1. Lecture 1.4: Rules of Inference CS 250, Discrete Structures, Fall 2014 Nitesh Saxena Adopted from previous lectures by Cinda Heeren

2. 4/21/2015Lecture 1.4 - Rules of Inference2 Course Admin Slides from previous lectures all posted Expect HW1 to be coming in around coming Monday Questions?

3. 4/21/2015Lecture 1.4 - Rules of Inference3 Outline Rules of Inference

4. 4/21/2015Lecture 1.4 - Rules of Inference4 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! How do we know it?

5. 4/21/2015Lecture 1.4 - Rules of Inference5 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. Given set of true statements or previously proved theorems Rules of inference Proof

6. What rules we study 1. Modus Ponens 2. Modus Tollens 3. Addition 4. Simplification 5. Disjunctive Syllogism 6. Hypothetical Syllogism 4/21/2015Lecture 1.4 - Rules of Inference6

7. 4/21/2015Lecture 1.4 - Rules of Inference7 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 What rule of inference can we use to justify it?

8. 4/21/2015Lecture 1.4 - Rules of Inference8 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. p p  q  q Tautology: (p  (p  q))  q Inference Rule: Modus Ponens

9. 4/21/2015Lecture 1.4 - Rules of Inference9 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.  q p  q  p p Tautology: (  q  (p  q))   p Inference Rule: Modus Tollens

10. 4/21/2015Lecture 1.4 - Rules of Inference10 Rules of Inference – Addition I am not a great skater.  I am not a great skater or I am tall. p  p  q Tautology: p  (p  q) Inference Rule: Addition

11. 4/21/2015Lecture 1.4 - Rules of Inference11 Rules of Inference – Simplification I am not a great skater and you are sleepy.  you are sleepy. p  q  p Tautology: (p  q)  p Inference Rule: Simplification

12. 4/21/2015Lecture 1.4 - Rules of Inference12 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! p  q  q  p Tautology: ((p  q)   q)  p Inference Rule: Disjunctive Syllogism

13. 4/21/2015Lecture 1.4 - Rules of Inference13 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. p  q q  r  p  r Tautology: ((p  q)  (q  r))  (p  r) Inference Rule: Hypothetical Syllogism

14. 4/21/2015Lecture 1.4 - Rules of Inference14 Examples Amy is a computer science major.  Amy is a math major or a computer science major. Addition If Ernie is a math major then Ernie is geeky. Ernie is not geeky!  Ernie is not a math major. Modus Tollens

15. 4/21/2015Lecture 1.4 - Rules of Inference15 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M

16. 4/21/2015Lecture 1.4 - Rules of Inference16 Complex Example: Rules of Inference 1. M  CGiven 2.  D   CGiven 3. D  SGiven 4.  MGiven 5. CDS (1,4) 6. DMT (2,5) 7. SMP (3,6) Ellen is smart!

17. 4/21/2015Lecture 1.4 - Rules of Inference17 Rules of Inference: Common Fallacies Rules of inference, appropriately applied give valid arguments. Mistakes in applying rules of inference are called fallacies.

18. 4/21/2015Lecture 1.4 - Rules of Inference18 Rules of Inference: Common Fallacies If I am Bonnie Blair, then I skate fast I skate fast!  I am Bonnie Blair Nope If you don’t give me $10, I bite your ear. I bite your ear!  You didn’t give me $10. Nope ((p  q)  q)  p Not a tautology.

19. 4/21/2015Lecture 1.4 - Rules of Inference19 Rules of Inference: Common Fallacies If it rains then it is cloudy. It does not rain.  It is not cloudy Nope If it is a car, then it has 4 wheels. It is not a car.  It doesn’t have 4 wheels. Nope ((p  q)   p)   q Not a tautology.

20. 4/21/2015Lecture 1.4 - Rules of Inference20 Today’s Reading Rosen 1.6 Please start solving the exercises at the end of each chapter section. They are fun.