CSE 240 Logic and Discrete Mathematics

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1 CSE 240 Logic and Discrete Mathematics Instructor: odd Sproull Department of Computer Science and Engineering Washington University in St. Louis 1Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 1 CSE 240 Course Information ime and Location Hillman 60 Monday and Wednesday 8:30AM 10 AM Website Instructor Address todd@wustl.edu Book Discrete Mathematics and Its Applications by Rosen 8 th edition Office Hours By Appointment Head A Andrew Mitra andrewmitra@wustl.edu What is this course about? An introduction to mathematical fundamentals needed by a Computer Scientist Why is mathematics needed? 2Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 2

2 Grading hree in class exams 18% of total grade per exam (54% total) ~10 Homework Assignments 36% of total grade Lowest score dropped ~10 Short Quizzes 10% of total grade Lowest score dropped Also Group Assignments that can only improve your grade No inal Exam 3Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 3 Some of the opics Covered Logic Is this statement rue or alse? Proofs Can you prove that your claim is rue? Direct, Indirection, and Proofs by Contradiction Mathematical Induction Counting How many ways can I pick n items from a set of m? How do I win at BlackJack? Probability Existence proofs Expected value, variance Graphs Can we model a collection of computers connected together? inite State Machines Can a computer carry out this task? How do we model computation? 4Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 4

3 Reading Assignment Read Sections by Wednesday 5Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 5 Section 1.1 Propositional Logic Proposition A declarative statement that is either true or false s of propositions = = 4 hree is odd our is odd Not propositions unny Apple What time is it? ypically represented with the letters p, q, r, and s Suppose p 6Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 6

4 Propositional Logic Negation Suppose p is a proposition he negation of p is p It is not the case that p _ Also represent by p hree is NO odd our is NO odd March does NO have 31 days Negate the following At least 10 inches of rain fell today Answer It is not the case that at least 10 inches of rain fell today Less than 10 inches of rain fell today ruth table for negation: p p 7Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 7 Conjunction A conjunction is equivalent to using the word AND Denoted by the Ù symbol Suppose p and q are propositions p Ù q A conjunction is RUE when both p and q are RUE his class is fun AND I like pizza ruth able for conjunction: p q p Ù q 8Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 8

5 Disjunction A disjunction is equivalent to the word OR Denoted by the symbol Ú p Ú q A disjunction is false when both p and q are false oday is riday OR it is raining today ruth table for Disjunction p q p Ú q 9Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 9 Exclusive Or he exclusive or is true when either p or q is RUE But not both p and q Denoted by the symbol Å I will EIHER pay attention in class OR fall asleep ruth able for exclusive or p q p Å q 10Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 10

6 Implications Denoted by the symbol p q corresponds to If p, then q or p implies q or q whenever p If I work hard in this class, then I will earn an A in CSE240 If today is riday, then = 5 (rue or alse? Why?) If today is riday, then = 6 (rue or alse? Why) An implication is also referred to as a conditional statement p q p q ruth able for implications? 11Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 11 ruth table for p q p q p q 12Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 12

7 More about p q Different English representations of p implies q q whenever p x is odd x + 1 is even If x is odd, it is true If x is NO odd, the whole statement is true Propositions can only be rue or alse No undefined p q p q 13Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 13 Logical Equivalence Denoted by the symbol p p corresponds to p is logically equivalent to NO NO p º Consider p v q? p q Are they logically equivalent? How can we prove that? º p q p p v q p q 14Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 14

8 Converse, Contrapositive, and Inverse ormed as variants of the conditional statement p q Converse q p Contrapositive q p Inverse p q s Consider the conditional statement he St. Louis Cardinals win whenever it is raining q whenever p If it is raining, then the Cardinals win if p, then q Converse If the Cardinals win, then it is raining Contrapositive If the Cardinals do not win, then it is not raining Inverse If it is not raining, then the Cardinals do not win 15Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 15 Biconditional Statement Denoted by the symbol «p «q corresponds to p if and only if q or p is necessary and sufficient for q or p iff q p«q is true only when p and q have the same truth values You can take the flight if and only if you buy a ticket p q p «q ruth able for biconditionalstatement 16Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 16

9 Compound Propositions We are able to combine multiple propositions together to build more complicated propositions Construct a truth table for the following proposition (p Ù q) ( p Ú q) p q p p Ù q p Úq (p Ù q) ( p Ú q) 17Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 17 Logic and Bit Operations Computers represent information using bits A bit is a symbol with values 0 and 1 Boolean Variables use values of rue or alse Boolean variables can be represented with a bit Logical operations can be performed by replacing and with 1 and 0 Substitute Ù, Ú, and Å with AND, OR, and XOR Example 0 OR 1 = 1 Bit Strings are a sequence of 0 or more bits length is string is the number of bits Able to perform bitwise operations on bit strings Bitwise OR Bitwise AND Bitwise XOR ruth Value I need some help with my math Bit Extensible - CSE 240 Logic Networking and Discrete Platform Mathematics 18

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