1.3 Other Basic Tools (Chap. 3)

Transcription

1 1.3 Other Basic Tools (Chap. 3) Summation Notation The Greek letter Σ (a capital sigma) is used to designate summation. For example, suppose an experimenter measured the performance of four subjects on a memory task. Subject 1's score will be referred to as X 1, Subject 2's as X 2, and so on. The scores are shown below: The way to use the summation sign to indicate the sum of all four X's is: This notation is read as follows: Sum the values of X from X 1 through X 4. The index i (shown just under the Σsign) indicates which values of X are to be summed. The index i takes on values beginning with the value to the right of the "=" sign (1 in this case) and continues sequentially until it reaches the value above the Σ sign (4 in this case). Therefore i takes on the values 1, 2, 3, and 4 and the values of X 1, X 2, X 3, and X 4 are summed ( = 26). In order to make formulas more general, variables can be used with the summation notation. For example, means to sum up values of X from 1 to N where N can be any number but usually indicates the sample size. 1

2 Often an abbreviated form of the summation notation is used. For example, ΣX means to sum all the values of X. When only a subset of the values of X is to be summed then the full version is required. Thus, the sum of all elements of X except the first and the last (the N'th) would be indicated as: which would be read as the sum of X with i going from 2 to N-1. Some formulas require that each number be squared before the numbers are summed. This is indicated by: and is equal to = 174. The abbreviated version is simply: ΣX 2. It is very important to note that it makes a big difference whether the numbers are squared first and then summed or summed first and then squared. The symbol (ΣX) 2 indicates that the numbers should be summed first and then squared. For the present example, this equals: ( ) 2 = 26 2 = 676. This, of course, is quite different from 174. Sometimes a formula requires that the sum of crossproducts be computed. For instance, if 3 subjects were each tested twice, they might each have a score on X and on Y. Subject X Y The sum of cross products (2 x 3) + (1 x 6) + (4 x 5) = 32 can be represented in summation notation simply as: ΣXY. 2

3 Basic Rules for Sums The following data will be used to illustrate the rules: X Y (1) Sum Rule: Σ(X + Y) = ΣX + ΣY Σ(X + Y) = = 21 ΣX = = 9 ΣY = = 12 ΣX + ΣY = = 21 (2) Constant Term Rule: ΣaX = aσx (a is a constant) For an example, let a = 2. ΣaX = (2)(3) + (2)(2) + (2)(4) = 18 a ΣX = (2)(9) = 18 (3) Square Rule: Σ(X-M) 2 = ΣX 2 - (ΣX) 2 /N (N is the number of numbers, 3 in this case, and M is the mean which is also equal to 3 in this case. Σ(X-M) 2 = (3-3) 2 + (2-3) 2 + (4-3) 2 = 2 ΣX 2 = = 29 (ΣX) 2 /N = 9 2 /3 = 27 ΣX 2 - (ΣX) 2 /N = = 2 (4) Scalar Multiple Rule: (5) Linearity Rule: 3

4 (6) Subtotal Rule: If then (7) Dominance Rule: If for all then Arithmetic Series Arithmetic Series is sums of terms of arithmetic progressions. We have some useful summations formulae from the next theorems. Theorem If k is a real number, then Real-World Applications A simple application of summations is in the field of the Time Value of Money (TVOM) in economics. The concept of the time value of money is the notion that money has a time value associated with it, independent of the effects of inflation, and that this value can be quantified. The example of the practical application of TVOM concepts can be applied in the valuation of a fixed income security (i.e. a bond). Example What price should you pay for a bond with an 8% semiannual coupon, a $1,000 face and 10 years to maturity if you want a yield of 10%? 4

5 1.3.2 Binomial Theorem In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. Its simplest version reads whenever n is any non-negative integer, the numbers are the binomial coefficients, and n! denotes the factorial of n. For example, here are the cases n = 2, n = 3 and n = 4: Formula (1) is valid for all real or complex numbers x and y, and more generally for any elements x and y of a semiring as long as xy = yx. Newton's generalized binomial theorem Isaac Newton generalized the formula to other exponents by considering an infinite series: where r can be any complex number (in particular r can be any real number, not necessarily positive and not necessarily an integer), and the coefficients are given by In case k = 0, this is a product of no numbers at all and therefore equal to 1, and in case k = 1 it is equal to r, as the additional factors (r 1), etc., do not appear. Another way to express this quantity is 5

