* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download CSE 20 - Lecture 14: Logic and Proof Techniques
Turing's proof wikipedia , lookup
History of the function concept wikipedia , lookup
Axiom of reducibility wikipedia , lookup
Modal logic wikipedia , lookup
Jesús Mosterín wikipedia , lookup
Gödel's incompleteness theorems wikipedia , lookup
History of logic wikipedia , lookup
Combinatory logic wikipedia , lookup
Sequent calculus wikipedia , lookup
Foundations of mathematics wikipedia , lookup
Propositional formula wikipedia , lookup
Naive set theory wikipedia , lookup
Laws of Form wikipedia , lookup
Interpretation (logic) wikipedia , lookup
Quantum logic wikipedia , lookup
Truth-bearer wikipedia , lookup
Mathematical logic wikipedia , lookup
Natural deduction wikipedia , lookup
Propositional calculus wikipedia , lookup
Intuitionistic logic wikipedia , lookup
Curry–Howard correspondence wikipedia , lookup
Law of thought wikipedia , lookup
CSE 20 Lecture 14: Logic and Proof Techniques CSE 20: Lecture 14 Midterm Review Representation of integers in base b Logic Proof systems: Direct Proof Proof by contradiction Contraposetive Sets Theory Functions CSE 20: Lecture 14 Midterm Review Representation of integers in base b Logic Proof systems: Direct Proof Proof by contradiction Contraposetive Sets Theory Functions NO CALCULATOR, NO CHEAT SHEET CSE 20: Lecture 14 Use of Propositional Logic Check the correctness of a statement: whether a sentence/paragraph/proposition is logically correct. CSE 20: Lecture 14 Use of Propositional Logic Check the correctness of a statement: whether a sentence/paragraph/proposition is logically correct. Make deductions CSE 20: Lecture 14 Use of Propositional Logic Check the correctness of a statement: whether a sentence/paragraph/proposition is logically correct. Make deductions Check if two propositions are equivalent. CSE 20: Lecture 14 Equivalence of statements/propositions If the truthtables of two statement/propositions are identical then the two statement/propositions are equivalent One can also use various rules for propositional logic. CSE 20: Lecture 14 Rules of Propositional Logic 1 Commutative law: (p ∨ q) = (q ∨ p) and (p ∧ q) = (q ∧ p) 2 Associative law: (p ∨ (q ∨ r)) = ((p ∨ q) ∨ r) and (p ∧ (q ∧ r)) = ((p ∧ q) ∧ r) 3 Distributive law: (p ∨ (q ∧ r)) = (p ∨ q) ∧ (p ∨ r) and (p ∧ (q ∨ r)) = (p ∧ q) ∨ (p ∧ r) 4 De Morgan’s Law: ¬(p ∨ q) = (¬p ∧ ¬q) and ¬(p ∧ q) = (¬p ∨ ¬q) CSE 20: Lecture 14 Equivalence of Statement: iclicker Question 1 Which of the following is equivalent to A =⇒ B A. B. C. D. E. (¬B ∨ A) = F alse (¬A ∧ B) = F alse (¬B ∧ A) = F alse ¬A =⇒ ¬B (¬B ∨ A) CSE 20: Lecture 14 Proof Techniques To prove statement B from A: CSE 20: Lecture 14 Proof Techniques To prove statement B from A: Direct Proof: A =⇒ B CSE 20: Lecture 14 Proof Techniques To prove statement B from A: Direct Proof: A =⇒ B Proof by contradiction: (¬B ∧ A) gives a contradiction CSE 20: Lecture 14 Proof Techniques To prove statement B from A: Direct Proof: A =⇒ B Proof by contradiction: (¬B ∧ A) gives a contradiction Proof by Contraposetive: A =⇒ B is same as proving ¬B =⇒ ¬A. CSE 20: Lecture 14 Sets Sets For example: Set of names of all students CSE 20: Lecture 14 Sets Sets For example: Set of names of all students Set of letters in the english alphabet CSE 20: Lecture 14 Sets Sets For example: Set of names of all students Set of letters in the english alphabet Set of digits. {0, 1, . . . , 9} or {0, 1} Size of a