Chapter One Logic and Sets 1.1 INTRODUCTION Given positive integers m and n, we say that m is a factor of n provided n = mq for some positive integer q. In particular, n is a factor of itself, since n = n 1. If m is a factor of n and m < n, then m is called a proper factor of n. For example, the proper factors of 6 are 1, 2, and 3, and the proper factors of 50 are 1, 2, 5, 10, and 25. The integer 6 has the interesting property that it is equal to the sum of its proper factors, that is, 6 = 1 + 2 + 3. Numbers having this property are called perfect numbers. In fact, 6 is the smallest perfect number. Are there any others? Recall that a positive integer p > 1 is said to be a prime number if 1 is its only proper factor. The ten smallest prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. Note that no prime number p can be perfect, since the only proper factor of p is 1, and 1 < p. In general, a positive integer n is called deficient provided it is greater than the sum of its proper factors; thus, every prime is deficient. What about a positive integer n that is the product of two distinct primes, such as 6 = 2 3, 10 = 2 5, and 15 = 3 5? We see that 6 is perfect, but that both 10 and 15 are deficient. In general, if n = p 1 p 2, where p 1 and p 2 are distinct primes with p 1 < p 2, then the proper factors of n are 1, p 1, and p 2, and 1 + p 1 + p 2 < p 1 p 2, unless p 1 = 2 and p 2 = 3. Thus, except for 6, any n that is the product of two distinct primes is deficient. Another easy result is that any number that is the square of a prime, such as 4, 9, 25, and 49, is deficient. In fact, we can state the following more general result, whose proof is left to Exercise 4. Theorem 1.1: Any power of any prime is deficient; that is, if n = p k for some prime p and some positive integer k, then n is deficient. It follows from what we have said so far that, of the numbers less than 30, each of the following is deficient: 2, 3, 4, 5, 7,8, 9, 10, 11, 13, 14, 15, 16, 17,19,21,22,23,25,26,27,29 (It probably also makes sense to classify 1 as deficient, since 1 has no proper factors.) So now, let s check the numbers between 1 and 30 not on the above list, besides 6, which we know is perfect. First of all, for 12, we see that its proper factors are 1, 2, 3, 4, and 6, and that 1+2+3+4+6 = 16 > 12.
2 Chapter 1 Logic and Sets So 12 falls into a third category numbers which are less than the sum of their proper factors. Such numbers are said to be abundant. Next, for 18, its proper factors are 1, 2, 3, 6, and 9, and 1 + 2 + 3 + 6 + 9 = 21 > 18. Hence, 18 is abundant. Similarly, 20 and 24 are abundant. Next comes 28. Its proper factors are 1, 2, 4, 7, and 14, and 1 + 2 + 4 + 7 + 14 = 28. Therefore, 28 is the second smallest perfect number! One could, perhaps with the aid of a computer, continue to search for perfect numbers. This has been done, with the following results: (1) 28 is the only 2-digit perfect number; (2) 496 is the only 3-digit perfect number; (3) 8128 is the only 4-digit perfect number; (4) there are no k-digit perfect numbers for k = 5, 6, 7; in fact, the next perfect number after 8128 is 33550336. So perfect numbers are rare indeed! Perfect numbers have been studied at least since the time of the Greek mathematician Pythagoras and his followers in the sixth century B.C. They thought that such numbers had magical properties, and certainly must have wondered whether there are infinitely many perfect numbers. In other words, is there a largest perfect number, or does the list of perfect numbers go on forever? This is an example of the kind of question with which mathematicians are concerned. Questions such as this may arise during seminars with colleagues or from reading journal articles. Often, questions arise from attempts to settle other questions, perhaps in the context of some application. As with any science, mathematicians use experimental evidence to help them form and test questions. Let s look at the four smallest perfect numbers and see if we can find any pattern to them. An obvious thing to try (for a mathematician!) is to factor each perfect number as a product of primes, since it can be shown that all of a number s proper factors are determined from its prime factors. Here s what we get: 6 = 2 1 3 28 = 2 2 7 496 = 2 4 31 8128 = 2 6 127 We see that each of these perfect numbers is the product of a power of 2 and a prime. Furthermore, focusing on the primes 3, 7, 31, and 127, we see that each of these is one less than a power of two; that is: 6 = 2 1 (2 2 1) 28 = 2 2 (2 3 1) 496 = 2 4 (2 5 1) 8128 = 2 6 (2 7 1) The above pattern was observed by the Greek mathematician Euclid [ca. 300 BC], and he is credited with the following result. Theorem 1.2: For a positive integer k, if 2 k 1 is prime, then n = 2 k 1 (2 k 1) is a perfect number. This result is not difficult to verify and is left to Exercise 10. About 2000 years later, the great Swiss mathematician Leonhard Euler (1707-1783) proved the following related result.