6 which is important when one is working with infinite series and would like to represent them in terms of generalized hypergeometric functions. The notation is the Pochhammer symbol. This form is vital in applied mathematics, for example, when evaluating the formulas that model the statistical properties of the phase-front curvature of a light wave as it propagates through optical atmospheric turbulence. A particularly handy but non-obvious form holds for the reciprocal power: The sum in (2) converges and the equality is true whenever the real or complex numbers x and y are "close together" in the sense that the absolute value x/y is less than one. The geometric series is a special case of (2) where we choose y = 1 and r = 1. Formula (2) is also valid for elements x and y of a Banach algebra as long as xy = yx, y is invertible and x/y < Double Summations Double summations arise in many contexts (as in the analysis of nested loops in computer programs). Example 1 Compute the following double sum. Example 2 Compute the following double sum. 6

7 Mathematical logic Mathematical logic is a subfield of mathematics that is concerned with formal systems in relation to the way that they encode intuitive concepts of mathematical objects such as sets and numbers, proofs, and computation. It is often divided into the subfields of model theory, proof theory, set theory and recursion theory. (1)Proposition Proposition is a mathematical statement such as "3 is greater than 4," "an infinite set exists," or "7 is prime." An axiom is a proposition that is assumed to be true. With sufficient information, mathematical logic can often categorize a proposition as true or false, although there are various exceptions (e.g., "This statement is false"). Mathematical propositions are not only characterized by the fact that they assert implications, but also by the fact that they contain variables. The notion of the variable is one of the most difficult with which Logic has to deal, and in the present work a satisfactory theory as to its nature, in spite of much discussion, will hardly be found. For the present, I only wish to make it plain that there are variables in all mathematical propositions, even where at first sight they might seem to be absent. Elementary Arithmetic might be thought to form an exception: 1+1=2 appears neither to contain variables nor to assert an implication. But as a matter of fact, the true meaning of this proposition is: If x is one and y is one, and x differs from y, then x and y are two. And this proposition both contains variables and asserts an implication. We shall find always, in all mathematical propositions, that the words any or some occur; and these words are the marks of a variable and a formal implication. Thus the above proposition may be expressed in the form: Any unit and any other unit are two units. The typical proposition 7

8 of mathematics is of the form φ(x, y, z,...) implies ψ(x, y, z,...), whatever values x, y, z,... may have; where φ(x, y, z,...) and ψ(x, y, z,...), for every set of values x, y, z,..., are propositions. It is not asserted that φ is always true, nor yet that ψ is always true, but merely that, in all cases, when φ is false as much as when φ is true, ψ follows from it. The distinction between a variable and a constant is somewhat obscured by mathematical usage. It is customary, for example, to speak of parameters as in some sense constants, but this is a usage which we shall have to reject. A constant is to be something absolutely definite, concerning which there is no ambiguity whatever. Thus, 1, 2, 3, e, π, Socrates, are constants; and so are man, and the human race, past, present and future, considered collectively. Proposition, implication, class, etc. are constants; but a proposition, any proposition, some proposition are not constants, for these phrases do not denote one definite object. And thus what are called parameters are simply variables. Take, for example, the equation ax + by + c = 0, considered as the equation for a straight line in a plane. Here we say that x and y are variables, while a, b, c are constants. But unless we are dealing with one absolutely particular line, say the line from a particular point in London to a particular point in Cambridge, our a, b, c are not definite numbers, but stand for any numbers, and are thus also variables. And in Geometry nobody does deal with actual particular lines; we always discuss any line. The point is that we collect the various couples x, y into classes of classes, each class being defined as those couples that have a certain fixed relation to one triad (a, b, c). But from class to class, a, b, c also vary, and are therefore properly variables. (2) Implications (3) Necessay and Sufficient A necessary and sufficient condition, then, is one which when working with others, must happen, and is all that needs to happen, for something else to be the case. A necessary condition is one that must be satisfied for the result to happen. Breathing is necessary to stay alive; if you did not breathe, you would not stay 8

