just known
that Boolean algebras (the algebra of sets is a Boolean algebra) have true inverses. I never thought to question it. Now it seems obvious that they don't. It's like finding out that the world is flat after all.
doctors and high blood pressure
Doctors are like mechanics. You often don't have enough knowledge of your own to decide if one is any good. But I've managed to gather enough evidence on my current doctor to make me decide I need a new one. It's annoying to look for a new doctor though.
My previous doctor retired and I just went to a new one for the first time about a month ago. I tried to describe to him a long-term treatment I have been taking from a previous cubed doctor. I described it to previous and previous squared, and they both either got it immediately or took my word for it (I don't know which). This guy seemed confused.
Actually, he seemed confused about a lot of things. He sort of acted like he had just gotten up from a nap. I wondered if he was on something.
Then, he took my blood pressure and it showed a little high. I asked him if he has a bigger cuff. He looked confused. I said that I've been measured high before by office staff, but then the doctor comes in and gets a larger cuff and measures it again, and I don't have high blood pressure. He brushed it off but didn't say any more about it.
I went back a few weeks later for some test results and he still looked like he'd been sleeping. Measured my blood pressure with the same cuff and told me it's high again. I explained the cuff-size thing again. Wouldn't you think a doctor would at least be curious enough to go on the internet
if a patient said something like that? Apparently, he hadn't bothered. I was a bit more insistent this time. I told him that this has happened to me five or six times, with at least two different doctors. My blood pressure measures lower with a bigger cuff.
He brushed me off again. He told me that cuff size doesn't matter. He told me that I need to cut down on salt and get more exercise. He hadn't bothered to ask me how much salt I eat or how much exercise I get. I swim 1000 yards or more about five days a week.
Then I looked it up on the internet and found out that my arm is over two inches larger than the largest size for a regular cuff. I mentioned it to a woman who is a former nurse and she said she could just look at me and know that I needed a large cuff. And this doctor had never heard of such a thing. So I need a new doctor.
I wonder how many people are going to be taking blood pressure medication and have other treatments over the career of this doctor because he is too arrogant to listen to patients.
UPDATE: It occurs to me that only logicians and computer scientists are likely to get the "previous squared" stuff. This is a common notation and pun in logic, computer science, and fields of math that deal with objects other than numbers: I have my previous doctor, and my previous previous doctor. "previous previous" looks like "previous multiplied by previous" or "previous squared".
Hmm. I think I should write a post on non-numeric algebras.
Welcome to the Dark Side, Donald
Donald Crankshaw at Back of the Envelope has always tended to be on the moderate, understanding side of the issues. He wasn't a mean ol' warmonger like me. Now he writes this
I know that some liberals worry that Bush will try to take these regimes out if he's re-elected. I'm worried that he won't. As far as I'm concerned, the regimes of North Korea and Iran are mad dogs that need to be put down--the sooner, the better, for their own people more than ours.
The attitude is spreading. BWAA HA HA HA.
BWAAAA HAA HAA HAA.
BWAAAA HAA HAA HAA.
where's 007 when you need him?
Al Qaeda is planning
a major political assassination to head off a series of terrorists attacks (from dgci
). I'm speculating that the target is Tony Blair.
There is a lot of opposition to the war in England and even in Blair's own party. Enough that it is reasonable to believe that if they killed Blair, England would be out of the War on Terror. This would be a tremendous victory for al Qaeda, dwarfing what they did in Spain.
By contrast, killing Bush would only piss Americans off. Al Qaeda can't be too dense to realize that. Faltering public support for the war would coalesce instantly behind Cheney.
Italy is another possibility, but it probably isn't any easier to kill Berlusconi than Blair (maybe harder because Political Correctness has gone so far in England and the Islamists have gained so much ground there) and the rewards for killing Blair are enormously higher.
Another clue is that when someone spilled the beans about the fact that Kahn was a double agent, a lot of arrests were made in England but nowhere else I heard of. That suggests that something big was going on in England.
Michelle Malkin is revealing some startling things about the Japanese internments of WWII. Things I've never heard before. Things that have apparently been covered up by leftist historians who don't want Americans to know the truth.
Did you know that before the US started interning Japanese Americans, there had already been an incident in which three Japanese Americans had aided Japan
against the US in the war? A downed Japanese pilot from the attack at Pearl Harbor asked some Japanese Americans for help and they helped him. Apparently there were only three he asked, and all three helped. Three for three Japanese Americans --two of whom were born in the US-- were more loyal to Japan than to the US. The motivation for the internment becomes clearer doesn't it?
Did you know that half the internees were Europeans
rather than Japanese? Did you know that when the Japanese received compensation and an apology decades later, the Europeans did not?
This is getting interesting enough that I may have to read Malkin's book.
You would think that after reading Ann Coulter's book Treason
where she reveals how McCarthy has been slandered, I wouldn't be surprised to find another instance where I have been misled by revisionist history. But surprised I am.
Christian Carnival XXX
Christian Carnival XXX
is up. I wish they'd announce these on the mailing list so I wouldn't forget to put up a post...
Check out the article by Back of the Envelope about God and science. It's a very interesting topic and Donald Crankshaw writes well about it. When I get some time to get back to serious blogging, I might write a response on Donald's post.
And I'm not going any jokes about triple-X ratings for Christian Carnival XXX because that would be just too crude and too obvious. I'm going to show some class and subtlety and not even mention it.
al Sadr -- budding democrat
There's a good article in the Washington Times
(from the Corner
) detailing the ongoing civil war in Najaf between al Sadr and the Iraqi government. Al Sadr, an Iranian puppet, is quoted as saying, "Our demand is for the American occupation to get out of Iraq. We want an independent, democratic, free country."
Al Sadr is a cleric, a religious leader who was fighting to establish sharia law a few months ago. Now he wants democracy? Clearly, a man like al Sadr will say whatever he thinks will help him win. Now, he thinks that what will help him win is appealing to democrats.
This brings up an interesting question: is this because al Sadr sees an upwelling of sentiment for democracy in Iraq, or is this just an appeal to the American president's opposition in the US and Europe?
science and religion
From Scott Harris of Lone Star Reality
, a quote by Churchill on Islam:
... Mohammedanism is a militant and proselytising faith. It has already spread throughout Central Africa, raising fearless warriors at every step; and were it not that Christianity is sheltered in the strong arms of science - the science against which it had vainly struggled - the civilisation of modern Europe might fall, as fell the civilisation of ancient Rome.
... Today we have a Europe that has embraced Science, but rejected Christianity. And now, Europe is in the process of falling prey to Islam. So despite Churchill's assertion that Christianity was being protected by science, I think the evidence shows that it was Science being protected by Christianity.