Week 2: Proof Strategies I mean the word “proof” not in the sense of the lawyers, who set two half proofs equal to a whole one, but in the sense of a mathematician, where 1 푝푟표표푓 = 0, and it is demanded for proof that every doubt becomes 2 impossible. Carl Friedrich Gauss The Main Techniques Just about any mathematical statement worth thinking about has a kind of “logical structure” – meaning that the statement can be broken down into smaller, simpler statements that are logically connected in some way. As mentioned before, the most common logical structure is “If 푃1, then 푃2.” 1 A direct proof of the proposition “If 푃1, then 푃2” has the following general form: Proof: Suppose 푃1. … … …. Therefore 푃2. ∎ The “… … …” represents some missing written explanation, possibly many paragraphs or pages long. As a first example, we previously wrote a direct proof of the fact that if 푚2 − 1 is prime (푚 being a positive whole number), then 푚 = 2. The contrapositive of the proposition “If 푃1, then 푃2” is “If 푃2 is not true, then 푃1 is not true.” As noted before, these statements are always equivalent. A proof by contraposition of the proposition “If 푃1, then 푃2” has the following general form: Proof: Suppose 푃2 is not true. … … …. Therefore 푃1 is not true. By contraposition, if 푃1, then 푃2. ∎ As an example, we previously wrote a proof by contraposition that if the square of a positive whole number 푛 is even, then 푛 must be even. A contradiction is a statement that absolutely must be false. A proof by contradiction of any simple proposition 푃 has the following general form: Proof: Suppose 푃 is not true. … … …. Therefore …, which is a contradiction. By this contradiction, 푃 must be true. ∎ As an example, we previously wrote a proof by contradiction that √2 is irrational. In the proof, we supposed that √2 is not irrational and ended up concluding that there is a 1 Note that the phrase “direct proof” is usually applied to any proof that proceeds without “indirect” logical techniques like proof by contraposition or proof by contradiction, which are explained next. Week 2 Page 2 reduced fraction whose numerator and denominator are both even. This is a contradiction, which proves that √2 must be irrational. The idea of proof by contradiction can be applied to a proposition in the “If 푃1, then 푃2” form. In the proof, we would suppose 푃1 is true and prove “by contradiction” that 푃2 must be true. In other words, a proof by contradiction of the proposition “If 푃1, then 푃2” would have the following general form: Proof: Suppose 푃1 is true, but 푃2 is not true. … … …. Therefore …, which is a contradiction. By this contradiction, if 푃1, then 푃2. ∎ Here is another good way to think about this last type of proof: Proving the proposition “If 푃1, then 푃2” can be accomplished by proving that it is impossible for both 푃1 to be true and 푃2 to be false at the same time. The “general forms” given above are meant to outline the overall structures of proofs. The logical structure of a proof is vital, and must be based on the logical structure of the mathematical statement being proved. However, the specific words used in a proof are a matter of personal taste, context, and the intended audience of the proof. Words like “suppose” and “therefore” are not mandatory, but they are popularly used in formal proofs, written for an audience at the appropriate reading level. The words “contradiction” and especially “contraposition” should be completely avoided in proofs written for an audience unfamiliar with those terms. Note carefully how we avoided these intimidating words in previous proofs. Exercise 1: Prove that if 푥 − 푦 is odd (with 푥 and 푦 being positive whole numbers), then 푥 + 푦 is odd. Exercise 2: Assuming 푎 and 푏 are positive whole numbers, prove that if 푎 ⋅ 푏 is odd, then both 푎 and 푏 are odd. Exercise 3: Prove that 푁2 + 푁 + 3 is always odd (as long as 푁 is a positive whole number). Exercise 4: Assuming 푛 is a positive whole number, prove that if 푛2 − 1 is not a multiple of 8, then 푛 is even. Exercise 5: Prove that the graphs of 푦 = 푥2 + 푥 + 2 and 푦 = 푥 − 2 do not intersect. (Do not use a graph in the proof.) Proof by Induction As noted before, there is a special logical technique that is sometimes useful for proving a statement that is supposed to be true for every positive whole number. The reasoning Week 2 Page 3 behind the