set is the number of elements in the set. Size of set A is denoted by |A|. CSE 20: Lecture 14 Operations on Sets Union, ∪. A ∪ B is the set of all elements that are in A OR B. Intesection, ∩ A ∩ B is the set of all elements that are in A AND B. Complement, Ac or A Ac is the set of elements N OT in A. Cartesan Product. CSE 20: Lecture 14 Rules of Set Theory Let p, q and r be sets. 1 Commutative law: (p ∪ q) = (q ∪ p) and (p ∩ q) = (q ∩ p) 2 Associative law: (p ∪ (q ∪ r)) = ((p ∪ q) ∪ r) and (p ∩ (q ∩ r)) = ((p ∩ q) ∩ r) 3 Distributive law: (p ∪ (q ∩ r)) = (p ∪ q) ∩ (p ∪ r) and (p ∩ (q ∪ r)) = (p ∩ q) ∪ (p ∩ r) 4 De Morgan’s Law: (p ∪ q)c = (pc ∩ q c ) and (p ∩ q)c = (pc ∪ q c ) CSE 20: Lecture 14 Set Theory and Propositional Logic Set Theory and Propositional Logic is two mathematical language which follow very similar rules. CSE 20: Lecture 14 iclicker Question 2 If A and B are two sets such that size of A is 10 and size of B is 12. If size of A ∪ B is 20 what is size of A ∩ B. A B C D E 2 3 10 22 Can’t be said. CSE 20: Lecture 14 iclicker Question 3 If A and B are two sets such that size of A is 10 and size of B is 12. If size of A ∩ B is 4 what is size of A\B. A B C D E 2 3 6 10 Can’t be said. CSE 20: Lecture 14 iclicker Question 4 If A and B are two sets such that size of A is 10 and size of B is 12. How many functions are there from A to B. A B C D E 1012 1210 120 12 None of the above. CSE 20: Lecture 14 Propositional Logic Every statement is either TRUE or FALSE There are logical connectives ∨, ∧, ¬, =⇒ and ⇐⇒ . Two logical statements can be equivalent if the two statements answer exactly in the same way on every input. To check whether two logical statements are equivalent one can do one of the following: Checking the Truthtable of each statement Reducing one to the other using reductions CSE 20: Lecture 14 Predicate Logic There are two important symbols: ∀ and ∃. Some statements can be defined using a variable. For example: Px = “4x2 + 3 is divisible by 5” We can have statements like: ∀x ∈ Z, 4x2 + 3 is divisible by 5. Or ∃x ∈ Z, 4x2 + 3 is divisible by 5. CSE 20: Lecture 14 Rules of negation ¬(∀x, Px ) = (∃x, ¬Px ). ¬(∃x, Px ) = (∀x, ¬Px ). CSE 20: Lecture 14 Negating a sentence: iclicker What is the negation of the sentence: “There is an university in USA where every department has at least 20 faculty and at least one noble laureate.” CSE 20: Lecture 14 Negating a sentence: iclicker What is the negation of the sentence: “There is an university in USA where every department has at least 20 faculty and at least one noble laureate.” There is an university in USA where every department has less than 20 faculty and at least one noble laureate. All universitis in USA where every department has at least 20 faculty and at least one noble laureate. For all universities in USA there is a department has less than 20 faculty or at most one noble laureate. For all universities in USA there is a department has less than 20 faculty and at least one noble laureate. CSE 20: Lecture 14 Is √ 2 a rational? In the√last class we used proof by contradiction to prove that 2 is not rational. CSE 20: Lecture 14 Proof Techniques To prove statement B from A: CSE 20: Lecture 14 Proof Techniques To prove statement B from A: Direct Proof: A =⇒ B Proof by contradiction: (¬B ∧ A) gives a contradiction Proof by Contraposetive: A =⇒ B is same as proving ¬B =⇒ ¬A. CSE 20: Lecture 14 Is √ 2+ √ 3 a rational? Prove that √ √ 2 + 3 is not rational. CSE 20: Lecture 14