L.A. Math: Romance, Crime, and Mathematics in the City of Angels
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L. A Math: Romance, Crime and Mathematics in the City of Angels By James D. Stein Supplementary Mathematical Material for Website 1 Mathematical Accompaniments Chapter 1 – Symbolic Logic Chapter 2 – Percentages Chapter 3 – Averages and Rates Chapter 4 – Sequences and Arithmetic Progressions Chapter 5 – Algebra; The Language of Quantitative Relationships Chapter 6 – Mathematics of Finance Chapter 7 – Set Theory Chapter 8 – The Chinese Restaurant Principle; Combinatorics Chapter 9 – Probability and Expectation Chapter 10 – Conditional Probability and Bayes’ Theorem Chapter 11 – Statistics Chapter 12 – Game Theory Chapter 13 – Elections Chapter 14 – Traveling Efficiently – and Other Algorithms 2 Chapter 1 - Symbolic Logic Before you dive into this section, let me repeat something I said earlier – you don’t have to read this! I think a fair amount of learning takes place by erosion – if you’re simply exposed to something often enough, it will sink in. Maybe not deeply, but enough to give you the idea. Hang around with musicians, you get some idea of what goes into music – maybe not enough to make you a musician yourself, but you’ll be a lot more knowledgeable about it than if you spent no time on it at all. However, I hope that you’ll try reading some of these sections. They’re not deep, and if you get fed up, you can always go on to the next story. Sherlock Holmes was fond of telling Watson that, when you eliminate the impossible, whatever remains, no matter how improbable, must be the truth. It's certainly a simple, insightful, and elegantly phrased remark. However, it does not come as a stunning surprise, because most of us are already aware of the inherent logic behind Holmes' statement. It is perhaps fitting that Sherlock Holmes, for whom logic was a sine qua non, was an Englishman, for it was another Englishman, George Boole, who was most responsible for the invention of symbolic logic. Prior to George Boole, mathematicians concerned themselves with mathematical objects such as numbers and geometric figures. A goal of mathematics then, as now, was to prove theorems - - about numbers, geometric figures, and such. To George Boole goes the honor of being the 3 first extensive investigator of the nature of proof (not to slight the Greek philosophers, who made the initial contributions in this area). One of the reasons that mathematics is so successful that it does is its relentless focus on concepts whose definitions are unambiguous. Boole focused his attention on statements or sentences which were either unambiguously true or unambiguously false. Such statements are called propositions. From now on, the letters P and Q will be used to denote propositions, and the letters T and F will be used as abbreviations for “true” and “false”, respectively. To each proposition there is an opposite, the proposition we shall denote by NOT P. A proposition can be negated simply by sticking the phrase "It is false that ... " in front of the proposition. For instance, if P is the proposition "Today is Thursday," the opposite of P (sometimes called the negation of P) is the proposition "It is false that today is Thursday." From simple propositions more complicated ones can be built up through the use of the logical connectives OR and AND. At this stage, let us introduce what mathematicians call a "convention" (this is probably short for 'conventional agreement'). We will use the words OR and AND to denote the logical connectives that enable us to construct complex propositions from simple ones according to the following rules. If P and Q are propositions, the proposition P OR Q is true if either P is true or Q is true, or possibly both are true. This is known as the inclusive “or” – if your waiter asks “Will you have coffee or dessert?” and you answer “yes”, you mean that you will either have coffee, or dessert, or maybe both. There is also the exclusive “or”, as in “I will either drive to San Diego tomorrow or take the plane,” obviously, you’re not going to do both. However, mathematicians decided to 4 use the inclusive “or”, just like they decided to use the symbol + for addition rather than something else. It’s important to have everyone in agreement about what terms mean. The proposition P AND Q, however, is true only when both P and Q are true. This information can be quickly summarized in tabular form. The layout that follows is known as a truth table. P Q P OR Q P AND Q T T T T T F T F F T T F F F F F With the four rows, we have covered all the possible true-false combinations for the two propositions P and Q, and the other columns give the truth values of the proposition at the top of the column for the truth values of P and Q in the same row. Here's a very informative, very short truth table. P NOT P P OR NOT P P AND NOT P T F T F F T T F 5 In words, for any proposition P, the proposition P OR NOT P is always true, and the proposition P AND NOT P is always false. Propositions which are always true are known as tautologies. Some truth tables are important, but not especially interesting. It is easy to show that P OR Q and Q OR P have the same truth table. This isn’t surprising, as if your waiter asks “Will you have dessert or coffee?”, it’s the same question as if he asked “Will you have coffee or dessert?” Similarly, P AND Q and Q AND P have the same truth table. It is possible to use parentheses to construct ever more complicated propositions, much as parentheses are used in arithmetic and algebra for exactly the same purpose. If P, Q, and R are propositions, we can construct the compound proposition P OR (Q AND R) by first constructing the proposition Q AND R, and then taking that proposition and OR-ing (as the computer folk are fond of saying) the proposition P with the proposition Q AND R. We can compute the truth value of a complicated proposition from the truth values of its components simply by stripping away levels of parentheses. Just as we compute the numerical expression (2+3) x (3 x (4+7)) by working from the inside out, we can do the same thing with complex propositions. Arithmetically, (2+3) x (3 x (4+7)) = 5 x (3 x 11) = 5 x 33 = 165. Suppose now that P is true, Q is false, and R is false. To compute the truth value of the following logical expression, (P OR NOT Q) AND (NOT P OR R), let's replace each proposition by T or F as we compute it. 1) (P OR NOT Q) AND (NOT P OR R) 2) (T OR NOT F) AND (NOT T OR F) 6 3) (T OR T) AND (F OR F) 4) T AND F 5) F Now let's return to Sherlock Holmes. How can we analyze his remark that, when you have eliminated the impossible, whatever remains, however improbable, must be true? Let's suppose that P and Q are propositions such that P OR Q is true. Suppose further that Q is false. How can we conclude that P must be true? One look at the truth table for P OR Q should make it fairly obvious. P Q P OR Q 1) T T T 2) T F T 3) F T T 4) F F F Since P OR Q is true, line (4) is eliminated. Since Q is false, lines (1) and (3) are likewise out. No matter how improbable P may be, it must be true, as line (2) is the only one remaining, and P is true in line (2). Now things get a little complicated. Boole decided to assess the validity of the argument IF P THEN Q on the basis of the true-false values of the propositions P and Q. What Boole decided 7 was that the important thing was to make sure that any argument which started with a true premise (the premise is the P in IF P THEN Q) and ended with a false conclusion (the conclusion is the Q in IF P THEN Q) would be labeled as false. After all, if you start with the truth and reach a false conclusion, your argument must be fallacious. In order to single out these fallacious arguments, Boole made all other IF P THEN Q statements true, by fiat. This resulted in the following truth table for IF P THEN Q. P Q IF P THEN Q T T T T F F F T T F F T Let's look at the compound proposition IF ((P OR Q) AND NOT Q) THEN P. So that everything will fit on one line, let R denote the proposition (P OR Q) AND NOT Q P Q NOT Q P OR Q (P OR Q) AND NOT Q IF R THEN P T T F T F T T F T T T T F T F T F T F F T F F T No matter what the truth values of P and Q, the proposition 8 IF ((P OR Q) AND NOT Q) THEN P is always true! Admittedly, when Sherlock Holmes used it, he assumed implicitly that either P or Q is true. Nonetheless, no matter what the truth values of P and Q, IF ((P OR Q) AND NOT Q) THEN P must be true.