CSCI 2824, Discrete Structures Fall 2017 Tony Wong. Lecture 1: Introduction and Binary Arithmetic

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1 CSCI 2824, Discrete Structures Fall 2017 Tony Wong Lecture 1: Introduction and Binary Arithmetic 1

2 What does Discrete Structures mean? 2

3 What does Discrete Structures mean? It s the computer sciency way of saying discrete math 3

4 What does Discrete Structures mean? It s the computer sciency way of saying discrete math Okay then what is discrete math? 4

5 What does Discrete Structures mean? It s the computer sciency way of saying discrete math Okay then what is discrete math? Well, there s continuous math - like derivatives and integrals 5

6 What does Discrete Structures mean? It s the computer sciency way of saying discrete math Okay then what is discrete math? Well, there s continuous math - like derivatives and integrals and then there s discrete math - like counting, sorting, enumeration 6

7 Graph theory: Bridges of Konigsberg The city of Konigsberg has two islands formed by a river with seven bridges connecting the islands and the mainland. Is there a path that traverses each bridge exactly once? 7

8 Logic Two doors riddle: 8

9 Logic Two doors riddle: - Two doors, guarded by two creatures. - One door goes where you want to, but the other leads to certain death. - One always lies, and one always tells the truth. - How can you ask only one of them only one question to discover which door is which? 9

10 Logic 100 prisoners and a lightbulb riddle: - There are 100 prisoners in separate cells. - There is a room they can visit one at a time. - There is a light in the room that they can turn on or off. - Any of the prisoners can claim that all of them have visited the room at least once. - If they re right, all the prisoners are set free. - If they re wrong, they re all executed. 10

- Any of the prisoners can claim that all of them have visited the room at least once.

11 Logic 100 prisoners and a lightbulb riddle: - There are 100 prisoners in separate cells. - There is a room they can visit one at a time. - There is a light in the room that they can turn on or off. - Any of the prisoners can claim that all of them have visited the room at least once. - If they re right, all the prisoners are set free. - If they re wrong, they re all executed. - How can the prisoners conspire just once at the beginning, and then develop a strategy to be set free? 11

12 Discrete Probability Monty Hall problem: Let s Make a Deal - There are three doors. - One has a nice prize behind it 12

13 Discrete Probability Monty Hall problem: Let s Make a Deal - There are three doors. - One has a nice prize behind it - and the other two have goats. - You get to pick a door and will be awarded the prize behind it. - Then the host reveals a goat behind one of the other two doors. - You now have the option to stick with your original door or switch. - Should you stick with your original door or switch? Or does it not matter? 13

14 Discrete Probability This is a good example of what makes discrete structures/math unique: That we re dealing with discrete events: - the cash is behind door 1, 2 or 3. As opposed to continuous events: - the cash is hiding at one of the real numbers between 0.5 and 3.5. (you d have pretty much 0 probability of winning that game, depending on the rounding ) 14

3. As opposed to continuous events: - the cash is hiding at one of the real

15 Discrete Probability This is a good example of what makes discrete structures/math unique: That we re dealing with discrete events: - the cash is behind door 1, 2 or 3. As opposed to continuous events: - the cash is hiding at one of the real numbers between 0.5 and 3.5. (you d have pretty much 0 probability of winning that game, depending on the rounding ) Real applications: (just some vague examples) Complexity of algorithms (100 prisoners and a light bulb) Anything related to data science 15

16 Recursion Tower of Hanoi Move the stack of discs from one peg to another without placing a larger disc on top of a smaller one. 16

17 Recursion Tower of Hanoi Move the stack of discs from one peg to another without placing a larger disc on top of a smaller one. Real applications: Recursions often yield more readable code, as well as elegant solutions Recursions are often frowned upon because of stack memory issues Still, recursions can shed some light on how a nice for loops can work 17

18 Number Theory Puzzles: Show that all four-digit palindromic numbers are divisible by 11 18

19 Number Theory Puzzles: Show that all four-digit palindromic numbers are divisible by 11 if you repeat a three-digit number twice to obtain a six-digit number, the result will be divisible by 7, 11, and 13. Furthermore, if you divide the six-digit number by 7, 11, and 13 you'll get back the original three-digit number. 19