9 alive. Breathing is not sufficient to stay alive, for if you did nothing but breathe, you could still die. A sufficient condition is one that, if it is satisfied, the result is certain to happen. Jumping is sufficient to leave the ground, since the act of jumping causes you to leave the ground. Jumping is not necessary to leave the ground however, since one could step onto a ladder and leave the ground in a way which isn't jumping. Sometimes conditions can be both necessary and sufficient. For example, "Being the First of July " is both necessary AND sufficient for "being the Canada National Day". "Necessary and sufficient" is another way of saying the logical statement "if and only if" (sometimes abbreviated to "iff"). Necessary conditions To say that P is necessary for Q is to say that "if P is not true, then Q is not true". By contraposition, this is the same thing as "whenever Q is true, so is P". The logical relation between them is expressed as "If Q then P" or "Q P" (Q implies P), and may also be seen as "P, if Q", "P whenever Q" or "P when Q". Example 1: Consider the statement "Being a rectangle is necessary for being a square." Here if you are not a rectangle then it is impossible for you to be a square. That is, if you are a square, then you are also automatically a rectangle. Example 2: Suppose that any lightning bolt causes thunder (however quiet the thunder may be) and suppose that by "thunder" we mean the sound caused by lightning (not any other loud rumbling). Then it might be said "thunder is necessary for lightning", for if there is absolutely no thunder, then there cannot be any lightning. That is, if lightning does occur, then it must create some thunder. Example 3: As an example of something NOT being a necessary condition, consider the rectangle/square example. Notice that being a square is NOT a necessary condition for being a rectangle, since there are rectangles which are not squares. 9

10 Sufficient conditions To say that P is sufficient for Q is to say that P being true forces Q to be true, or whenever P occurs, Q occurs. The logical relation is expressed as "If P then Q" or "P Q", and may also be seen as "P implies Q." Example 1: For simplicity, let us suppose everyone is biologically male or female, and that a "father" is a biological male who has fathered a child. Then "being a father is sufficient for being male". Example 2: As in the previous section, let us define "thunder" as the sound that lightning creates. Then "thunder is sufficient for lightning." For if one hears thunder, then some lightning must have occurred in order to create the thunder. Example 3: As an example of a condition being NOT sufficient, consider the "male/father" example. Being male is NOT sufficient for being a father, since there are males which are not yet fathers. Relationship between "Necessary" and "Sufficient" The statement that "P is sufficient for Q" is the same as "Q is necessary for P", for both statements are the same as "P implies Q". Example: Recall that "Being a rectangle is necessary for being a square". Also, "being a square is sufficient for being a rectangle." Necessary and sufficient conditions To say that P is necessary and sufficient for Q is to say two things: (i) P is necessary for Q (Q P) (ii) P is sufficient for Q (P Q) For example, if Alice always eats steak on Monday, but never on any other day, it can be said "being Monday is a necessary condition for Alice eating steak." This is so since 10

11 Alice does not eat steak on days that are not Monday. Also, "being Monday is a sufficient condition for Alice eating steak." This is true since Alice always eats steak on Monday. Consider the thunder/lightning example as outlined in previous sections. "Thunder is necessary for lightning", since absolutely no thunder means there isn't any lightning to create any noise. "Thunder is sufficient for lightning" since thunder (as we have narrowly defined it) must have originated from some lightning. The relationship between being a square and being a rectangle is one which is NOT "necessary and sufficient" despite the ordering of the conditions "square" and "rectangle". "Being a rectangle is necessary for being a square", yet "being a rectangle is NOT sufficient for being a square". "Being a square is sufficient for being a rectangle", yet "being a square is NOT necessary for being a rectangle." "P is necessary and sufficient for Q" expresses the same thing as "P if and only if Q" (P Q). (3) Mathematical Proof In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. A proof is a logical argument, not an empirical one. That is, one must demonstrate that a proposition is true in all cases before it is considered a theorem of mathematics. An unproven proposition for which there is some sort of empirical evidence is known as a conjecture. Some common proof techniques are: Direct proof: where the conclusion is established by logically combining the axioms, definitions and earlier theorems Proof by induction: where a base case is proved, and an induction rule used to prove an (often infinite) series of other cases 11

