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Lecture 1.4: Rules of Inference

Lecture 1.4: Rules of Inference. CS 250, Discrete Structures, Fall 2015 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 Questions?. Outline. Rules of Inference.

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Lecture 1.4: Rules of Inference

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  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

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