Oh Yeah? Well, Prove It.

Transcription

1 Oh Yeah? Well, Prove It. MT 43A - Abstract Algebra Fall 009 A large part of mathematics consists of building up a theoretical framework that allows us to solve problems. This theoretical framework is built upon a set of axioms. Axioms are unproven assumptions that are the foundation of all mathematics. (Not everyone agrees on what axioms should be used, but that s a whole other story.) These basic axioms are combined using the rules of logic into more complicated statements called theorems. If the axioms are true and the logic is sound, then the theorems must also be true. New theorems are created by combining axioms and known theorems. A proof of a theorem is simply an explanation of why the theorem is true. Different people will require different amounts of explanation. It will take some practice to get used to how much explanation I expect. In this class we will not attempt to build all of mathematics from a set of axioms (an extremely difficult proposition!). Instead, we will start with some known mathematical objects (like the integers), and see how some basic definitions, axioms and known theorems can be built into a useful theoretical framework. In order to do this, we need to learn how to prove new theorems. One of the first definitions we will use is that of an even integer. I m sure you all think you know what an even integer is, but many of you will hesitate if asked whether zero is even or odd. The following definition makes this obvious. Definition: An integer is called even if it is divisible by, and called odd otherwise. In other words, the even numbers are those that can be written as k for some integer k, and the odd numbers are those that can be written as k + 1 for some integer k. It should be clear upon a moment s reflection that zero is an even number. A statement that has not been proved is called a conjecture. Conjectures can usually be written in the form If P, then Q. This can also be written as P = Q, which is read P implies Q. P consists of one or more statements called the hypotheses and Q is a statement called the conclusion. Example 1: The statement The sum of two even integers is even. can be written as the statement If a and b are even integers, then a + b is an even integer. The hypotheses are a and b are even integers and the conclusion is a + b is an even integer. Some conjectures are written in the form P if and only if Q. This is also written as P Q, or P iff Q. In this case it is said that Q is a necessary and sufficient condition for P. To prove an if and only if statement, we must prove two statements: If P, then Q (Q is necessary for P ), and If Q, then P (Q is sufficient for P ) One direction is frequently significantly easier to prove than the other.

2 Example : The square of an integer n is even if and only if n is even. P is the statement: The square of an integer n is even, and Q is the statement: n is even. To prove this statement we must prove both If n is even, then n is even. If n is even, then n is even. A proof that P = Q is a logically correct argument that shows that if the hypotheses are true, then the conclusion must also be true. The argument may use axioms, definitions and previously proved statements. Once a statement has been proved, it is called a theorem. A theorem whose main purpose is to prove other more important theorems is called a lemma. A theorem that easily follows from a more important theorem is called a corollary. There is no algorithm, or single list of instructions, for finding proofs, but here are some tips to help you get started. 1. Understand what you are trying to prove. Look up definitions of all the terms. Even though you may know what the terms mean, it is important to have the official definitions in front of you, as these will frequently give you a clue about how you might proceed.. Understand what the hypotheses are. Write down definitions of all the terms involved in the hypotheses. This is almost always the first step in a proof. 3. It is often helpful to look at examples that illustrate the conjecture. This can give you a better understanding of the problem. No collection of specific examples constitues a general proof that a conjecture is true. 4. It is sometimes useful to work backward from the conclusion. Ask yourself, What would I have to know in order to know that the conclusion is true. Then, What would I have to know in order to know that these statements are true. Keep going until you see a way to connect the hypotheses with the conclusion. 5. Once you have discovered your proof, write it up in a clear correct logical form. State the main steps from the hypotheses to the conclusion with some justification for each step. Avoid excessive detail, but provide enough detail so that your argument is convincing. There are several standard methods of proof. Here is a description of some of the most common methods of proof, with examples of each. DIRECT PROOF: Begin with the hypotheses and follow a direct line of reasoning to the conclusion. Theorem: If a and b are even integers, then a + b is an even integer.

3 Proof: If a is an even integer, then a can be written as an integer multiple of (this is the definition of an even integer). Thus, we can write a = m for some integer m. Similarly, since b is even, we can write b = n for some integer n (not necessarily the same integer as we used to write a, of course). Now look at a + b. We can write a + b as m + n = (m + n), and we see that a + b is an integer multiple of. Therefore, a + b is an even integer. It is important to mark the end of every proof. The box,, is commonly used for this. Another common way to mark the end of a proof is with the letters QED. (In fact, the box is often called a QED-box.) QED is an abbreviation of the latin phrase Quod Erat Demonstrandum - which was to be demonstrated. COUNTER-EXAMPLE: To prove that a statement is false in general, it is enough to find one specific case for which the statement is false. Such a statement is called a counter-example. Find a counter-example to the following conjecture: Conjecture: If a, b and d are integers and d divides the product ab, then d divides a or d divides b. CONTRADICTION: Assume the hypothesis is true and the conclusion is false, and show that this contradicts something known to be true. Theorem: There are infinitely many prime numbers. Proof: This theorem is unusual in that there are no hypotheses explicitly stated! We only have basic definitions and previous theorems to go on. We will prove this by contradiction, using the Fundamental Theorem of Arithmetic (Theorem 0.3 on page 6). Every integer greater than one is prime or a product of primes. In order to arrive at a contradiction, assume that there is a finite number of prime numbers. If the number of prime numbers is finite (say there are n of them), then we can list them. p 1, p,... p n Consider the (very large) number we get if we multiply all these primes together and add 1. V = p 1 p... p n + 1 Notice that p k does not divide V for any k from 1 to n (since no primes divide 1). In other words, none of the primes on the list divide V. The Fundamental Theorem of Arithmetic, however, tells us that V can be written uniquely as a product of prime numbers. In other words, some prime numbers have to divide V. But all the primes are on the list, and none of them divide V. This is a contradiction! Contradictions are sometimes denoted with the symbol.

