Lecture 1.4: Rules of Inference, and Proof Techniques*

Lecture 1.4: Rules of Inference, and Proof Techniques* CS 250, Discrete Structures, Fall 2011 Nitesh Saxena *Adopted from previous lectures by Cinda Heeren

Course Admin • Slides from previous lectures all posted • Expect HW1 to be coming in around coming Monday • Thanks for taking the competency exam • Your graded exams will be returned to you next week • Your scores will not affect your grade in this course in any way Lecture 1.4 - Rules of Inference, and Proof Techniques

Outline • Rules of Inference (contd.) • Other Proof techniques Lecture 1.4 - Rules of Inference, and Proof Techniques

Complex Example: Rules of Inference Here’s what you know: Ellen is a math major or a CS major. If Ellen does not like discrete math, she is not a CS major. If Ellen likes discrete math, she is smart. Ellen is not a math major. Can you conclude Ellen is smart? M  C D  C D  S M Lecture 1.4 - Rules of Inference, and Proof Techniques

Ellen is smart! Complex Example: Rules of Inference 1. M  C Given 2. D  C Given 3. D  S Given 4. M Given 5. C DS (1,4) 6. D MT (2,5) 7. S MP (3,6) Lecture 1.4 - Rules of Inference, and Proof Techniques

Rules of Inference: Common Fallacies Rules of inference, appropriately applied give valid arguments. Mistakes in applying rules of inference are called fallacies. Lecture 1.4 - Rules of Inference, and Proof Techniques

If I am Bonnie Blair, then I skate fast I skate fast! If you don’t give me $10, I bite your ear. I bite your ear!  I am Bonnie Blair  You didn’t give me $10. Nope Nope ((p  q)  q)  p Not a tautology. Rules of Inference: Common Fallacies Lecture 1.4 - Rules of Inference, and Proof Techniques

If it rains then it is cloudy. It does not rain. If it is a car, then it has 4 wheels. It is not a car.  It is not cloudy  It doesn’t have 4 wheels. Nope Nope ((p  q)  p)  q Not a tautology. Rules of Inference: Common Fallacies Lecture 1.4 - Rules of Inference, and Proof Techniques

HUH? 7 = 3 mod 4 37 = 1 mod 4 94 = 2 mod 4 16 = 0 mod 4 7 = 111 mod 4 37 = 61 mod 4 94 = 6 mod 4 16 = 1024 mod 4 Direct Proofs A totally different example: Prove that if n = 3 mod 4, then n2 = 1 mod 4. Lecture 1.4 - Rules of Inference, and Proof Techniques

Direct Proofs Coming back to our Theorem: If n = 3 mod 4, then n2 = 1 mod 4. Proof: If n = 3 mod 4, then n = 4k + 3 for some int k. This means that: n2 = (4k + 3)(4k + 3) = 16k2 + 24k + 9 = 16k2 + 24k + 8 + 1 = 4(4k2 + 6k + 2) + 1 = 4j + 1 for some int j = 1 mod 4.

Direct Proofs: another example Theorem: If n is an odd natural number, then n2 is also an odd natural number. Proof: If n is odd, then n = 2k + 1 for some int k. This means that: n2 = (2k+1)(2k+1) = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1 = 2j + 1 for some int j  n2 is odd

Proofs by Contraposition Recall: Contrapositive: p  q and q  p Ex. “If it is noon, then I am hungry.” “If I am not hungry, then it is not noon.” We also know that: p  q  q  p Therefore, if establishing a direct proof (p  q) is difficult for some reason, we can instead prove its contraposition (q  p), which may be easier. Lecture 1.4 - Rules of Inference, and Proof Techniques

Proofs by Contraposition: example Theorem: If 3n + 2 is an odd natural number, then n is also an odd natural number. Proof: If n is not odd, then n = 2k for some int k. This means that: 3n+2 = 3(2k) + 2 = 2(3k) + 2 = 2(3k + 1) = 2j for some int j  3n+2is not odd Lecture 1.4 - Rules of Inference, and Proof Techniques

Proofs by Contraposition: another example Theorem: If N = ab where a and b are natural numbers, then a <= sqrt(N) or b <= sqrt(N). Proof: If a > sqrt(N) AND b > sqrt(N), then by multiplying the two inequalities, we get ab > N This negates the proposition N=ab Lecture 1.4 - Rules of Inference, and Proof Techniques

Proofs by Contradiction Recall: Contradiction is a proposition that is always False To prove that a proposition p is True, we try to find a contradiction q such that p  q is True. If p  q is True and q is False, it must be the case that p is True. We suppose that p is False and use this to find a contradiction of the form r  r Lecture 1.4 - Rules of Inference, and Proof Techniques

Proofs by Contradiction: example Theorem: Every prime number is an odd number Proof: A: n is prime B: n is odd We need to show that A  B is true We need to find a contradiction q such that:  (A  B)  q We know:  (A  B)  (A  B)AB This means that we suppose (n is prime) AND (n is even) is True But, if n is even, it means n has 2 as its factor, and this means that n is not prime. This is a contradiction because (n is prime) AND (n is not prime) is True

Proofs by Contradiction: example Theorem: If 3n + 2 is an odd natural number, then n is also an odd natural number. Proof: A: 3n + 2 is odd B: n is odd We need to show that A  B is true We need to find a contradiction q such that:  (A  B)  q We know:  (A  B)  (A  B) AB This means that we suppose that (3n + 2 is odd) AND (n is even) is True. But, if n is even, it means n = 2k for some int k, and this means that 3n + 2 = 6K+2 = 2(3K+1)  even. This is a contradiction: (3n + 2 is odd) AND (3n +2 is even)

Disproving something: counterexamples If we are asked to show that a proposition is False, then we just need to provide one counter-example for which the proposition is False In other words, to show that x P(x) is False, we can just show x P(x) = x P(x) to be True Example: “Every positive integer is the sum of the squares of two integers” is False. Proof: counter-examples: 3, 6,… Lecture 1.4 - Rules of Inference, and Proof Techniques

Today’s Reading • Rosen 1.7 • Please start solving the exercises at the end of each chapter section. They are fun. Lecture 1.4 - Rules of Inference, and Proof Techniques

Lecture 1.4: Rules of Inference, and Proof Techniques*