20 Number Theory Puzzles: Show that all four-digit palindromic numbers are divisible by 11 if you repeat a three-digit number twice to obtain a six-digit number, the result will be divisible by 7, 11, and 13. Furthermore, if you divide the six-digit number by 7, 11, and 13 you'll get back the original three-digit number. if you choose a digit from 1-9 and repeat it three times to form a three-digit number, the result divided by the sum of the three digits will 20 always be 37

21 Number Theory Real applications: Public key encryption/rsa security Psuedorandom number generation super important, it turns out Hash functions Showing off to your parents/friends (depending on how cool your friends are) 21

22 Changing gear: Course Logistics What you need Mathematical curiosity or at least not math-averse. Do some programming Expect most homework assignments to include some Python component 22

23 Changing gear: Course Logistics What you need Mathematical curiosity or at least not math-averse. Do some programming Expect most homework assignments to include some Python component Some form of the book Old/international versions/pdf are okay, but make sure you have matched the appropriate sections 23

24 Changing gear: Course Logistics Structure: Weekly homework assignments (40%) Half online via Moodle, half written Participation (10%) Tutorial problems Midterm exam (20%) Final exam (30%) 24

25 Changing gear: Course Logistics Structure: Weekly homework assignments (40%) Half online via Moodle, half written Participation (10%) Tutorial problems Midterm exam (20%) Final exam (30%) Moodle enrollment keys for our classes: csci (9a) key is csci2824_f17_002 csci (11a) key is csci2824_f17_003 25

26 Changing gear: Course Logistics Structure: Weekly homework assignments (40%) Half online via Moodle, half written Participation (10%) Tutorial problems Midterm exam (20%) Final exam (30%) Must have combined exam average > 50% to get a C- or better in the class. 26

27 Changing gear: Course Logistics Structure: Weekly homework assignments (40%) Half online via Moodle, half written Participation (10%) Tutorial problems Midterm exam (20%) Must have combined exam average > 50% to get a C- or Final exam (30%) better in the class. Keep track of things via the course webpage (Piazza) Read the syllabus for more details! Especially regarding due date and collaboration policies Let us know about special needs as soon as possible 27

28 Communicating with Piazza We will use Piazza to manage all communications: Announcements and course materials (notes, supplemental material, the syllabus and other logistical notes) will be posted here 28

Communicating with Piazza We will use Piazza to manage all communications: https://piazza.

29 Communicating with Piazza We will use Piazza to manage all communications: Announcements and course materials (notes, supplemental material, the syllabus and other logistical notes) will be posted here You (yes, you!) can ask questions in an open forum and responses can be posted by other students or instructors/cas/graders Discuss assigned work (but of course do not post answers/vital code) Read the other questions before posting your own! If you have a question specific to one instructor, be sure to use the appropriate salutation (e.g., Hey Chris, or Hey Tony, ) Do not send us ! Send us message via Piazza instead. 29

30 Expectations Come to class prepared, at least skim the material beforehand Be willing to contribute To classroom discussion As well as on Piazza (students helping one another is one of the reasons why we use it!) Be on time. If you are late, do not be disruptive If you use a laptop, sit in the back so you do not distract others (unless you have a documented special need) More on this in a moment Handwritten notes and notes on a tablet are okay anywhere. 30

31 Expectations Let s talk laptops. Results showed that students who used laptops in class spent considerable time multitasking and that the laptop use posed a significant distraction to both users and fellow students. Most importantly, the level of laptop use was negatively related to several measures of student learning, including self-reported understanding of course material and overall course performance. Also: And: (... and others ) 31

About Me Call me Tony. Or Dr. Wong if you re more comfortable with that. First year teaching in CS. Before this, I was a postdoc at Penn State s Earth and Environmental Systems Institute.

32 About Me Call me Tony. Or Dr. Wong if you re more comfortable with that. First year teaching in CS. Before this, I was a postdoc at Penn State s Earth and Environmental Systems Institute. And taught Earth Science. Taught throughout grad school in APPM. Research interests: applying statistics to climate science Office: ECOT 737 Office hours: W Th 12:30-2pm / by appointment For class: through Piazza only (this keeps things organized) If necessary, be sure to make it clear to which instructor your Piazza message is specific to in the salutation. 32

33 Enough chit chat. Let s get to work! Binary Arithmetic There is a nice overview of binary arithmetic available here: 33

34 Enough chit chat. Let s get to work! Binary Arithmetic There is a nice overview of binary arithmetic available here: We are most used to using decimal arithmetic (base-10) Counting on your fingers or toes? Why do we care? Computers are good are knowing whether something is on (1) or off (0) Two options base-2 system 34

Enough chit chat. Let s get to work! Binary Arithmetic There is a nice overview of binary arithmetic available here: http://www.cs.colorado.edu/~srirams/courses/csci2824-spr14/binarylecture.