1.1 Introduction 3 Theorem 1.3: If n is an even perfect number, then n = 2 k 1 (2 k 1) for some positive integer k, with 2 k 1 a prime. In view of the above theorems, the search for even perfect numbers reduces to the search for prime numbers of the form 2 k 1. Such primes are called Mersenne primes, after the French number theorist and friar, Marin Mersenne (1588-1638). Mersenne knew that, in order for 2 k 1 to be prime, it is necessary that k be prime (see Exercise 6). In a book that he wrote he stated without proof that 2 k 1 is prime for the following values of k: 2, 3, 5, 7, 13,17,19, 31, 67, 127, 257 and that these were the only values of k 257 for which 2 k 1 is prime. It turns out that Mersenne made five errors (remember, he had to do all of his calculations by hand!): 2 61 1 is prime; 2 67 1 is not prime; 2 89 1 is prime; 2 107 1 is prime; 2 257 1 is not prime. Thus, before computers, only twelve Mersenne primes, and hence only twelve even perfect numbers, were known. Then, in 1952, five more perfect numbers were found, corresponding to the Mersenne primes 2 521 1, 2 607 1, 2 1279 1, 2 2203 1, 2 2281 1 Another was found in 1957, and two more in 1961, using an IBM 7090 mainframe computer. In 1997, Englishman Gordon Spence, using an algorithm developed by George Woltman, a 39-year-old programmer from Florida, found the Mersenne prime 2 2976221 1 This prime has 895932 digits; if printed, it would fill a 450-page paperback book! According to Wikipedia, as of August 2007, there were 44 known Mersenne primes, and hence a total of 44 known even perfect numbers. The largest known prime number is a Mersenne prime, namely 2 32582657 1 It was discovered via the Great Internet Mersenne Prime Search (GIMPS), a distributed computing project on the Internet that anyone with a computer can participate in. If you re interested, check out www.mersenne.org/works.htm What about odd perfect numbers? Well, so far, none have been found. Through a combination of theory and computation, it is known that there are no odd perfect numbers having less than 300 digits! When a mathematical assertion is thought to be true, and when the evidence supports this belief, then the assertion is called a conjecture. For example, it is conjectured that there are infinitely many Mersenne primes, and hence that there are infinitely many even perfect numbers. Also, it is conjectured that there are no odd perfect numbers. We would like to state the conjecture about Mersenne primes more precisely in order to facilitate our discussion of it. Conjecture 1.4: For every positive integer n, there exists a Mersenne prime greater than n. Suppose this conjecture has been checked for all values of n up to n = 10 100, or even up to n = 10 1000. Does this mean it is true? Not to a mathematician, and this is what separates
4 Chapter 1 Logic and Sets mathematics from the other sciences. In order for a conjecture to be accepted as true, namely, for it to attain the higher status of a theorem, the statement must be deduced logically from basic assumptions and other accepted facts. That is, theorems must be proved. Logic furnishes a set of ground rules for analyzing mathematical assertions and for determining whether a proposed proof of an assertion is valid. These rules are one aspect of proof that can be learned; much of the business of proof involves intuition, creativity, imagination, and instinct. The deductive method is important in computer science, as well, where logic is applied to the process of designing, coding, and testing software systems, with the aim of ensuring that such systems perform as specified. There is also a relatively new programming methodology, called logic programming, that explicitly embodies the ideas of mathematical logic; one popular language for logic programming is called Prolog. It has been used to program expert systems programs that simulate the deductive analysis of the human expert in some narrow domain, such as that of medical diagnosis. As logic is introduced in this chapter, emphasis will be placed on certain common forms of exposition and reasoning, apart from any particular applications. Students who learn these well will find it easier to follow the development of this and subsequent courses in the mathematical sciences. Knowing the rules of logic the rules of the mathematical game allows students to focus on the content of courses and not be thrown off track or distracted by the logical forms being applied. Hereafter, the term argument is used in its mathematical sense, as a logical discussion that establishes the validity of some mathematical fact. We ask, then, What is the logic of an argument? Roughly speaking, the logic of an argument is what is left over when the particular meaning of the argument is ignored. In other words, the logic of an argument is its form or syntax. As an example, let us return to Conjecture 1.4. This statement has the form where P(n) represents the statement For every n, P(n). There exists a Mersenne prime greater than n. This P(n) is an example of a propositional function, and the variable n is allowed to have any value from some set of permissible values, in this case from the set of positive integers. Lots of other statements have this same logical form. For instance, an instructor returning an examination to a class might make the following statement: Every student missed the third question. This statement again has the form For every n, P(n). where P(n) represents the statement n missed the third question. Here the variable n may be replaced by (the name of) any student in the class. What would it mean for a statement of this form to be false? Let s take the instructor s statement first. The assertion that every student missed the third question on the exam is false provided one or more students got the question right. Namely, the statement is false provided the statement Every student missed the third question. There is some student who got the third question right.
1.1 Introduction 5 is true. We say that the two statements are logical negations of each other. In general, as will be discussed later in this chapter, the statement is false provided the statement For every n,p(n). There is some n such that P(n). is true, where the notation P(n) denotes the logical negation of P(n). Returning to Conjecture 1.4, recall that it also has the logical form For every n,p(n). Thus, Conjecture 1.4 is false provided the statement There is some n such that P(n). is true. In this case it can be determined that P(n) is the statement It is not the case that there exists a Mersenne prime greater than n. Again using techniques to be developed later in this chapter, we can simplify P(n) as follows: Every Mersenne prime is less than or equal to n. Putting this all together, we have determined that the negation of the Conjecture 1.4 is the following statement: There is some positive integer n such that every Mersenne prime is less than or equal to n. which is just a long-winded way of saying that there are only finitely many Mersenne primes. The point to be made from the above discussion is that the abstract logical form of a mathematical statement or argument is independent of its particular content. The rules of logic provide a means for analyzing statements and for determining whether mathematical arguments are valid or not. It is important to understand the meaning of the term statement, as it is used in symbolic logic. Sentences such as Everyone will pass this class. or The number 72 is positive. are declarative sentences each makes an assertion. On the other hand, the question or the exclamation Is 437 a prime number? Holy cow! are not declarative sentences. For purposes of mathematical logic, our interest is in declarative sentences that are either true or false; these are called statements or propositions. If a statement is true, then its truth-value is denoted T, whereas the truth-value of a false statement is denoted F.
