CSE 311: Foundations of Computing I. Lecture 1: Propositional Logic
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1 CSE 311: Foundations of Computing I Lecture 1: Propositional Logic
2 Some Perspective Computer Science and Engineering Programming CSE 14x Theory Hardware CSE 311
3 About the Course We will study the theory needed for CSE: Logic: How can we describe ideas precisely? Formal Proofs: How can we be positive we re correct? Number Theory: How do we keep data secure? Relations/Relational Algebra: How do we store information? Finite State Machines: How do we design hardware and software? Turing Machines: Are there problems computers can t solve?
4 About the Course It s about perspective! Tools for reasoning about difficult problems Tools for communicating ideas, methods, objectives Tools for automating difficult problems Fundamental structures for computer science
5 About the Course It s about perspective! Tools for reasoning about difficult problems Tools for communicating ideas, methods, objectives Tools for automating difficult problems Fundamental structures for computer science This is NOT a programming course!
6 Instructors Paul Beame Kevin Zatloukal Section B MWF 10:30-11:20 in MUE 153 Office Hours: MF 11:30-12:00 and M 3:00-4:00 CSE 668 Section A MWF 1:30-2:20 in SIG 134 Office Hours: WF 2:30-3:00 and M 12:00-1:00 CSE 212 Office hours are for students in both sections
7 TAs and Administrivia Teaching Assistants: Darin Chin Daniel Fuchs YuqiHuang Joy Ji KaiyuZheng Section: Thursdays starting this week Office Hours: TBA Joshua Fan Kush Gupta Sean Jaffe Cheng Ni Homework: Due WED at 11:59 pm online Write up individually Extra Credit Grading (roughly): 50% Homework 15-20% Midterm 30-35% Final Exam (Optional) Book: Rosen: Readings for 6 th (used) or 7 th (cut down) editions. Good for practice with solved problems All Course cs.uw.edu/311
8 Administrivia All Course cs.uw.edu/311
9 Administrivia Midterm: Wed, May 2 in class Final Exam: Mon, Jun 4 A section 2:30-4:20 B section 4:30-6:20 Not at 8:30-10:20 time in exam schedule Location TBA All Course cs.uw.edu/311
10 Logic: The Language of Reasoning Why not use English? Turn right here Buffalo buffalo Buffalo buffalo buffalo buffalo Buffalo buffalo We saw her duck
11 Logic: The Language of Reasoning Why not use English? Turn right here Does right mean the direction or now? Buffalo buffalo Buffalo buffalo buffalo buffalo Buffalo buffalo This means Bison from Buffalo, that bison from Buffalo bully, themselves bully bison from Buffalo. We saw her duck Does duck mean the animal or crouch down?
12 Logic: The Language of Reasoning Why not use English? Turn right here Does right mean the direction or now? Buffalo buffalo Buffalo buffalo buffalo buffalo Buffalo buffalo This means Bison from Buffalo, that bison from Buffalo bully, themselves bully bison from Buffalo. We saw her duck Does duck mean the animal or crouch down? Language like Java or English Words, sentences, paragraphs, arguments Today is about wordsand sentences
13 Why Learn A New Language? Logic, as the language of reasoning, will help us Be more precise Be more concise Figure out what a statement means more quickly
14 Propositions A propositionis a statement that has a truth value, and is well-formed If I were to ask you out, would your answer to that question be the same as your answer to this one?
15 Are These Propositions? = 5 The home page renders correctly in Chrome. Turn in your homework on Wednesday. This statement is false. Akjsdf! Who are you? Every positive even integer can be written as the sum of two primes.
16 Are These Propositions? = 5 This is a proposition. It s okay for propositions to be false. The home page renders correctly in Chrome. This is a proposition. It s okay for propositions to be false. Turn in your homework on Wednesday. This is a command which means it doesn t have a truth value. This statement is false. This statement does not have a truth value! (If it s true, it s false, and vice versa.) Akjsdf! This is not a proposition because it s gibberish. Who are you? This is a question which means it doesn t have a truth value. Every positive even integer can be written as the sum of two primes. This is a proposition. We don t know if it s true or false, but we know it s one of them!
17 Propositions A propositionis a statement that has a truth value, and is well-formed We need a way of talking about arbitraryideas Propositional Variables: Truth Values:
18 Propositions A propositionis a statement that has a truth value, and is well-formed We need a way of talking about arbitraryideas Propositional Variables:,,,, Truth Values: Tfor true Ffor false
19 A Proposition You can get measles and mumps if you didn t have the MMR vaccine, but if you had the MMR vaccine then you can t get either. We d like to understand what this proposition means. This is where logic comes in. There are pieces that appear multiple times in the phrase (e.g., you can get measles ). These are called atomic propositions. Let s list them:
20 A Proposition You can get measles and mumps if you didn t have the MMR vaccine, but if you had the MMR vaccine then you can t get either. We d like to understand what this proposition means. This is where logic comes in. There are pieces that appear multiple times in the phrase (e.g., you can get measles ). These are called atomic propositions. Let s list them: Measles: You can get measles Mumps: You can get mumps MMR: You had the MMR vaccine
21 Putting Them Together You can get measles and mumps if you didn t have the MMR vaccine, but if you had the MMR vaccine then you can t get either. Measles: You can get measles Mumps: You can get mumps MMR: You had the MMR vaccine Now, we put these together to make the sentence: ((Measlesand Mumps) if not MMR) but (if MMRthen not (Measlesor Mumps)) ((Measlesand Mumps) if not MMR) and (if MMRthen not (Measlesor Mumps)) This is the general idea, but now, let s define our formal language.
