After that, we will introduce more ideas from Chapter 5: Number Theory. Your quiz in recitation tomorrow will involve writing proofs like those.

Transcription

1 Wednesday, Oct 17 Today we will finish Course Notes 3.2: Methods of Proof. After that, we will introduce more ideas from Chapter 5: Number Theory. The exercise generator Methods of Proof, 3.2 (also includes concepts from Chapter 5) creates multiple choice or true/false questions that get at the correct way to start and structure a proof. The Get a Clue link then provides access to the complete proof. Your quiz in recitation tomorrow will involve writing proofs like those. There is no online quiz this week. The date for Test 2 has been shifted to Thursday, Oct 25.

2 Existence Proofs A. True or B. False!n! 2n 2 1 = 5n + 2 We are asked to prove the existence of an object (in this case, a certain unique integer). We will perform a calculation to see if we can produce a unique integer n such that 2n 2 1 = 5n + 2. If we are able to produce such a number, we will have employed a constructive existence proof.

3 The Pigeonhole Principle The pigeonhole principle is an idea that underlies certain indirect existence proofs that involve claims about counting: If n+1 objects (pigeons) are distributed among n containers (pigeonholes), at least one container must receive two or more objects.

4 An example related to the pigeonhole principle Suppose we are going to use an indirect proof to prove the following claim: In a classroom, seats are arranged in a rectangular array of 8 rows, with 10 seats in each row. If 52 people are seated in that room, at least one row must have more than six people. Select an appropriate assumption with which to begin the proof. A. Assume that at least one row has more than six people. B. Assume that no row has more than six people. C. Assume that every row has more than six people. D. Assume that every row has at least six people. E. Assume than every person sits in at least six rows.

5 To avoid losing points on a proof exercise: 1. Avoid using terminology or notation that gives the reader the impression that you don t know what you are talking about. For instance, an informed writer would not use terminology such as x is a real integer or x is a rational integer when he/she is trying to say x is an integer. 2. Never include the conclusion in your opening assumption. This gives the impression that you are using circular reasoning, or arguing from the converse, which are illogical. Such an error will cause you to lose at least half of the possible points for the exercise. For example, suppose we are asked to use the definition of even number to prove: If n is even, then n 2 is even. Then the direct proof should begin like this: Assume n is an arbitrary even number You would lose at least half credit if your wrote something like: Assume n is an arbitrary even number and n 2 is even or Assume n 2 is even

6 3. Never try to use specific cases to prove a general conjecture. For example, suppose, again, we are asked to use the definition of even number to prove: If n is even, then n 2 is even. If your proof looked something like this: 2 is even and 2 2 = 4 is even 6 is even and 6 2 = 36 is even 10 is even and 10 2 = 100 is even and so on, you would earn 0 points credit. 4. Use words A proof is an explanation. In order for your proof to be readable, it must contain words, along with the appropriate mathematical manipulations. If your proof is just a splat of equations, then it isn t a proof.

7 Return to Chapter 5 In discussing Methods of Proof (Chapter 3, Section 2) we introduced the divisibility relation from Chapter 5: Let a, b, be integers; then a b if and only if there is an integer k such that ak = b. We also introduced the modular congruence relation from Chapter 5: For integers a, b, and positive integer m, a b (mod m) if and only if m a b. These relations, and further concepts from number theory, are at the heart of some fundamental ideas in computing, including cryptography.

8 The Division Algorithm Let a be an integer and d be a positive integer. Then there are unique integers q and r, with 0 r<d, such that a = qd + r. In the relation a = qd + r, a is called the, d is called the, q is called the, and r is called the. Note that the Division Algorithm isn t really an algorithm, but that is the commonly used name for this theorem. As the following examples suggest, finding q and r for a particular a and d can be done using an algorithmic approach, which is probably the basis for the name.

9 EXAMPLE Let a = 61, d = 13; find q and r according to the division algorithm. (You may have already completed this exercise in your head; the following systematic approach indicates the association between this theorem and the word algorithm.) To find q, and then nonnegative r, we can carry out the following sequence of calculations of the form kd for integers k until kd exceeds a = 1 13 = 2 13 = 3 13 = 4 13 = 5 13 =

10 EXAMPLE Let a = 21, d = 6; find q and r according to the division algorithm. True/false: q = 3 The approach taken in the previous example can be helpful in the case where a is negative; remember that r must be nonnegative and less than d (= 6). 0 6 = 1 6 = 2 6 = 3 6 = 4 6 =

11 An operation derived from the Division Algorithm In the expression a = qd + r we say that r = a modulo d or r = a mod d That is, a mod d is the remainder according to the Division Algorithm when the integer a is divided by the positive integer d. Referring to the results of the previous examples, we say that 61 mod 13 = 21 mod 6 = Beware: most calculators and programming languages have a feature is that like the mod operation (frequently denoted a % d ); exactly how these functions work can vary from one device or language to another, but they tend to not align with our mathematical definition of a mod d, which insists, among other things, that d must be positive and a mod d must be nonnegative and less than d. Note that a mod d is an operation, unlike a b (mod m), which is a relation. The two concepts are connected to one another however.

12 Theorem Let a, b, be integers and let m be a positive integer. a b (mod m) if and only if a mod m = b mod m. Here is an example of what this theorem is stating. Let m = 5, a = 28, b = 53. Then a mod m = 28 mod 5 = b mod m = 53 mod 5 = Also note that a b (mod m) because