PPTX
advertisement
Welcome to CS/MATH 320L – Applied Discrete Mathematics Spring 2015 Instructor: Marc Pomplun January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets 1 Instructor – Marc Pomplun Office: S-3-171 Lab: S-3-135 Office Hours: Tuesdays 4:00pm – 5:30pm Thursdays 7:00pm – 8:30pm Phone: 287-6443 (office) 287-6485 (lab) E-Mail: marc@cs.umb.edu Website: http://www.cs.umb.edu/~marc/cs320/ January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets 2 The Visual Attention Lab Eye movement research January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets 3 The new EyeLink-2K System January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets 4 Example: Distribution of Visual Attention January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets 5 Selectivity in Complex Scenes January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets 6 Selectivity in Complex Scenes January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets 7 Selectivity in Complex Scenes January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets 8 Selectivity in Complex Scenes January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets 9 Selectivity in Complex Scenes January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets 10 Selectivity in Complex Scenes January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets 11 Modeling of Brain Functions January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets 12 Modeling of Brain Functions unit and connection in the interpretive network layer l +1 unit and connection in the gating network unit and connection in the top-down bias network layer l layer l -1 January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets 13 Computer Vision: January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets 14 Human-Computer Interfaces: January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets 15 Now back to CS 320L: Course Kit: Kenneth H. Rosen, Discrete Mathematics and its Applications 7th Edition (Available at the UMB Bookstore) On the Web: http://www.cs.umb.edu/~marc/cs320/ (contains all kinds of course information and also my slides in PDF and PPT formats, updated after each session) January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets 16 Your Evaluation • 4 sets of exercises each set 5% 20% (only individual submissions allowed) • midterm (75 minutes) 35% • final exam (2.5 hours) 45% January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets 17 Grading For the assignments, exams and your course grade, the following scheme will be used to convert percentages into letter grades: 95%: A 90%: A- 86%: B+ 82%: B 78%: B- 74%: C+ 70%: C 66%: C- 62%: D+ 56%: D 50%: D- 50%: F January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets 18 Academic Dishonesty You are allowed to discuss problems regarding your homework with other students in the class. However, you have to do the actual work (computing values, writing algorithms, drawing graphs, etc.) by yourself. You cannot copy anything from other sources (Wikipedia, other students’ work, etc.) The first violation will result in zero points for the entire homework or exam (and official notification). The second violation will result in failing the course. January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets 19 Complaints about Grading If you think that the grading of your assignment or exam was unfair, • write down your complaint (handwriting is OK), • attach it to the assignment or exam, • and give it to me or put it in my mailbox. I will re-grade the whole exam/assignment and return it to you in class. January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets 20 Why Care about Discrete Math? • Digital computers are based on discrete “atoms” (bits). • Therefore, both a computer’s – structure (circuits) and – operations (execution of algorithms) can be described by discrete math. January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets 21 Syllabus • • • • • • • • • • Logic and Set Theory Functions and Sequences Algorithms Applications of Number Theory Mathematical Reasoning Counting Probability Theory Relations and Equivalence Relations Graphs and Trees Boolean Algebra January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets 22 Mathematical Appetizers Useful tools for discrete mathematics: • • • • Logic Set Theory Functions Sequences January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets 23 Logic • Crucial for mathematical reasoning • Used for designing electronic circuitry • Logic is a system based on propositions. • A proposition is a statement that is either true or false (not both). • We say that the truth value of a proposition is either true (T) or false (F). • Corresponds to 1 and 0 in digital circuits January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets 24 The Statement/Proposition Game “Elephants are bigger than mice.” Is this a statement? yes Is this a proposition? yes What is the truth value of the proposition? true January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets 25 The Statement/Proposition Game “520 < 111” Is this a statement? yes Is this a proposition? yes What is the truth value of the proposition? false January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets 26 The Statement/Proposition Game “y > 5” Is this a statement? yes Is this a proposition? no Its truth value depends on the value of y, but this value is not specified. We call this type of statement a propositional function or open sentence. January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets 27 The Statement/Proposition Game “Today is January 29 and 99 < 5.” Is this a statement? yes Is this a proposition? yes What is the truth value of the proposition? false January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets 28 The Statement/Proposition Game “Please do not fall asleep.” Is this a statement? no It’s a request. Is this a proposition? no Only statements can be propositions. January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets 29 The Statement/Proposition Game “If elephants were red, they could hide in cherry trees.” Is this a statement? yes Is this a proposition? yes What is the truth value of the proposition? probably false January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets 30 The Statement/Proposition Game “x < y if and only if y > x.” Is this a statement? yes Is this a proposition? yes … because its truth value does not depend on specific values of x and y. What is the truth value of the proposition? January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets true 31 Combining Propositions As we have seen in the previous examples, one or more propositions can be combined to form a single compound proposition. We formalize this by denoting propositions with letters such as p, q, r, s, and introducing several logical operators. January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets 32 Logical Operators (Connectives) We will examine the following logical operators: • Negation • Conjunction • Disjunction • Exclusive or • Implication • Biconditional (NOT) (AND) (OR) (XOR) (if – then) (if and only if) Truth tables can be used to show how these operators can combine propositions to compound propositions. January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets 33 Negation (NOT) Unary Operator, Symbol: January 29, 2015 P P true false false true Applied Discrete Mathematics Week 1: Logic and Sets 34 Conjunction (AND) Binary Operator, Symbol: January 29, 2015 P Q PQ true true true true false false false true false false false false Applied Discrete Mathematics Week 1: Logic and Sets 35 Disjunction (OR) Binary Operator, Symbol: January 29, 2015 P Q PQ true true true true false true false true true false false false Applied Discrete Mathematics Week 1: Logic and Sets 36 Exclusive Or (XOR) Binary Operator, Symbol: January 29, 2015 P Q PQ true true false true false true false true true false false false Applied Discrete Mathematics Week 1: Logic and Sets 37 Implication (if - then) Binary Operator, Symbol: January 29, 2015 P Q PQ true true true true false false false true true false false true Applied Discrete Mathematics Week 1: Logic and Sets 38 Biconditional (if and only if) Binary Operator, Symbol: January 29, 2015 P Q PQ true true true true false false false true false false false true Applied Discrete Mathematics Week 1: Logic and Sets 39 Statements and Operators Statements and operators can be combined in any way to form new statements. P true Q P Q (P)(Q) true false false false true false false true true false true true true false false false true January 29, 2015 true Applied Discrete Mathematics Week 1: Logic and Sets true 40 Statements and Operations Statements and operators can be combined in any way to form new statements. P Q true true PQ (PQ) (P)(Q) true false false true false false true true false true false true true false false false true true January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets 41 Equivalent Statements P true Q (PQ) (P)(Q) (PQ)(P)(Q) true false false true true false true true true false true true true true false false true true true The statements (PQ) and (P)(Q) are logically equivalent, because (PQ)(P)(Q) is always true. January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets 42 Tautologies and Contradictions A tautology is a statement that is always true. Examples: • R(R) • (PQ)(P)(Q) If ST is a tautology, we write ST. If ST is a tautology, we write ST. January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets 43 Tautologies and Contradictions A contradiction is a statement that is always false. Examples: • R(R) • ((PQ)(P)(Q)) The negation of any tautology is a contradiction, and the negation of any contradiction is a tautology. January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets 44 Exercises We already know the following tautology: (PQ) (P)(Q) Nice home exercise: Show that (PQ) (P)(Q). These two tautologies are known as De Morgan’s laws. January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets 45
