Chapter 2 Section 2.1: Proofs Proof Techniques. CS 130 Discrete Structures

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1 Chapter 2 Section 2.1: Proofs Proof Techniques CS 130 Discrete Structures

2 Some Terminologies Axioms: Statements that are always true. Example: Given two distinct points, there is exactly one line that contains them. Definitions: Used to create new concepts in terms of existing ones Theorem: A proposition that has been proved to be true. Two kinds of theorems: Lemma and Corollary Lemma: A theorem that is usually not too interesting in its own right but is useful in proving another theorem. Corollary: A theorem that follows quickly from another theorem CS 130 Discrete Structures 2

3 Inductive Reasoning v.s. Deductive Reasoning Inductive reasoning: draw a conclusion based on experience You observe a number of cases in which whenever P is true, Q is also true. On the basis of these experiences, you may formulate a conjecture: P Q The more cases you find where Q follows from P, the more confident your are in your conjecture But no matter how reasonable the conjecture sounds, you will not be satisfied Deductive reasoning: verify the truth or falsity of a conjecture Either produce a proof for P Q Or find a counterexample that disproves the conjecture, where P is true and Q is false CS 130 Discrete Structures 3

4 Proof Techniques Disproof by counterexample Exhaustive proof Direct proof Proof by contraposition Proof by contradiction CS 130 Discrete Structures 4

5 Disproof By Counterexample Prove ( x)p(x) is NOT true, thus [( x)p(x)] is true by showing ( x)[p(x)] Example Disprove the following conjectures: All animals living in the ocean are fish. Pf. For example, whales live in the ocean and are not fish. All input to a computer is provided by the keyboard. Pf. For example, mice provide input to a computer and are not keyboards. For a positive integer n, n factorial is defined as n(n- 1)(n-2) 1 and is denoted by n!. Prove or disprove the conjecture For every positive integer n, n! <= n 2. CS 130 Discrete Structures 5

6 Exhaustive Proof If dealing with a finite domain in which the proof is to be shown to be valid, then using the exhaustive proof technique, one can go over all the possible cases for each member of the finite domain. Final result of this exercise: you prove or disprove the theorem but you could be definitely exhausted. Example 1: If an integer between 1 and 20 is divisible by 6, then it is also divisible by 3. Example 2: For any positive integer less than or equal to 5, the square of the integer is less than or equal to the sum of 10 plus 5 times the integer. CS 130 Discrete Structures 6

7 Direct Proof Prove P directly, using generalized methods of deduction Example 1: show that every integer divisible by 6 is divisible by 3 Pf. Let x be an integer divisible by 6. There exists an integer y S.T. x = 6 y. Therefore x = 3 2 y, and x is divisible by 3. Example 2: the product of two even integers is even CS 130 Discrete Structures 7

8 Proof By Contraposition Use the variants of P Q to prove the conjecture. (Q P ) is equivalent to (P Q) Prove P Q is true by showing Q P Example 1: prove that if the square of an integer is odd, then the integer must be odd. Example 2: Prove that every integer divisible by 6 is divisible by 3. Example 3: Prove that the product xy is odd if and only if both x and y are odd integers. CS 130 Discrete Structures 8

9 Indirect Proof: Proof By Contradiction By constructing a truth table, we can see: (P ^ Q 0 ) (P Q) To prove P Q by proving (P ^ Q 0 ) Example 1: Prove If a number added to itself gives itself, then the number is 0. Prove P is true by showing P 0, P implies a contradiction Example 2: Prove The sum of even integers is even. CS 130 Discrete Structures 9

10 Summary Proof Technique Approach to prove P Q Remarks Exhaustive Proof Demonstrate P Q for all cases May only be used for finite number of cases Direct Proof Assume P, deduce Q The standard approachusually the thing to try Proof by Contraposition Proof by Contradiction Assume Q, derive P Assume P Λ Q, deduce a contradiction Use this Q if as a hypothesis seems to give more ammunition then P would Use this when Q says something is not true CS 130 Discrete Structures 10

11 Serendipity Serendipity means proving by luck A tennis tournament has 342 players. A single match involves 2 players. The winner of a match plays the winner of another match in the next round, while losers are eliminated from the tournament. The 2 players who have won all previous rounds play in the final game, and the winner wins the tournament. Prove that the total number of matches to be played is 341. CS 130 Discrete Structures 11

12 Class Exercises 1. Provide counterexamples to the following statements: The number n is an odd integer if and only if 3n + 5 is an even integer. 2. Product of any 2 consecutive integers is even 3. The sum of 3 consecutive integers is even 4. Product of 3 consecutive integers is even 5. If a number x is positive, so is x+1 (do a proof by contraposition) 6. The square of an odd integer equals 8k+1 for some integer k 7. The sum of two rational numbers is rational 8. For a positive integer x, x+1/x 2 CS 130 Discrete Structures 12

13 Chapter 2 Section 2.2: Induction CS 130 Discrete Structures

14 A Motivating Example Climbing a ladder We assume the following two assertions: You can reach the first rung Once you get to a rung, you can always climb to the next rung (This is an implication) If we can prove the above two assertions are true, then we can conclude that: You can reach as high as you want Induction: a very useful proof technique in computer science CS 130 Discrete Structures 14

15 First Principle of Induction Two assertions: P(1) is true For all k, if P(k) is true, then P(k+1) is true Conclusion: P(n) is true for all positive integers n Inductive proof: P(1) is the basis step P(k) P(k+1) is the inductive step We assume P(k) is true to prove the inductive step, P(k) is called the inductive assumption CS 130 Discrete Structures 15

16 Steps To Prove By Induction Step 1: Prove the basis step Step 2: Assume P(k) is true Step 3: Prove P(k+1) is true CS 130 Discrete Structures 16

17 Examples Prove the following for any (n >= 1): (2n-1) = n n = 2 n n > n 2 2n 1 is divisible by 3 n 2 > 3n for n >= 4 2 n+1 < 3 n for all n > 1 CS 130 Discrete Structures 17

18 Second Principle of Induction Assertions: P(1) is true For all k, P(r) is true for all r, where 1 <= r <= k Conclusion: P(n) is true for all positive integers n More powerful than the first principle of induction Steps: Step 1: Prove P(1) is true Step 2: Assume P(r) is true for all r, where 1 <= r <= k Step 3: Prove P(k+1) is true CS 130 Discrete Structures 18

19 Examples Prove that for every n >= 2, n is a prime number or a product of prime numbers. Prove that any amount of postage greater than or equal to 8 cents can be built using only 3-cent and 5-cent stamps. CS 130 Discrete Structures 19

20 Class Exercises Use mathematical induction to prove that the statements are true for every positive integer n (4n-2) = 2n 2 Prove that n 2 > n+1 for n >= 2 Prove that n < n 2 for n > 1 Prove that the statements are true for every positive integer. 2 3n 1 is divisible by 7 CS 130 Discrete Structures 20