technique is this: if we know that a statement applies to the number 1, and we know that whenever the statement applies to a whole number, it must also apply to the next whole number, then the statement must apply to every positive whole number. This line of reasoning is usually called the “principle of mathematical induction.” A proof by induction that a certain statement applies to every positive whole number has the following general form: Proof: The statement is true for 1, because …. Now, suppose the statement is true for a whole number 푛. … … …. Therefore the statement is true for 푛 + 1. By induction, the statement is true for every positive whole number. ∎ Let’s apply the principle of mathematical induction to some new statements about positive whole numbers. Proposition: 푛 + 3 < 5푛2 for any positive whole number 푛. Proof: This inequality is true for 1, since 1 + 3 < 5 ⋅ 12. Now, suppose the inequality 푛 + 3 < 5푛2 is true for some positive whole number 푛. Then (푛 + 1) + 3 < 5푛2 + 1, which is certainly less than 5푛2 + 1 + 10푛 + 4 since 푛 is positive. So (푛 + 1) + 3 < 5(푛 + 1)2. In other words, the inequality is true for 푛 + 1. By induction, the inequality is true for any positive whole number 푛. ∎ Proposition: No matter what the positive whole number 푛 is, the number 4푛 − 1 is always a multiple of 3. Proof: The statement is true when 푛 = 1, because 41 − 1 = 3, which is a multiple of 3. Suppose we have a positive whole number 푛 for which 4푛 − 1 is a multiple of 3. This means that there is a whole number 푚 so that 4푛 − 1 = 3푚. Therefore 4푛+1 − 1 = 4 ⋅ 4푛 − 1 − 4 + 4 = 4(4푛 − 1) + 3 = 4 ⋅ 3푚 + 3 = 3(4푚 + 1), which is a multiple of 3. The statement is true for 푛 + 1. By induction, the statement is true for all positive whole numbers! ∎ The principle of mathematical induction is easily adapted to claims made about all whole numbers after a certain point. For example – Proposition: 푛! > 2푛+2 is true for every whole number 푛 ≥ 6. Proof: This formula is correct for 푛 = 6 because 6! = 720, which is larger than 28 = 256. Now, suppose the formula is correct for some whole number 푛 ≥ 6. Note that (푛 + 1)! can be rewritten as 푛! ⋅ (푛 + 1), and since 푛! > 2푛+2, we can say that 푛! ⋅ (푛 + 1) > 2푛+2(푛 + 1). Then, since 푛 + 1 is larger than 2, we know that 2푛+2(푛 + 1) > 2푛+2 ⋅ 2, which is the same as 2(푛+1)+2. Week 2 Page 4 This shows that the formula is correct for 푛 + 1. By induction, the formula is correct for every whole number 푛 ≥ 6. ∎ Earlier, we found that the sum of the first 푛 odd numbers is always simply 푛2. What about the product of those numbers? There is a “formula” for that product, but it’s not so simple. (2푛)! Proposition: The product of the first 푛 odd numbers is always . 푛! 2푛 (2⋅2)! Proof: This is true for the product of the first two odd numbers: 1 ⋅ 3 = = 3. 2! 22 Now, suppose we have a certain positive whole number 푛 so that the product of the first 푛 (2푛)! odd numbers is . Then the product of the first 푛 + 1 odd numbers is 푛! 2푛 (2푛)! (2푛 + 1)! 1 ⋅ 3 ⋅ … ⋅ (2푛 − 1) ⋅ (2푛 + 1) = (2푛 + 1) = 푛! 2푛 푛! 2푛 (2푛 + 1)! 2푛 + 2 (2푛 + 1)! 2푛 + 2 (2푛 + 2)! (2(푛 + 1))! = ⋅ = ⋅ = = 푛! 2푛 2푛 + 2 푛! 2푛 2(푛 + 1) (푛 + 1)! 2푛+1 (푛 + 1)! 2푛+1 This shows that the formula is true for the first 푛 + 1 odd numbers. By induction, we have completed the proof. ∎ Exercise 6: Prove that the sum of the first 푛 positive cubic numbers is always 푛2(푛 + 1)2/4. Exercise 7: Prove that for every positive whole number 푛, the number 푛3 + 5푛 + 6 is always a multiple of 3. Exercise 8: Prove that if 푛 is any whole number greater than 1, then the sum of the first 푛 square root reciprocals is greater than √푛. That is, 1/√1 + 1/√2 + ⋯ + 1/√푛 > √푛. Exercise 9: Prove that the following formula is true for every positive whole number 푛: 3 4 5 푛 + 2 (푛 + 1)(푛 + 2) ⋅ ⋅ ⋅ … ⋅ = . 12 22 32 푛2 2푛! Figurate Numbers Some of the earliest Greek mathematicians took a strong interest in numbers that, when visualized as a group of dots, can be organized into a pleasant geometric pattern. For example, a “square number” can be visualized as a group of dots in a regularly-spaced square pattern. Week 2 Page 5 A “triangular number,” when visualized with dots, can be organized into a regularly-space triangular pattern. For example, the number 3 is obviously a triangular number, because 3 dots can be placed to form the corners of an equilateral triangle.