6 Chapter 1 Logic and Sets and For instance, each of the sentences The quotient obtained when 7 is divided by 3 is an odd integer. The integer 6 is a factor of the product of 117 and 118. is a statement. In fact, the first is false and the second is true. Next, consider the sentences The number x is positive. and He is a baseball pitcher. These are not statements because, as they are presented, we cannot determine the truth-value of either one. If we replace x by 3 in the first sentence, then we obtain the (false) statement The number 3 is positive. Similarly, we may replace He in the second sentence by Josh Beckett to obtain the (true) statement Josh Beckett is a baseball pitcher. Symbols like x in the example above are called variables; such symbols are used to represent any one of a number of permissible values. Later on we will say more about sentences with variables that become statements when the variables are given particular values. Lest the reader be misled, it should be pointed out that some sentences with variables are statements. For instance, the sentence is a (true) statement. We also accept as statements sentences like or For every real number x, if x > 2, then x 3 > 8. The billionth digit in the decimal expansion of 2 is 7. Every even integer greater than 2 can be expressed as the sum of two primes. We take the attitude that such sentences are either true or false, although we may not know the truth-value at the present time. The truth-value of the first sentence can, at least in theory, be computed. The second sentence is another example of a mathematical conjecture; it is called the Goldbach conjecture. There exists a great deal of empirical evidence for its truth, but as yet it has not been proven. We have said that an understanding of logic is central to the process of doing mathematics. Another central idea that we shall use throughout is that of a set. A set can be thought of as a collection of objects. The Greek alphabet, a baseball team, and the Euclidean plane are all examples of sets. The Greek alphabet consists of letters, a baseball team is composed of its players, and the Euclidean plane is made up of points. In general, the objects that make up a set are referred to as its elements or members. It is common practice to think of the terms set and member as undefined or primitive notions. Most of us have an intuitive understanding (based on examples like those just given) of what these terms mean.
1.1 Introduction 7 Sets are the building blocks for most mathematical structures. For example, Euclidean plane geometry is based on the interpretation of the Euclidean plane as a set of points. Other important examples arise in the field of abstract algebra, which studies the properties of sets on which one or more binary operations are defined. For example, ordinary addition is a binary operation on the set of positive integers. Certain special sets of numbers are used frequently in mathematics, so some special notation has been developed for them. The numbers 1, 2, 3,... are called the positive integers, and the notation Z + is used to denote the set of positive integers. The positive integers, together with the number 0 and the negative integers 1, 2, 3,... form the set of integers; this set is denoted by Z. A rational number is any number expressible in the form m/n, where m and n are integers and n 0. For example, 2/3, 5/11, 17 = 17/1, and.222... = 2/9 are rational numbers. The set of rational numbers is denoted by Q. The numbers 2, 3 5, and π are not rational; in general, such numbers are called irrational numbers. The rational numbers and the irrational numbers together make up the set of real numbers, which is denoted R. To repeat, then, we adopt the following notational conventions: Z + = the set of positive integers Z = the set of integers Q = the set of rational numbers R = the set of real numbers Another notation that is commonly used is N to denote the set of natural numbers. Unfortunately, mathematical scientists are not in agreement as to what this set is. To some, the set of natural numbers is the same as the set of positive integers; that is, the natural numbers are the counting numbers 1, 2, 3.... Others want to include 0 as a natural number. For example, in the Ada programming language there is a predefined type NATURAL (actually, a subtype of the type INTEGER) whose domain is the set 0, 1, 2, 3,.... Our approach in this book is to simply avoid using the notation N and the terminology natural number. But we do want to alert you to the potential confusion with this terminology. The basic relationship between an element x and a set A is that of membership. If x is an element of A, then we write x A, and if x is not an element of A we write x / A. For instance, 3 Z, 3 / Z +, 2 R, and 2 / Q. Except for certain special sets like R, sets are denoted in this book by upper case italic letters such as A, B, C and elements by lower case italic letters such as a, b, c. If a set A consists of a small number of elements, then we can exhibit A by explicitly listing its elements between braces. For example, if A is the set of odd positive integers less than 16, then we write A = {1, 3, 5, 7, 9, 11, 13, 15} However, some sets contain too many elements to be listed in this way. In many such cases, the three-dot (or ellipsis) notation is used to mean and so on or and so on up to, depending on the context. For instance, the set Z + can be exhibited as the set Z as and the set B of integers between 17 and 93 as Z + = {1, 2, 3,...} Z = {..., 3, 2, 1, 0, 1, 2, 3,...} B = {17, 18, 19,..., 93}