35 Enough chit chat. Let s get to work! Binary Arithmetic There is a nice overview of binary arithmetic available here: We are most used to using decimal arithmetic (base-10) Counting on your fingers or toes? Why do we care? Computers are good are knowing whether something is on (1) or off (0) Two options base-2 system Example: holding down Ctrl (0/1)? Alt (0/1)? Del (0/1)? Abort only if (111) 35

36 For example: The decimal number implies that we have: For another example: The binary number follows the same pattern, but with powers of 2 instead of 10. So what is this in decimal form? 36

37 For example: The decimal number implies that we have: 5 x x x x x 104 = For another example: The binary number follows the same pattern, but with powers of 2 instead of 10. So what is this in decimal form? 37

38 For another example: The binary number follows the same pattern, but with powers of 2 instead of 10. So what is this in decimal form? We have: 38

39 For another example: The binary number follows the same pattern, but with powers of 2 instead of 10. So what is this in decimal form? We have: 1 x x x x x 24 = = 27 39

40 Some general rules: 1. We chop off leading 0s a. Just like the decimal number 0123 is written as 123, the binary number is written as Gosh, it would be cumbersome to keep writing the decimal number this and the binary number that. Surely there must be an easier way! 40

41 Some general rules: 1. We chop off leading 0s a. Just like the decimal number 0123 is written as 123, the binary number is written as Gosh, it would be cumbersome to keep writing the decimal number this and the binary number that. Surely there must be an easier way! a. There is! We write the base (10 for decimal, 2 for binary, e.g.) as a subscript next to the number. b. So from the previous example, 41

42 Converting the number N from decimal to binary Start from right (the ones place) and move to the left. 42

43 Converting the number N from decimal to binary Start from right (the ones place) and move to the left. If the first digit is odd, then the first bit is a 1. Otherwise, it s a 0. Reset N (N-1) / 2 if N was odd. Reset N N/2 if N was even. 43

44 Converting the number N from decimal to binary Start from right (the ones place) and move to the left. If the first digit is odd, then the first bit is a 1. Otherwise, it s a 0. Reset N (N-1) / 2 if N was odd. Reset N N/2 if N was even. Move to the next bit to the left. Repeat procedure until N either is 0 or 1. 44

45 Converting the number N from decimal to binary Start from right (the ones place) and move to the left. If the first digit is odd, then the first bit is a 1. Otherwise, it s a 0. Reset N (N-1) / 2 if N was odd. Reset N N/2 if N was even. Move to the next bit to the left. Repeat procedure until N either is 0 or 1. Example: Convert 2824 from decimal to binary. 45

46 Converting the number N from decimal to binary Example: Convert 2824 from decimal to binary. First digit is even, so first bit is a 0. Next iteration, we have 2824/2 =

47 Converting the number N from decimal to binary Example: Convert 2824 from decimal to binary. 47

48 Converting the number N from decimal to binary Example: Convert 2824 from decimal to binary. First digit is even, so first bit is a 0. Next iteration, we have 2824/2 = First digit is even, so second bit is a 0. Next iteration, we have 1412/2 = 706. First digit is even, so third bit is 0. Next iteration, we have 706/2 = 353. First digit is odd, so fourth bit is 1. Next iteration, we have (353-1)/2 = 176. First digit is even, so fifth bit is 0. Next iteration, we have 176/2 = 88. First digit is even, so sixth bit is 0. eventually, find =

49 Representing fractions as binary We can! First bit right of the radix point represents ½. Second bit on the right represents ¼. Third bit represents ⅛. and so on. 49

50 Representing fractions as binary We can! First bit right of the radix point represents ½. Second bit on the right represents ¼. Third bit represents ⅛. and so on. Example: Convert 0.75 from decimal to binary. 50

51 Representing fractions as binary We can! First bit right of the radix point represents ½. Second bit on the right represents ¼. Third bit represents ⅛. and so on. Example: Convert 0.75 from decimal to binary = ¾ = ½ + ¼ =

52 Representing fractions as binary A tougher example: Convert from decimal to binary. We need a better algorithm, like for the big numbers: Multiply N by 2. Next bit is the digit in the ones place. Proceed with new N as the part to the right of the decimal point. Continue until you have 0 remaining. 52