4 This contradiction means that our initial assumption that there is a finite number of prime numbers must be false. Therefore there must be infinitely many prime numbers. Some (uptight) mathematicians don t like to use proof by contradiction. They insist that any statement proved by contradiction can be proved directly. This may or may not be true, but it is true that proof by contradiction is often a much simpler and more elegant way to prove a statement that is difficult to prove directly. Always try a direct proof first, but if you get stuck, a proof by contradiction will sometimes work. CONTRAPOSITIVE: Sometimes the contrapositive of a statement is easier to prove than the statement. The contrapositive of the statement is the statement If P, then Q If not Q, then not P. A statement and its contrapositive are logically equivalent. If one is true the other is necessarily true, and vice versa. For example, the statement If I roll a 7, then I win. is logically equivalent to the statement If I don t win, then I don t roll a 7. Either they are both true or both false. Theorem: If n is even, then n is even. Proof: We prove the contrapositive: If n is odd, then n is odd. If n is odd, then n can be written as k + 1 for some integer k. Then n = (k + 1) = 4k + 4k + 1 = (k + k) + 1. Since n is of the form x + 1 for the integer x = k + k, n is an odd integer. INDUCTION: Induction is a technique for proving that a statement is true for all positive integers. Proof by induction is like knocking over dominos - prove that you can knock over the first domino, then prove that if one domino falls then it knocks over the next domino. If you can prove both those things, then all the dominos will fall. More formally, if you wish use induction to prove a statement S n true for all integers n n 0, you must Prove a base case: prove the statement S n is true for n = n 0.

5 Prove the inductive step: assume the statement is true for case n, and use this to prove the statement is true for case n + 1. In other words prove that S n = S n+1 S n is called the inductive hypothesis. You assume S n is true, and use it to show S n+1 must also be true.

6 Theorem: The sum of the the first n positive integers is n(n+1). In other words, we wish to prove the statement S n : n = for all n 1. (Note that S n is simply a name for the above equation.) Proof: (By induction) Base case: S 1 is the statement 1 = 1(1+1). This is true. The inductive step: Assume S n is true. That is, We wish to use this to prove S n+1 : n = n + (n + 1) = (n + 1)(n + ) To see this, we add (n + 1) to both sides of equation S n and get n + (n + 1) = + (n + 1) Getting a common denominator and simplifying the right side produces We see that if S n is true then + (n + 1) = = n + 3n + (n + 1)(n + ) = + (n + 1) (n + 1)(n + ) n + (n + 1) = is also true, but this is statement S n+1. So S n implies S n+1. Therefore, since we have shown a base case and proved the inductive step, the statement S n is true for all n 1. In other words, the sum of the the first n positive integers is n(n+1). There you have it - a quick and dirty introduction to methods of proof. I hope you find it helpful. Of course the only way to really learn how to write proofs is to grit your teeth, try to write some, get stuck, get frustrated (but don t stay frustrated!), ask for help, make mistakes, learn from them, and repeat all of these steps as often as necessary. Many of you will need to use office hours more than you are used to. I ll be able to answer some questions in class, but the best help will come one on one, or in small groups. Don t be afraid to ask for help. As long as you keep trying, I ll be willing to help as best as I can.

7 EXERCISES: 1. Use the following outline to prove by induction that for all integers greater than or equal to 1, the sum of the first n odd integers is n. Since the n th odd integer is n 1, the symbolic form for this statement is S n : (n 1) = n (a) Check that the statement is true in the case n = 1. (b) Write the inductive hypothesis, S n. This is what you assume to be true. (c) Write the expression S n+1. This is what you hope to show is true. (d) What simple algebraic operation would you have to do to the left side of S n to get the left side of S n+1? Of course, if you do this to the left side of equation S n, you must do it to the right side of S n, as well. Do it! (e) Does the left side of the expression from part (d) equal the left side of S n+1? (It better!) (f) Do some simple algebra to see if the right side of part (d) equals the right side of S n+1. If it does, you have shown that S n = S n+1, and thus completed the proof by induction.. Prove that if a, b and c are integers and a divides b and a divides c, then a divides b + c and a divides b c. 3. Prove that if a, b and c are integers and a divides b and b divides c, then a divides c. 4. Prove that the product of two even integers is divisible by Prove that the product of two odd integers has remainder 1 when divided by Prove that the square of any integer is congruent to 0 or 1 mod Prove that if a, b and c are integers such that a + b = c, then either a or b is even. (This is an inclusive or - either a or b is even, or possibly both are even.) 8. Use induction to prove that n = n+1 1 for n The Fibonacci numbers are defined recursively by the formulas F 1 = 1 F = 1 F n = F n 1 + F n calculate the first ten Fibonacci numbers. Use induction to prove that for all n 1. F 1 + F + F F n = F n F n+1