Discrete Structures Rules of inference Dr Muhammad Humayoun

Discrete Structures Rules of inference Dr. Muhammad Humayoun Assistant Professor COMSATS Institute of Computer Science, Lahore. mhumayoun@ciitlahore. edu. pk https: //sites. google. com/a/ciitlahore. edu. pk/dstruct/ 1

Rules of Inference Valid Arguments in Propositional Logic l Assume you are given the following two statements: § “if you are in this class, then you will get a grade” § “you are in this class” Therefore, § “You will get a grade” 2

Modus Ponens (Latin for “the way that affirms by affirming” • • If it snows today, then we will go skiing Hypothesis: It is snowing today By modus ponens, the conclusion is: We will go skiing 3

• If I smoke, then I cough • I Smoke ____________ • I cough 4

Modus Tollens (Latin for "the way that denies by denying") ● Assume you are given the following two statements: § “you will not get a grade” § “if you are in this class, you will get a grade” ● Let p = “you are in this class” ● Let q = “you will get a grade” ● By Modus Tollens, you can conclude that you are not in this class 5

Addition • If you know that p is true, then p q will ALWAYS be true i. e. p → p q 6

Addition • 7

Simplification • If p q is true, then p will ALWAYS be true i. e. p q→p 8

Simplification • If p q is true, then p will ALWAYS be true i. e. p q→p • p: “It is below freezing” • q: “It is raining now” • p q : It is below freezing and raining now. • p q → p: It is below freezing and raining now implies that it is below freezing 9

Hypothetical syllogism • 10

Disjunctive syllogism • 11

Resolution • Computer programs have been developed to automate the task of reasoning and proving theorems. • Many of these programs make use resolution 12

Rules of Inference to Build Arguments • 13

Rules of Inference to Build Arguments • 14

Rules of Inference to Build Arguments • 15

Definitions •

Rules of Inference to Build Arguments • 17

Rules of Inference to Build Arguments • 18

Rules of Inference to Build Arguments • 19

Rules of Inference to Build Arguments • 20

• If you send me an e-mail message, then I will finish writing the program • If you do not send me an e-mail message, then I will go to sleep early • If I go to sleep early, then I will wake up feeling refreshed __________________ • If I do not finish writing the program, then I will wake up feeling refreshed 21

Fallacies • 29

Example • If you do every problem in this book, then you will learn discrete mathematics. • You learned discrete mathematics. • Therefore, you did every problem in this book. • p: You did every problem in this book • q: You learned discrete mathematics 30

• If you do every problem in this book, then you will learn discrete mathematics. • You learned discrete mathematics. ___________________ • Therefore, you did every problem in this book. • p: You did every problem in this book • q: You learned discrete mathematics 31

Rules of Inference for Quantified Statements 34

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Example • Show that the premises: • "Everyone in this discrete mathematics class has taken a course in computer science" and “Aslam is a student in this class" • Imply the conclusion “Aslam has taken a course in computer science. “ 37

• • D(x): x is in this discrete mathematics class C(x): x has taken a course in computer science Premises: ∀x(D(x) → C(x)) and D(Aslam) Conclusion: C(Aslam) 38

• • D(x): x is in this discrete mathematics class C(x): x has taken a course in computer science Premises: ∀x(D(x) → C(x)) and D(Aslam) Conclusion: C(Aslam) • Steps • ∀x(D(x) → C(x)) • D(Aslam) → C(Aslam) Reason Premise Universal instantiation 39

• • D(x): x is in this discrete mathematics class C(x): x has taken a course in computer science Premises: ∀x(D(x) → C(x)) and D(Aslam) Conclusion: C(Aslam) • • Steps ∀x(D(x) → C(x)) D(Aslam) → C(Aslam) D(Aslam) Reason Premise Universal instantiation Premise 40

Example • Show that the premises “A student in this class has not read the book, ” and “Everyone in this class passed the first exam” imply the conclusion “Someone who passed the first exam has not read the book. ” 42

• Show that the premises “A student in this class has not read the book, ” and “Everyone in this class passed the first exam” imply the conclusion “Someone who passed the first exam has not read the book. ” • C(x): “x is in this class” • B(x): “x has read the book” • P(x): “x passed the first exam” • Premises: ? ? ? 43

• Show that the premises “A student in this class has not read the book, ” and “Everyone in this class passed the first exam” imply the conclusion “Someone who passed the first exam has not read the book. ” • C(x): “x is in this class” • B(x): “x has read the book” • P(x): “x passed the first exam” • Premises: ∃x(C(x)∧￢B(x)) and ∀x(C(x) → P(x)). • The conclusion? ? ? 44

• Show that the premises “A student in this class has not read the book, ” and “Everyone in this class passed the first exam” imply the conclusion “Someone who passed the first exam has not read the book. ” • C(x): “x is in this class” • B(x): “x has read the book” • P(x): “x passed the first exam” • Premises: ∃x(C(x) ∧￢B(x)) and ∀x( C(x) → P(x) ). • The conclusion: ∃x( P(x) ∧￢B(x) ) 45

• Premises: ∃x(C(x) ∧￢B(x)) and ∀x( C(x) → P(x) ). • The conclusion: ∃x( P(x) ∧￢B(x) ) 46

• Premises: ∃x(C(x) ∧￢B(x)) and ∀x( C(x) → P(x) ). • The conclusion: ∃x( P(x) ∧￢B(x) ) 47

• Premises: ∃x(C(x) ∧￢B(x)) and ∀x( C(x) → P(x) ). • The conclusion: ∃x( P(x) ∧￢B(x) ) 48

• Premises: ∃x(C(x) ∧￢B(x)) and ∀x( C(x) → P(x) ). • The conclusion: ∃x( P(x) ∧￢B(x) ) 49

• Premises: ∃x(C(x) ∧￢B(x)) and ∀x( C(x) → P(x) ). • The conclusion: ∃x( P(x) ∧￢B(x) ) 50

• Premises: ∃x(C(x) ∧￢B(x)) and ∀x( C(x) → P(x) ). • The conclusion: ∃x( P(x) ∧￢B(x) ) 51

• Premises: ∃x(C(x) ∧￢B(x)) and ∀x( C(x) → P(x) ). • The conclusion: ∃x( P(x) ∧￢B(x) ) 52

• Premises: ∃x(C(x) ∧￢B(x)) and ∀x( C(x) → P(x) ). • The conclusion: ∃x( P(x) ∧￢B(x) ) 53