8 Chapter 1 Logic and Sets Often a set A is described as consisting of those elements x in some set B that satisfy a specified property. As an example, let E be the set of even integers. Then we may write This set can also be described in the form E = {..., 4, 2, 0, 2, 4,...} E = {m Z m is even} We read {m Z m is even} as the set of all m in the set Z such that m is even. Here, the symbol translates as such that or for which. Note that every positive integer is an integer, every integer is a rational number, and every rational number is a real number. In general, for two sets A and B, it is possible that each element of A is also an element of B. Definition 1.1: For sets A and B, A is called a subset of B, denoted A B, provided every element of A is also an element of B. Given a subset S of the set of real numbers, it is frequently useful to be able to denote the set of positive elements of S, the set of negative elements of S, and the set of nonzero elements of S. For this we adopt the following notation: S + = {x S x > 0} S = {x S x < 0} S = {x S x 0} Thus, for example, we use Z + to denote the set of positive integers, Q + to denote the set of positive rational numbers, R to denote the set of negative real numbers, and R to denote the set of nonzero real numbers. Also, given a subset S of the real numbers and a real number c, we adopt the following useful notation: c + S = {c + x x S} So, for example, cs = {cx x S} 2Z = 2 {..., 3, 2, 1, 0, 1, 2, 3,...} = {..., 6, 4, 2, 0, 2, 4, 6,...} denotes the set of even integers, and 1 + 2Z = 1 + {..., 6, 4, 2, 0, 2, 4, 6,...} = {..., 5, 3, 1, 1, 3, 5, 7,...} denotes the set of odd integers. Notice then that an integer m is even provided m = 2q for some integer q. Similarly, an integer m is odd provided m = 2k + 1 for some integer k. In your previous courses in mathematics, you have already encountered the special subsets of R called intervals. These are used frequently in this book and we review the standard notation for them here. Let a and b be real numbers with a < b. Then
1.1 Introduction 9 (a, b) = {x R a < x < b} [ a, b ] = {x R a x b} (a, b ] = {x R a < x b} [ a, b) = {x R a x < b} The interval (a, b) is called an open interval, [ a, b ] is called a closed interval, and both (a, b ] and [ a, b) are called half-open intervals. In each case a and b are called the endpoints of the interval. Intervals can also be unbounded; the possible forms (where the symbol denotes infinity ) are (a, ) = {x R a < x} [ a, ) = {x R a x} (, b) = {x R x < b} (, b ] = {x R x b} (, ) = R In a particular mathematical discussion involving sets, it is usually assumed or understood that all sets under consideration are subsets of some specified set U. This set U is called a universal set. In a calculus class, for instance, it may be that U is the set of real numbers, whereas in a combinatorics lecture it may be that U is the set of integers. Exercise Set 1.1 1. Indicate which of the following are statements. (a) The integer 24 is even. (b) Is the integer 3 15 1 even? (c) The product of 2 and 3 is 7. (d) The sum of x and y is 3. (e) If the integer x is odd, is x 2 odd? (f) It is not possible for 3 15 1 to be both even and odd. (g) The product of x 2 and x 3 is x 6. (h) The integer 2 524287 1 is prime. 2. Verify that the following formula holds for any number r 1 and any positive integer k: 1 + r + r 2 + + r k 1 = rk 1 r 1 (Hint: Let s denote the sum on the left-hand side of the formula, and consider rs s.) 3. Write each of the sets by listing its elements. (a) A = {m Z 4 < m < 5} (b) B = {n Z + 4 < n < 5} (c) C = {x R x 3 x 2 2x = 0} (d) D = {x Q x 4 6x 2 + 8 = 0} 4. Prove Theorem 1.1. (Hint: Use the result of Exercise 2.) 5. Write each of the sets in the form {m Z p(m)}, where p(m) is some property of the integer m; for example, {..., 4, 2, 0, 2, 4,...} = {m Z m is even}.
10 Chapter 1 Logic and Sets (a) {..., 3, 2, 1} (b) {0, 1, 4, 9, 16,...} (c) {..., 27, 8, 1, 0, 1, 8, 27,...} (d) {..., 8, 4, 0, 4, 8,...} (e) {..., 15, 9, 3, 3, 9, 15,... } 6. Let x denote a real number and let a, b, d, and k denote positive integers. (a) Apply the result of Exercise 2 to show that (b) Use the result of part (a) to verify that x d 1 = (x 1)(x d 1 + x d 2 + + x + 1) 2 ab 1 = (2 a 1)(2 a(b 1) + 2 a(b 2) + + 2 a + 1) (c) Use the result of part (b) to show that, if k is not prime, then 2 k 1 is not prime. 7. Write each of the sets using the notations S +, S, cs, and/or c + S. (a) {..., 3, 2, 1} (b) {2, 4, 6, 8,...} (c) {..., 9, 4, 1, 6, 11,...} (d) {..., 8, 4, 0, 4, 8,...} (e) {..., 15, 9, 3, 3, 9, 15,... } 8. Verify that 496 is perfect as follows. First, (use the result of Exercise 2 to) find the sum s 1 of the factors of 496 that are powers of 2. Second, find the sum s 2 of the factors of 496 that have the form of a power of 2 times 31 (including 496 itself). Third, verify that s 1 + s 2 = 2(496) = 992. 9. Write each of the following subsets of Z by listing the elements. (a) {m Z m [ 3, 3 ] } (b) {m Z m [ 3, 3)} (c) {m Z m ( 3, )} (d) {m Z m (, 3 ] } 10. Prove Theorem 1.2: If m = 2 k 1 is prime, then n = m(m + 1)/2 is a perfect number. To do this, generalize the procedure used in Exercise 8 for the special case k = 5. 11. Write each of the sets in an alternate, yet equivalent, form. (a) 3 + Z + (c) πz (b) 1 + Z (d) 2Q 12. Verify that 2 11 1 is not prime. 13. Write each of the following subsets of R using interval notation. (a) {x 2 < x < 3} (b) {x 4 < x 9} (c) {x 1 x 5} (d) {x 1 < x} (e) {x x < 0} 14. Find, explicitly, the 7 smallest perfect numbers.
