Mathematically speaking, my life is fairly straightforward this weekend (famous last words).
My kid has worked on “Boolean Reasoning” (Frank Markham Brown, Dover) this morning. When I stopped, I think we had just figured out how to find the “consensus” of two logical products. Oh, I know how to do it by hand – the challenge is to get Mathematica® to do it.
What’s that mean?
If, for example, I have two logical products (using juxtaposition for conjunction, i.e. for “and”)
x y z’ , x w z,
their consensus is found by taking their conjunction…
x y z’ x w z,
dropping both of the paired symbols z and z’…
x y x w,
and then dropping one of the repeated symbol x…
x y w.
The biggest hassle is that the conjunction we form in the first step is always false in this example – as it must be in any case where we have both a symbol and its negation:
z z’ = False,
so the entire conjunction is also false. This is useful: if the conjunction is not always false, then I do not need to look for a pair consisting of a variable and its negation; I would only need to drop repeated symbols.
The biggest problem with the book “Boolean Reasoning” is that it has almost no examples. Basically, it has one example of each computation.
There is, however, a Schaum’s Outline, “Boolean algebra and switching circuits” by Elliott Mendelson… and, of course, it has lots of examples. More importantly, it includes the concepts of “consensus” and “prime implicant”. I’ll get to test out my understanding and my code. And I won’t have to show you every example in Brown’s book.
I also got an interesting looking engineering book, and it may come in handy down the road if I want to know more about the design of logic circuits, but for now my interest is in logic per se.
I am waiting for a used copy of the book that was the primary reference for “Boolean reasoning”, namely “Boolean functions and equations”, by Rudeanu. It might come in the mail today. I probably won’t – shouldn’t! – get to work on it today… but nothing will stop me from at least looking through it as soon as it arrives.
After I post this, I am going to work on a technical post for the blog. Right now, it appears that I will finally tackle a color post about the nonlinear behavior of my monitor.
On the other hand, it has me up to my eyeballs in numbers… shades of physics lab! (I wish I could say that I always knew I should major in math, but the fact is that the sophomore physics lab at Caltech drove me out of physics; it was way, way harder than physics lab at Northeastern, and I hadn’t had the freshman lab at Caltech to prepare me. On the other hand, it was no accident that I was already taking the one sophomore course required of math majors.)
I might joke that one of the reasons I like linear algebra is that I can throw all of the numbers into matrices, and have fewer numbers and symbols to keep track of.
Just because I love mathematics doesn’t mean I am a computing whiz. The existence of Mathematica makes my mathematical life a whole lot easier. In fact, it makes things easy that I would otherwise find effectively impossible.
Just because I love mathematics doesn’t mean I love computing. I love answers, but not if it’s excruciating to get them. I love putting together a successful computation – I’m just glad that I don’t have to do every little thing myself, but work instead in a rather high level language.
So we’ll see how this monitor post goes, with its plethora of numbers to keep straight… and if it goes badly, I will switch to the next scheduled logic post instead (“universal generalization” etc.).
And if both of these plans fail, I do have plan C: I have a post (simple projectile motion) at stage IV – that is, all of the mathematics is done and all that remains is to translate the post into LaTeX. Worst-case, that’s the post that will be finished Sunday evening to go out Monday evening.
I would prefer, however, to hold the projectile post in reserve; in fact, I’d like to build up a reserve of a few posts.
Of course, there’s always plan D: no technical post at all– but unless something goes wrong this weekend, I should be able to get a post out.
So let me quit gabbing and get about my math….