22 Logical Connectives Negation (not) Conjunction (and) Disjunction (or) Exclusive Or Implication Biconditional
23 Logical Connectives Negation (not) Conjunction (and) Disjunction (or) Exclusive Or Implication Biconditional Measles: Mumps: MMR: You can get measles You can get mumps You had the MMR vaccine You can get measles and mumps if you didn t have the MMR vaccine, but if you had the MMR vaccine then you can t get either. ((Measlesand Mumps) if not MMR) and (if MMRthen not (Measlesor Mumps))
24 Logical Connectives Negation (not) Conjunction (and) Disjunction (or) Exclusive Or Implication Biconditional Measles: Mumps: MMR: You can get measles You can get mumps You had the MMR vaccine You can get measles and mumps if you didn t have the MMR vaccine, but if you had the MMR vaccine then you can t get either. ((Measlesand Mumps) if not MMR) and (if MMRthen not (Measlesor Mumps)) ((Measles Mumps) if MMR) (if MMRthen (Measles Mumps))
25 Some Truth Tables p p p q p q p q p q p q p q
26 Some Truth Tables p p T F F T p q p q T T T T F F F T F F F F p q p q T T T T F T F T T F F F p q p q T T F T F T F T T F F F
27 Implication If it s raining, then I have my umbrella It s useful to think of implications as promises. That is Did I lie? p q p q T T T T F F F T T F F T I have my umbrella I do not have my umbrella It s raining It s not raining
28 Implication If it s raining, then I have my umbrella It s useful to think of implications as promises. That is Did I lie? p q p q T T T T F F F T T F F T I have my umbrella I do not have my umbrella It s raining No Yes It s not raining No No The only lieis when: (a) It s raining AND (b) I don t have my umbrella
29 Implication If it s raining, then I have my umbrella Are these true? = 4 earth is a planet p q p q T T T T F F F T T F F T = 5 26 is prime
30 Implication If it s raining, then I have my umbrella Are these true? = 4 earth is a planet p q p q T T T T F F F T T F F T The fact that these are unrelated doesn t make the statement false! = 4 is true; earth is a planet is true. T T is true. So, the statement is true = 5 26 is prime Again, these statements may or may not be related = 5 is false; so, the implication is true. (Whether 26 is prime or not is irrelevant). Implication is not a causal relationship!
31 (1) I have collected all 151 Pokémon if I am a Pokémon master (2) I have collected all 151 Pokémon only if I am a Pokémon master These sentences are implications in opposite directions:
32 (1) I have collected all 151 Pokémon if I am a Pokémon master (2) I have collected all 151 Pokémon only if I am a Pokémon master These sentences are implications in opposite directions: (1) Pokémon masters have all 151 Pokémon (2) People who have 151 Pokémon are Pokémon masters So, the implications are: (1) IfI am a Pokémon master, theni have collected all 151 Pokémon. (2) IfI have collected all 151 Pokémon, theni am a Pokémon master.
33 Implication: pimplies q whenever pis true qmust be true if pthen q qif p pis sufficient for q ponly if q q is necessary for p p q p q T T T T F F F T T F F T
34 Biconditional: piff q pis equivalent to q pimplies qand qimplies p p is necessary and sufficient for q p q p q
35 Back to our Vaccine Sentence Translation... You can get measles and mumps if you didn t have the MMR vaccine, but if you had the MMR vaccine you can t get either. ((Measles Mumps) if MMR) (if MMRthen (Measles Mumps)) ( MMR (Measles Mumps)) (MMR (Measles Mumps))
36 Understanding the Vaccine Sentence You can get measles and mumps if you didn t have the MMR vaccine, but if you had the MMR vaccine you can t get either. ((Measles Mumps) if MMR) (if MMRthen (Measles Mumps)) ( MMR (Measles Mumps)) (MMR (Measles Mumps)) Define shorthand : MMR : Measles : Mumps ( ) ( )
37 Analyzing the Vaccine Sentence with a Truth Table ( ) ( ) ( ) ( ( )) T T T T T F T F T T F F F T T F T F F F T F F F
38 Analyzing the Vaccine Sentence with a Truth Table ( ) ( ) ( ) ( ( )) T T T F T T T F F F T T F F F T T F F F T F T F F T T F F F T F F F F T F T T T F T T T T T T F T T F T F T F F T F T F F F T T F F T F T F F F F T F F F T T F
39 Biconditional: piff q pis equivalent to q pimplies qand qimplies p p is necessary and sufficient for q p q p q T T T T F F F T F F F T
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