1.2 Logical Connectives 11 1.2 LOGICAL CONNECTIVES In research seminars, in classrooms, or at roadside pubs, mathematicians and their students (and perhaps other people, as well) are frequently interested in determining the truth-value of some given mathematical statement. Many mathematical statements are formed by using the words not, or, and, if then, and if and only if to combine simpler statements. These are called the logical connectives (or simply connectives) and are defined in this section. In this chapter, lower-case italic letters, such as p, q, r, or subscripted versions of these letters, such as p 1, q 2, r 3, are used to denote or represent statements; these are called propositional variables. There are two types of statements which one deals with in logic. A simple (or primitive) statement is any statement which contains neither logical connectives nor any other statement as a constituent part. Statements which are not simple are called compound statements; in other words, a compound statement is one which contains either logical connectives or at least two simple statements. The symbolic representation of a primitive or compound statement is called a formula. Definition 1.2: Let p and q be formulas. 1. The disjunction of p and q is the compound statement p or q It is true provided at least one of p or q is true; otherwise it is false. We denote the disjunction of p and q by p q 2. The conjunction of p and q is the compound statement p and q It is true provided both p and q are true; otherwise it is false. We denote the conjunction of p and q by p q 3. The negation of p is the statement not p It has truth-value opposite that of p and is denoted by p. (Two alternate notations for p are p and p.) The truth-value of a given statement is determined from the truth-values of the simple statements of which it is composed. We will demonstrate how a truth table is used to examine a given statement. Such a table contains a column for each propositional variable in the statement and a column for the whole statement. The table has a row corresponding to each possible combination of truth-values for the propositional variables involved. If the statement is particularly complex, additional propositional variables may be introduced to represent parts of it, and then the table contains a column for each of these as well.
12 Chapter 1 Logic and Sets Tables 1.1 (a), (b), and (c) show the truth tables for disjunction, conjunction, and negation. Note that there are four possible combinations for the truth-values of two propositional variables p and q. (a) p q p q T T T T F T F T T F F F (b) p q p q T T T T F F F T F F F F (c) p T F p F T Tables 1.1 Truth tables for p or q, p and q, and for not p Example 1.1: For which integers m is the condition satisfied? m > 2 and m < 3 Solution: The condition m > 2 is satisfied by the integers 1, 0, 1, 2, 3, 4,..., while the condition m < 3 is satisfied by..., 3, 2, 1, 0, 1, 2. We want those integers that satisfy both conditions, namely, those that are both greater than 2 and less than 3. There are four such values of m: 1, 0, 1, and 2. Example 1.2: The Boston Red Sox, Cleveland Indians, Detroit Tigers, and New York Yankees are professional baseball teams. Suppose on a given night the Red Sox play the Yankees and the Indians play the Tigers. Both games are completed, and the next morning someone makes the statement Find the negation of this compound statement. The Tigers or the Red Sox won last night. Solution: The given compound statement is of the form p q, where p represents the statement and q represents the statement The Tigers won last night. The Red Sox won last night. According to its definition, p q is false provided p is false and q is false. Thus, the negation of p q is true provided both p and q are false, that is, provided both p and q are true. It follows that the negation of the given compound statement is Both the Tigers and the Red Sox lost last night. In the next section we show that, in general, the negation of the formula p q is the formula p q.
1.2 Logical Connectives 13 Consider next the compound statement If p, then q. How is the truth-value of such a statement determined from the truth-values of the statements p and q? An example should help to clarify the situation and motivate the general definition. Example 1.3: Consider the statement If you score 70 or better on the final exam, then you will pass the course. which an instructor might make to a student. We let p represent the statement and q represent the statement The student scored 70 or better on the final exam. The student passed the course. Then the instructor s statement (in its past-tense form) is represented by the formula If p, then q. Let us analyze this statement for each of the four possible truth-value combinations of p and q. (Assume the semester is over and the student s final exam score and course grade are known.) Case 1: Both p and q are true. In this case the student scored 70 or better on the final exam and did pass the course, just as the instructor promised. Certainly the instructor s statement is true in this case. Case 2: p is true and q is false. Here, the student scored 70 or better on the final exam, but for some reason did not pass. Perhaps the instructor made an error in recording the grade. At any rate, based on the evidence, we must conclude that the instructor s statement is false. Case 3: p is false and q is true. The student scored less than 70 on the final exam but passed anyway. Perhaps the student got a 69 and the instructor, being in a generous mood, decided to give the student a break. The point is that the facts do not contradict the instructor s statement. In a sense, the statement has not been tested. (The instructor did not say what would happen if the student scored less than 70 on the final.) Thus we take the instructor s statement to be true. Case 4: Both p and q are false. The student scored less than 70 on the final and did not pass the course. This is much like the previous case, in that the instructor s statement has not been tested. So we take the instructor s statement to be true. We see in this example that the only case in which the formula If p, then q. is false is when p is true and q is false. Examples like this one serve to motivate the next definition. Definition 1.3: Let p and q be propositional variables. The formula If p, then q.
14 Chapter 1 Logic and Sets is called the implication of q by p and is denoted by p q. It is false only when p is true and q is false, and is true otherwise. It is common to read p q as p implies q, and to refer to a statement with this logical form as a conditional statement or simply as an implication. In such a statement, p is called the hypothesis and q the conclusion. The truth table for the implication is shown in Table 1.2. p q p q T T T T F F F T T F F T Table 1.2 Truth table for p q Next, consider the following statements: (a) If the Buffalo Bills scored more than 20 points, then they won the game. (b) If the Buffalo Bills won the game, then they scored more than 20 points. which someone might utter before hearing the score of a football game played earlier that day. It is clear that these statements are different yet related. They might or might not have the same truthvalue, depending on the truth values of the two simple statements of which they are composed. For example, if the Bills scored 24 points and won the game, then both (a) and (b) are true; if the Bills scored 17 points and won, then (a) is true and (b) is false. Consider also the following statement: (c) If the Buffalo Bills lost the game, then they scored 20 points or less. As will be seen in the next section, statements (a) and (c) must have the same truth-value. Note that these three statements, taken together, have the following logical forms: (a) If p, then q. (b) If q, then p. (c) If not q, then not p. Definition 1.4: Let p and q be propositional variables and let u represent the formula If p, then q. The implication If q, then p. is called the converse of u, and the implication If not q, then not p. is called the contrapositive of u.
