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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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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?

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

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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?

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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

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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

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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?

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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

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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

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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

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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

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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

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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

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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

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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 MM

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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!

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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.

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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.

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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.

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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.

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