Lecture 1.4: Rules of Inference CS 250, Discrete Structures, Fall 2012 Nitesh Saxena Adopted from previous lectures by Cinda Heeren

Lecture 1.4 - Rules of Inference Course Admin Slides from previous lectures all posted Expect HW1 to be coming in around coming Monday Thanks for taking the competency exam Your graded exams will be returned to you next week Again, your scores will NOT affect your grade in this course in any way 5/3/2019 Lecture 1.4 - Rules of Inference

Lecture 1.4 - Rules of Inference Outline Rules of Inference 5/3/2019 Lecture 1.4 - Rules of Inference

Lecture 1.4 - Rules of Inference 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? 5/3/2019 Lecture 1.4 - Rules of Inference

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 5/3/2019 Lecture 1.4 - Rules of Inference

Lecture 1.4 - Rules of Inference What rules we study Modus Ponens Modus Tollens Addition Simplification Disjunctive Syllogism Hypothetical Syllogism 5/3/2019 Lecture 1.4 - Rules of Inference

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? 5/3/2019 Lecture 1.4 - Rules of Inference

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 Inference Rule: Modus Ponens Tautology: (p  (p  q))  q 5/3/2019 Lecture 1.4 - Rules of Inference

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 Inference Rule: Modus Tollens Tautology: (q  (p  q))  p 5/3/2019 Lecture 1.4 - Rules of Inference

Rules of Inference – Addition I am not a great skater.  I am not a great skater or I am tall. p  p  q Inference Rule: Addition Tautology: p  (p  q) 5/3/2019 Lecture 1.4 - Rules of Inference

Rules of Inference – Simplification I am not a great skater and you are sleepy.  you are sleepy. p  q  p Inference Rule: Simplification Tautology: (p  q)  p 5/3/2019 Lecture 1.4 - Rules of Inference

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 Inference Rule: Disjunctive Syllogism Tautology: ((p  q)  q)  p 5/3/2019 Lecture 1.4 - Rules of Inference

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 Tautology: ((p  q)  (q  r))  (p  r) p  q q  r  p  r 5/3/2019 Lecture 1.4 - Rules of Inference

Lecture 1.4 - Rules of Inference Examples Addition 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. Modus Tollens 5/3/2019 Lecture 1.4 - Rules of Inference

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 5/3/2019 Lecture 1.4 - Rules of Inference

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) Ellen is smart! 5/3/2019 Lecture 1.4 - Rules of Inference

Rules of Inference: Common Fallacies Rules of inference, appropriately applied give valid arguments. Mistakes in applying rules of inference are called fallacies. 5/3/2019 Lecture 1.4 - Rules of Inference

Rules of Inference: Common Fallacies If I am Bonnie Blair, then I skate fast I skate fast!  I am Bonnie Blair Nope ((p  q)  q)  p Not a tautology. If you don’t give me $10, I bite your ear. I bite your ear!  You didn’t give me $10. Nope 5/3/2019 Lecture 1.4 - Rules of Inference

Rules of Inference: Common Fallacies If it rains then it is cloudy. It does not rain.  It is not cloudy Nope ((p  q)  p)  q Not a tautology. If it is a car, then it has 4 wheels. It is not a car.  It doesn’t have 4 wheels. Nope 5/3/2019 Lecture 1.4 - Rules of Inference

Lecture 1.4 - Rules of Inference Today’s Reading Rosen 1.6 Please start solving the exercises at the end of each chapter section. They are fun. 5/3/2019 Lecture 1.4 - Rules of Inference