1.2 Logical Connectives 15 It happens frequently in mathematics that we need to examine a statement with the logical form (p q) (q p) that is, we need to examine the conjunction of an implication and its converse. connective is used for just this situation. A special logical Definition 1.5: Let p and q be propositional variables. The formula (p q) (q p) is called the biconditional, and it is denoted by p q. A statement with this form is read p if and only if q and is often written using the shorthand form p iff q The truth table for the biconditional, shown in Table 1.3, is not difficult to obtain from the truth tables for the conjunction and the implication. Note that p q is true only when p and q have the same truth-value. p q p q q p p q T T T T T T F F T F F T T F F F F T T T Table 1.3 Truth table for p q This is a good spot to discuss the use of parentheses in a formula. For example, consider the formula not p or q Does it mean p q or does it mean p q? In fact, convention has it that the first formula p q is the correct interpretation. A similar problem occurs with the formula p or q and r (even if we express it symbolically as p q r). Do we want the disjunction of p with the conjunction of q and r, that is, p (q r), or do we want the conjunction of the disjunction of p and q with r, that is, (p q) r? Difficulties of this sort can be resolved by employing parentheses. For example, in the formula (p q) r the or is to be applied first, and then the and. Another common approach is to adopt some basic precedence rules that will allow many formulas to be written without parentheses.
16 Chapter 1 Logic and Sets Precedence Rules for the Logical Connectives: In a parenthesis-free formula, the logical connectives are to be applied in the following order: Connective not and or implies iff Precedence first second third fourth fourth We say that negation (not) has higher precedence than both conjunction (and) and disjunction (or), and that conjunction has higher precedence than disjunction. The connective or has higher precedence than both the implication (implies) and the biconditional (iff ), and these latter two connectives have the same (lowest) precedence level. Thus, if a formula involves both implies and iff, then parentheses must be used to make the formula unambiguous. Example 1.4: We use the precedence rules to write several formulas. Exercise Set 1.2 Formula Written Using the Precedence Rules (a) (not p) or q not p or q (b) (p and q) or r p and q or r (c) (not q) (not p) not q not p (d) (p q) (p q) (p q) p q (e) (p q) (p q) p q (p q) 1. Let p, q, and r represent the following statements: p: Ralph read the New York Times. q: Ralph watched the Daily Show. r: Ralph jogged 3 miles. Give a formula for each of these statements. (Use the symbols for the logical connectives.) (a) Ralph read the New York Times and watched the Daily Show. (b) Ralph read the New York Times or jogged 3 miles. (c) If Ralph read the New York Times, then he did not watch the Daily Show. (d) Ralph read the New York Times if and only if he jogged 3 miles. (e) It is not the case that if Ralph jogged 3 miles then he read the New York Times. (f) Ralph watched the Daily Show or jogged 3 miles, but not both. 2. Define the propositional variables p, q, and r as in Exercise 1. Write out the statement corresponding to each of these formulas.
1.2 Logical Connectives 17 (a) p r (c) (p q) r (e) p q (b) q r (d) p q (f) q r 3. Consider the following statements p and q: p: Roger Clemens had a sore arm in 1995. q: The Red Sox won the 1995 World Series. The statement p is true. Represent each of these statements by a formula. What is the statement s truth-value if q is true? What if q is false? (a) Roger Clemens had a sore arm in 1995 or the Red Sox won the 1995 World Series. (b) Roger Clemens had a sore arm in 1995 and the Red Sox won the 1995 World Series. (c) If Roger Clemens had a sore arm in 1995, then the Red Sox did not win the 1995 World Series. (d) If the Red Sox did not win the 1995 World Series, then Roger Clemens had a sore arm that year. (e) The Red Sox won the 1995 World Series if and only if Roger Clemens did not have a sore arm that year. 4. For each of these compound statements, first identify the simple statements p, q, r, and so on, of which it is composed. Then represent the statement by a formula. (a) If 2709 is an integer, then either 2709 is even or 2709 is odd. (b) If 53 is prime and 53 is greater than 2, then 53 is odd. (c) If 3 7 is not negative and its square is less than 4, then either 3 7 = 0 or 3 7 is positive and less than 2. (d) If 26 is even and 26 is greater than 2, then 26 is not prime. 5. Express each of these compound statements symbolically. (a) If triangle ABC is equilateral, then it is isosceles. (b) The integer n = 3 if and only if 3n 4 = 5. (c) If π π is a real number, then either π π is rational or π π is irrational. (d) The product xy = 0 if and only if either x = 0 or y = 0. (e) If 47089 is greater than 200, then, if 47089 is prime it is greater than 210. (f) If line k is perpendicular to line m and line m is parallel to line n, then line k is perpendicular to line n. (g) If x 3 3x 2 + x 3 = 0, then either x is positive or x is negative or x = 0. (h) If 7 and 23 are integers and 23 0, then 7/23 is a rational number. 6. In mathematics, the connective or is used inclusively, meaning one or the other or both. However, in everyday language, or is often used in the exclusive sense, as in the sentence With your order you may have french fries or potato salad. Used in this way, the or is interpreted as one or the other, but not both. Using the symbol to represent the connective exclusive or, construct the truth table for the formula p q.