12 Proof by contradiction (also known as reductio ad absurdum): where it is shown that if some statement were false, a logical contradiction occurs, hence the statement must be true. Proof by construction: (also known as proof by example) constructing a concrete example with a property to show that something having that property exists. Proof by exhaustion: where the conclusion is established by dividing it into a finite number of cases and proving each one separately An existence or nonconstructive proof is one that establishes that a certain mathematical object must exist (e.g. "Some X satisfies f(x)"), without explaining how such an object can be found. Often, this takes the form of a proof by contradiction in which the nonexistence of the object is proven to be impossible. In contrast, a constructive proof establishes that a particular object exists by providing a method of finding it. Deductive vs. Inductive Reasoning "Deductive reasoning" refers to the process of concluding that something must be true because it is a special case of a general principle that is known to be true. For example, if you know the general principle that the sum of the angles in any triangle is always 180 degrees, and you have a particular triangle in mind, you can then conclude that the sum of the angles in your triangle is 180 degrees. Deductive reasoning is logically valid and it is the fundamental method in which mathematical facts are shown to be true. "Inductive reasoning" (not to be confused with "mathematical induction" or and "inductive proof", which is something quite different) is the process of reasoning that a general principle is true because the special cases you've seen are true. For example, if all the people you've ever met from a particular town have been very strange, you might then say "all the residents of this town are strange". That is inductive reasoning: constructing a general principle from special cases. It goes in the opposite direction from deductive reasoning. 12

13 Inductive reasoning is not logically valid. Just because all the people you happen to have met from a town were strange is no guarantee that all the people there are strange. Therefore, this form of reasoning has no part in a mathematical proof. However, inductive reasoning does play a part in the discovery of mathematical truths. For example, the ancient geometers looked at triangles and noticed that their angle sums were all 180 degrees. After seeing that every triangle they tried to build, no matter what the shape, had an angle sum of 180 degrees, they would have come to the conclusion that this is something that is true of every triangle. Then they would have looked for a way to prove it using deductive reasoning; that is, deduce it as a consequence of other known general properties of triangles. In summary, then: inductive reasoning is part of the discovery process whereby the observation of special cases leads one to suspect very strongly (though not know with absolute logical certainty) that some general principle is true. Deductive reasoning, on the other hand, is the method you would use to demonstrate with logical certainty that the principle is true Set Theory Set Theory is the mathematical science of the infinite. It studies properties of sets, abstract objects that pervade the whole of modern mathematics. The language of set theory, in its simplicity, is sufficiently universal to formalize all mathematical concepts and thus set theory, along with Predicate Calculus, constitutes the true Foundations of Mathematics. The objects of study of Set Theory are sets. (1) Set A set is a collection of objects considered as a whole. The objects of a set are called elements or members. The elements of a set can be anything: numbers, people, letters of the alphabet, other sets, and so on. Sets are conventionally denoted with capital letters, A, B, C, etc. Two sets A and B are said to be equal, written A = B, if they have the same members. 13

14 As opposed to a multiset, a set cannot contain multiple copies of an element. Some sets may be described in words, for example: A is the set whose members are the first four positive whole numbers. B is the set whose members are the colors of the French flag. By convention, a set can also be defined by explicitly listing its elements between braces (sometimes called curly brackets or curly braces), for example: C = {4, 2, 1, 3} D = {red, white, blue} Two different descriptions may define the same set. For example, for the sets defined above, A and C are identical, since they have precisely the same members. The shorthand A = C is used to express this equality. Similarly, for the sets defined above, B = D. Set identity does not depend on the order in which the elements are listed, nor on whether there are repetitions in the list. For example, {6, 11} = {11, 6} = {11, 11, 6, 11}. For sets with many elements, an abbreviated list is sometimes used. For example, the first one thousand positive whole numbers can be described using the symbolic shorthand: {1, 2, 3,..., 1000}, where the ellipsis (...) indicates that the list continues in the obvious way. Similarly the set of even numbers can be described by the notation: {2, 4, 6, 8,... }. More complicated sets are sometimes described by a different notation. For example the set F, whose members are the first twenty numbers which are four less than a square integer, can be described using the following: 14

