Lecture 1.4: Rules of Inference. CS 250, Discrete Structures, Fall Nitesh Saxena. Adopted from previous lectures by Cinda Heeren
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1 Lecture 1.4: Rules of 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? 9/2/2015 Lecture Rules of 2 1
2 Outline Rules of 9/2/2015 Lecture Rules of 3 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? 9/2/2015 Lecture Rules of 4 2
3 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 9/2/2015 Lecture Rules of 5 What rules we study 1. Modus Ponens 2. Modus Tollens 3. Addition 4. Simplification 5. Disjunctive Syllogism 6. Hypothetical Syllogism 9/2/2015 Lecture Rules of 6 3
4 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? 9/2/2015 Lecture Rules of 7 Rules of 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 q p q (p (p q)) q Modus Ponens 9/2/2015 Lecture Rules of 8 4
5 Rules of 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 p q ( q (p q)) p Modus Tollens 9/2/2015 Lecture Rules of 9 Rules of Addition I am not a great skater. I am not a great skater or I am tall. p p q p (p q) Addition 9/2/2015 Lecture Rules of 10 5
6 Rules of Simplification I am not a great skater and you are sleepy. you are sleepy. p p q (p q) p Simplification 9/2/2015 Lecture Rules of 11 Rules of 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 p q q ((p q) q) p Disjunctive Syllogism 9/2/2015 Lecture Rules of 12 6
7 Rules of 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 ((p q) (q r)) (p r) Hypothetical Syllogism 9/2/2015 Lecture Rules of 13 Examples Amy is a computer science major. Addition 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. Modus Tollens 9/2/2015 Lecture Rules of 14 7
8 Complex Example: Rules of 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 9/2/2015 Lecture Rules of 15 Complex Example: Rules of 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) Ellen is smart! 9/2/2015 Lecture Rules of 16 8
9 Rules of : Common Fallacies Rules of inference, appropriately applied give valid arguments. Mistakes in applying rules of inference are called fallacies. 9/2/2015 Lecture Rules of 17 Rules of : 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! ((p q) q) p Not a tautology. You didn t give me $10. Nope 9/2/2015 Lecture Rules of 18 9
10 Rules of : 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. ((p q) p) q Not a tautology. It doesn t have 4 wheels. Nope 9/2/2015 Lecture Rules of 19 Today s Reading Rosen 1.6 Please start solving the exercises at the end of each chapter section. They are fun. 9/2/2015 Lecture Rules of 20 10
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