18 Chapter 1 Logic and Sets 7. In racquetball, it is important to know which player is serving, because a player scores a point only if that player is serving and wins a volley. If the serving player loses the volley, then the other player gets to serve. Thus, to keep score in a racquetball game between players A and B, it may be useful to define propositional variables p and q, where p is true if player A is serving and false if player B is serving, while q is true if player A wins the current volley and false if player B wins it. (a) Give a formula that is true if player A scores a point and is false otherwise. (b) Give a formula that is true if player B scores a point and is false otherwise. (c) Give a formula that is true if the serving player loses the current volley and is false otherwise. (d) What should happen to the value of p in order to change the serving player? 8. The implication If not p, then not q. is called the inverse of the implication p q. Let p represent the statement and let q represent the statement The Buffalo Bills scored more than 20 points. The Buffalo Bills won the game. Find the truth-value of both p q and its inverse p q in each of these cases. (a) p is true and q is true (c) p is false and q is true (b) p is true and q is false (d) p is false and q is false 9. Let the propositional variable p represent some statement, and let the variable p 1 represent the statement p is true. Show that p and p 1 have the same truth value. (Sometimes the phrase is true is appended to a statement for emphasis; for example, we might say p implies q is true instead of just p implies q. ) 1.3 LOGICAL EQUIVALENCE In mathematics, as in other subjects, there may be several different ways to say the same thing. In this section we formally define what this means for logical statements. Definition 1.6: Let u and v be formulas. We say that u is logically equivalent to v, denoted u v, provided u and v have the same truth-value for every possible choice of truth-values for the propositional variables involved. The two examples of logical equivalence that follow are important in that both are used often in mathematics.
1.3 Logical Equivalence 19 Example 1.5: To show that p q and p q are logically equivalent, we construct a truth table and compare the columns labeled by these two formulas. This truth table is shown in Table 1.4. Since the columns headed by p q and p q agree in each row, these formulas are logically equivalent. p q p q p q p q p q T T F F T F F T F F T T F F F T T F T F F F F T T F T T Table 1.4 Example 1.6: Use a truth table to verify that p q p q. Solution: See Table 1.5. p q p p q p q T T F T T T F F F F F T T T T F F T T T Table 1.5 Example 1.7: Recall that the contrapositive of the implication p q is the implication q p. In words, the contrapositive states If not q, then not p. We wish to show that an implication and its contrapositive are logically equivalent. A truth table can be used for this, but it is instructive to do it without a truth table. We need several facts. First, if u, v, and w are formulas with u v and v w, then clearly u w (in words, we say that logical equivalence is transitive). Second, from Example 1.6 we have that s t s t (1) It is also not difficult to see that and that s t t s (2) s s (3) (We use the propositional variables s and t here to avoid confusion with the variables p and q, which appear in the logical equivalence we are trying to establish.) Now, by (1), p q p q
20 Chapter 1 Logic and Sets Next, by (2), p q q p Finally, the hard step by (1), with s replaced by q and t replaced by p, we have Finally by (3), q q, so we obtain Thus it follows that q p q p q p q p p q q p Some formulas have the property that they are always true, namely, they are logically equivalent to the constant truth-value T. Definition 1.7: A formula that is true for all possible truth-values of its constituent propositional variables is called a tautology and is denoted by the constant formula T. A formula that is false for all possible truth-values of its constituent propositional variables is called a contradiction and is denoted by the constant formula F. Example 1.8: Table 1.6 shows the truth table for the formula p q p q Since this formula is always true, it is a tautology. p q p q p q p q p q T T T T T T F F T T F T F T T F F F F T Table 1.6 Truth table showing that p q p q is a tautology Example 1.9: Verify that the following formula is a tautology: [ (p q) r ] [ r (p q) ] Solution: Here, we have our first instance of a formula that involves three propositional variables. As illustrated by Table 1.7, there are eight possible truth-value combinations that must be considered. For convenience, we let u and v denote the formulas (p q) r and r (p q), respectively. Alternate Solution: It is also possible to verify that u v is a tautology by discussing what might be called the important or essential cases. We know that u v is true when u is false or
1.3 Logical Equivalence 21 when v is true. In other words, the only case in which u v is false is when u is true and v is false; we want to show that this can t happen. Suppose that v is false; then it must be that r is true and p q is false. Thus, it must be that r is false and p and q are both true; that is, r is false and p q is true. Therefore, u is false. So whenever v is false, u is also false. It follows that u v is a tautology. p q r p q r p q p q u v u v T T T F F F T F T T T T T F F F T T F F F T T F T F T F F T T T T T F F F T T F T T T T F T T T F F F T T T T F T F T F T F T T T T F F T T T F F T T T T F F F T T T F T T T T Table 1.7 Truth table showing that [(p q) r] [r (p q)] is a tautology Example 1.10: Express the formula p q as a conjunction. Solution: From Example 1.6, we recall that p q p q It follows that p q p q p q (by Example 1.5) p q Note that in the last step that the equivalence p p is used again. A note about logical equivalence at this point. In a later section we discuss what it means to prove a mathematical statement or theorem. The notion of logical equivalence will turn out to be useful in this regard. For suppose u and v are formulas representing mathematical statements and we wish to prove u. If u v, then it suffices to prove v. For example, suppose we want to prove a mathematical statement whose form is the implication p q. Since p q is logically equivalent to its contrapositive q p, we may instead prove q p. This technique, called proof by contrapositive, is a standard proof technique in mathematics, and we explore it further in Section 1.6. Here is a list of some of the important properties of logical equivalence that are frequently encountered and used in mathematics.