15 F = {n 2 4 : n is an integer; and 0 n 19} In this description, the colon (:) means "such that", and the mathematician interprets this description as F is the set of numbers of the form n 2 4, such that n is a whole number in the range from 0 to 19 inclusive. (Sometimes the pipe notation is used instead of the colon.) (2) Set membership If something is or is not an element of a particular set then this is symbolised by respectively. So, for example, with respect to the sets defined above: and and (since 285 = 17² 4); but and. (3) Cardinality of a set Each of the sets described above has a definite number of members; for example, the set A has four members, while the set B has three members. A set can also have zero members. Such a set is called the empty set (or the null set) and is denoted by the symbol ø. For example, the set A of all three-sided squares has zero members, and thus A = ø. Like the number zero, though seemingly trivial, the empty set turns out to be quite important in mathematics. A set can also have an infinite number of members; for example, the set of natural numbers is infinite. (4) Subsets If every member of the set A is also a member of the set B, then A is said to be a subset of B, written, also pronounced A is contained in B. Equivalently, we can write, read as B is a superset of A, B includes A, or B contains A. The relationship between sets established by is called inclusion or containment. 15

16 If A is a subset of but not equal to B, then A is called a proper subset of B, written (A is a proper subset of B) or (B is proper superset of A). However, in some literature these symbols are read the same as and, so it's often preferred to use the more explicit symbols and for proper subsets and supersets. Examples: A is a subset of B The set of all men is a proper subset of the set of all people. The empty set is a subset of every set and every set is a subset of itself: (5) Set Operations Unions There are several ways to construct new sets from existing ones. Two sets can be "added" together. The union of A and B, denoted by A B, is the set of all things which are members of either A or B. The union of A and B 16

17 Some basic properties of unions: A B = B A A is a subset of A B A A = A A ø = A Intersections A new set can also be constructed by determining which members two sets have "in common". The intersection of A and B, denoted by A B, is the set of all things which are members of both A and B. If A B = ø, then A and B are said to be disjoint. Some basic properties of intersections: A B = B A A B is a subset of A A A = A A ø = ø Complements The intersection of A and B Two sets can also be "subtracted". The relative complement of A in B (also called the set theoretic difference of B and A), denoted by B \ A, (or B A) is the set of all elements which are members of B, but not members of A. Note that it is valid to "subtract" members of a set that are not in the set, such as removing green from {1,2,3}; doing so has no effect. In certain settings all sets under discussion are considered to be subsets of a given universal set U. In such cases, U A, is called the absolute complement or simply complement of A, and is denoted by A. 17

18 The relative complement of A in B Mathematical Induction The complement of A in U Mathematical induction is a technique for proving a statement -- a theorem, or a formula -- that is asserted about every natural number. For example, n = ½ n (n + 1) This asserts that the sum of consecutive numbers from 1 to n is given by the formula on the right. Now we can test the formula for any given number, say n = 3: = ½ 3 4 = 6 -- which is true. It is also true for n = 4: = ½ 4 5 = 10 But how are we to prove this rule for every natural number? The method of proof follows from the following, which is called the principle of mathematical induction. If 1) when a statement is true for any specific natural number k, then it is also true for its successor, k + 1, 18

19 and 2) the statement is true for 1, then the statement is true for every natural number. For, when the statement is true for 1, then according to 1), it will also be true for 2. But that implies it will be true for 3; which implies it will be true for 4. And so on. It will be true for every natural number. To use the principle of induction, here is the procedure: To prove a statement that is asserted about every natural number n, prove two things: 1) If the statement is true for n = k, then it will be true for its successor, n = k ) The statement is true for n = 1. The hypothesis of Step 1) -- "The statement is true for n = k" -- is called the induction assumption, or the induction hypothesis. It is what we assume when we prove a theorem by induction. Example 1. Prove that the sum of the first n natural numbers is given by this formula: S(n) = n = n(n + 1) 2 We will call this statement S(n), because it depends on n. Example 2. Prove this rule of exponents: for every natural number n. (ab) n = a n b n, 19

20 Example 3. Prove this formula for the sum of consecutive cubes: = n²(n + 1)² 4 20