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

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Presentation on theme: "Lecture 1.4: Rules of Inference CS 250, Discrete Structures, Fall 2014 Nitesh Saxena Adopted from previous lectures by Cinda Heeren."— Presentation transcript:

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 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 Rules of Inference3 Outline Rules of Inference

4 4/21/2015Lecture 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 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 Rules of Inference6

7 4/21/2015Lecture 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 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 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 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 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 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 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 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 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 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 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 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 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 Rules of Inference20 Today’s Reading Rosen 1.6 Please start solving the exercises at the end of each chapter section. They are fun.


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