22 Chapter 1 Logic and Sets Properties of Logical Equivalence: 1. The commutative properties: (a) p q q p (b) p q q p 2. The associative properties: (a) (p q) r p (q r) (b) (p q) r p (q r) 3. The distributive properties: (a) p (q r) (p q) (p r) (b) p (q r) (p q) (p r) 4. The idempotent laws: (a) p p p 5. DeMorgan s laws: (a) p q p q 6. Law of the excluded middle: (a) p p is a tautology. (b) p p is a contradiction. (b) p p p (b) p q p q 7. An implication and its contrapositive are logically equivalent: p q q p 8. The converse and inverse of the implication p q are logically equivalent: q p p q 9. Let T denote any tautology and F denote any contradiction. Then: (a) p T T (b) p T p (c) p F p (d) p F F 10. An implication can be expressed as a disjunction, and the negation of an implication can be expressed as a conjunction: (a) p q p q (b) p q p q A brief comment concerning formulas such as (p q) r or ((p q) r) s. In view of the fact that (p q) r p (q r), it makes good sense to define p q r to be (p q) r. It follows that p q r p (q r), so we can insert parentheses into p q r in both valid ways without loss of meaning. Similarly, there are five meaningful ways to parenthesize the formula p q r s and, since all the resulting formulas are logically equivalent, we agree to let p q r s denote any one of them. Analogous remarks apply for disjunction. Example 1.11: Suppose we know the following statements are true: (a) π is an irrational number. (b) If π is an irrational number, then π + 4 is an irrational number.
1.3 Logical Equivalence 23 Then we should be able to conclude that the statement π + 4 is an irrational number is true. We can combine statements (a) and (b) and the conclusion drawn from them into the following single statement: If both π is an irrational number and π is an irrational number implies π + 4 is an irrational number, then π + 4 is an irrational number. The general form of this last statement is (p (p q)) q This formula has a Latin name: modus ponens. It represents any statement of the form If both p and p implies q, then q. Use properties of logical equivalence to show that modus ponens is a tautology. Solution: We apply the properties as indicated: (p (p q)) q p (p q) q 10(a) (p p q) q (p (p q)) q ((p p) (p q)) q (T (p q)) q) (p q) q p (q q) p T T 5(b) 10(b) 3(a) 6(a),1(a) 9(b),1(b) Frequently used in mathematical discussions are different (yet still equivalent) forms of the implication and biconditional. We next discuss these alternate forms. The implication p q has two interpretations in words thus far: and If p, then q. p implies q. We know that p q is logically equivalent to its contrapositive q p, which is read If not q, then not p. Assume p q is true. Then q p is also true, which means that if q is false, then p must also be false. Thus, p is true only under the condition that q is true. This last statement is written p only if q. 2(a) 6(a) 9(a)
24 Chapter 1 Logic and Sets and it is another way of saying p implies q. Another common phrase in mathematics is p is sufficient for q. It states that the condition that p holds is enough to guarantee that q holds. Hence, it is just another way of saying that p implies q. Finally, one last phrase equivalent to p q is q is necessary for p. This says that in order for p to be true, q must be true, so that q being falseimplies that p is false, again giving us the contrapositive q p. To summarize, then, the following statements are equivalent: If p, then q. p implies q. p only if q. p is sufficient for q. q is necessary for p. Example 1.12: Consider the following statement from calculus: In order that the derivative of a function f is 0, it is necessary that the function f is a constant. Rewrite this statement in four equivalent ways. Solution: The given statement is of the form q is necessary for p. where q represents the statement The function f is a constant. and p represents the statement The derivative of the function f is 0. Hence the given statement is equivalent to each of the following: If the derivative of the function f is 0, then the function f is constant. That the derivative of the function f is 0 implies that the function f is constant. The derivative of the function f is 0 only if the function f is constant. That the derivative of the function f is 0 is sufficient for the function f to be constant. Now consider the biconditional p if and only if q. Recall that this is, by definition, a shorthand for (p q) (q p)
1.3 Logical Equivalence 25 One way to read the above formula is This last statement is usually shortened to p is sufficient for q and p is necessary for q. p is necessary and sufficient for q. and is an alternative way of saying p if and only if q. For instance, the statements and The derivative of the function f is 0 if and only if f is constant. The derivative of the function f is 0 is necessary and sufficient for f to be constant. are equivalent. In view of the fact that p q and q p are equivalent, this last statement could be rephrased as The function f is a constant is necessary and sufficient for the derivative of f to be 0. Exercise Set 1.3 1. Use a truth table to verify DeMorgan s law 5(b). 2. Use truth tables to verify the associative properties 2(a) and 2(b). 3. Verify property 8 that the inverse and converse of the implication p q are logically equivalent. Try to do this in three different ways: (a) using a truth table (b) using property 10(a) (c) using property 7 4. Each implication given concerns integers x and y. Find (i) its converse, (ii) its contrapositive, (iii) its inverse, and (iv) its negation. (a) If x = 2, then x 4 = 16. (b) If y > 0, then y 3. (c) If x is odd and y is odd, then xy is odd. (d) If x 2 = x, then either x = 0 or x = 1. (e) If x = 17 or x 3 = 8, then x is prime. (f) If xy 0, then both x 0 and y 0. 5. Find (i) the converse, and (ii) the contrapositive of each of the following implications. (a) If quadrilateral ABCD is a rectangle, then ABCD is a parallelogram. (b) If triangle ABC is isosceles and contains an angle of 45 degrees, then ABC is a right triangle. (c) If quadrilateral ABCD is a square, then it is both a rectangle and a rhombus. (d) If quadrilateral ABCD has two sides of equal length, then it is a rectangle or a rhombus. (e) If polygon P has the property that P is equiangular if and only if P is equilateral, then P is a triangle.