Lecture 1.4: Rules of Inference

1. Lecture 1.4: Rules of Inference CS 250, Discrete Structures, Fall 2015 Nitesh Saxena Adopted from previous lectures by Cinda Heeren

2. Course Admin • Slides from previous lectures all posted • Expect HW1 to be coming in around coming Monday • Questions? Lecture 1.4 - Rules of Inference

3. Outline • Rules of Inference Lecture 1.4 - Rules of Inference

4. How do we know it? Proofs – How do we know? The following statements are true: If I am Mila, then I am a great swimmer. I am Mila. What do we know to be true? I am a great swimmer! Lecture 1.4 - Rules of Inference

5. Given set of true statements or previously proved theorems Rules of inference Proof Proofs – How do we know? A theorem is a statement that can be shown to be true. A proof is the means of doing so. Lecture 1.4 - Rules of Inference

6. What rules we study • Modus Ponens • Modus Tollens • Addition • Simplification • Disjunctive Syllogism • Hypothetical Syllogism Lecture 1.4 - Rules of Inference

7. What rule of inference can we use to justify it? Proofs – How do we know? The following statements are true: If I have taken MA 106, then I am allowed to take CS 250 I have taken MA 106 What do we know to be true? I am allowed to take CS 250 Lecture 1.4 - Rules of Inference

8. p p  q Tautology: (p  (p  q))  q  q Rules of Inference – Modus Ponens I have taken MA 106. If I have taken MA 106, then I am allowed to take CS 250.  I am allowed to take CS 250. Inference Rule: Modus Ponens Lecture 1.4 - Rules of Inference

9. q p  q Tautology: (q  (p  q))  p  p Rules of Inference – Modus Tollens I am not allowed to take CS 250. If I have taken MA 106 , then I am allowed to take CS 250.  I have not taken MA 106. Inference Rule: Modus Tollens Lecture 1.4 - Rules of Inference

10. p Tautology: p  (p  q)  p  q Rules of Inference – Addition I am not a great skater.  I am not a great skater or I am tall. Inference Rule: Addition Lecture 1.4 - Rules of Inference

11. p  q Tautology: (p  q) p  p Rules of Inference – Simplification I am not a great skater and you are sleepy.  you are sleepy. Inference Rule: Simplification Lecture 1.4 - Rules of Inference

12. p  q q Tautology: ((p  q)  q)  p  p Rules of Inference – Disjunctive Syllogism I am a great eater or I am a great skater. I am not a great skater.  I am a great eater! Inference Rule: Disjunctive Syllogism Lecture 1.4 - Rules of Inference

13. p  q q  r Tautology: ((p  q)  (q  r))  (p  r)  p  r Rules of Inference – Hypothetical Syllogism If you are an athlete, you are always hungry. If you are always hungry, you have a snickers in your backpack.  If you are an athlete, you have a snickers in your backpack. Inference Rule: Hypothetical Syllogism Lecture 1.4 - Rules of Inference

14. Addition Modus Tollens Examples Amy is a computer science major.  Amy is a math major or a computer science major. • If Ernie is a math major then Ernie is geeky. • Ernie is not geeky! •  Ernie is not a math major. Lecture 1.4 - Rules of Inference

15. 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

16. 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

17. 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

18. 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

19. 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

20. Today’s Reading • Rosen 1.6 • Please start solving the exercises at the end of each chapter section. They are fun. Lecture 1.4 - Rules of Inference