53 Representing fractions as binary A tougher example: Convert from decimal to binary. We need a better algorithm, like for the big numbers: Multiply N by 2. Next bit is the digit in the ones place. Proceed with new N as the part to the right of the decimal point. Continue until you have 0 remaining. Quick check: does this work for = 0.112? 53

54 Representing fractions as binary A tougher example: Convert from decimal to binary. We need a better algorithm, like for the big numbers: Multiply N by 2. Next bit is the digit in the ones place. Proceed with new N as the part to the right of the decimal point. Continue until you have 0 remaining * 2 = 1.75, so first bit right of the radix point is a * 2 = 1.5, so next bit is a 1. Continue with Continue with * 2 = 1.0, so next bit is a 1. Continue with 0. Exit Left with =

55 Representing fractions as binary Example: Convert from decimal to binary. 55

56 Representing fractions as binary Example: Convert from decimal to binary. Do the part left of the decimal first. 123 Odd, so first bit is a 1. Proceed with (123-1)/2 =

57 Representing fractions as binary Example: Convert from decimal to binary. Do the part left of the decimal first. 123 Odd, so first bit is a 1. Proceed with (123-1)/2 = Odd, so next bit is a 1. Proceed with (61-1)/2 = Even, so next bit is 0. Proceed with 30/2 = Odd, so next bit is a 1. Proceed with (15-1)/2 = 7. 7 Odd, so next bit is a 1. Proceed with (7-1)/2 = 3. 3 Odd, so next bit is a 1. Proceed with (3-1)/2 = 1. 1 is a 1, so last bit is a 1. Put it all together: =

58 Representing fractions as binary Example: Convert from decimal to binary. Do the part right of the decimal now * 2 = first bit is a 0 58

59 Representing fractions as binary Example: Convert from decimal to binary. Do the part right of the decimal now * 2 = first bit is a * 2 = next bit is a * 2 = next bit is a * 2 = next bit is a * 2 = next bit is a * 2 = next bit is a * 2 = next bit is a 1 eventually find = so =

60 Representing fractions as binary Example: Convert 0.1 from decimal to binary. 60

61 Representing fractions as binary Example: Convert 0.1 from decimal to binary. 0.1 * 2 = 0.2, so first bit is a * 2 = 0.4, so next bit is a * 2 = 0.8, so next bit is a * 2 = 1.6, so next bit is a * 2 = 1.2, so next bit is a * NOW HOLD ON A SECOND! Restarts the pattern from here. 61

62 Representing fractions as binary Example: Convert 0.1 from decimal to binary. 0.1 * 2 = 0.2, so first bit is a * 2 = 0.4, so next bit is a * 2 = 0.8, so next bit is a * 2 = 1.6, so next bit is a * 2 = 1.2, so next bit is a * NOW HOLD ON A SECOND! Restarts the pattern from here. So =

63 Representing fractions as binary Example: Convert 0.1 from decimal to binary. So = So computers would need to store an infinite number of bits in order to store exactly. obviously, computers can t do that. They truncate at a certain point, leading to some error. 63

64 Representing fractions as binary Example: Convert 0.1 from decimal to binary. So = So computers would need to store an infinite number of bits in order to store exactly. obviously, computers can t do that. They truncate at a certain point, leading to some error. How bad can this error be? 64

65 Representing fractions as binary How bad can this error be? Consider the function f(a,b) = b6 + a2(11a2b2 - b6-121b4-2) + 5.5b8 + a/2b where a=77617 and b=

66 Representing fractions as binary How bad can this error be? Consider the function f(a,b) = b6 + a2(11a2b2 - b6-121b4-2) + 5.5b8 + a/2b where a=77617 and b= If you run this on a 64-bit machine, you ll find f(a,b) = , or maybe f(a,b) = e+21 (depending on the way the program you code in truncates) But the true answer is more like f(a,b) =

67 Introduction and Binary Arithmetic Recap: Today, we Confirmed that we are in the right classroom at the right time Learned what the goals for this class are Learned how to represent decimal numbers as binary, and vice versa Next time: We talk logic! 67

When using computers, it should have a minimum number of easily identifiable states.

When using computers, it should have a minimum number of easily identifiable states. EET 3 Chapter Number Systems (B) /5/4 PAGE Number Systems (B) Number System Characteristics (Tinder) What s important in choosing a number system? Basically, there are four important characteristics desirable