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CS 173: Discrete Mathematical Structures

CS 173: Discrete Mathematical Structures. Cinda Heeren heeren@cs.uiuc.edu Rm 2213 Siebel Center Office Hours: W 9:30-11:30a. CS 173 Announcements. Midterm 1: 10/05/06, 7-9p, location on web. Class on Thursday cancelled.

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CS 173: Discrete Mathematical Structures

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  1. CS 173:Discrete Mathematical Structures Cinda Heeren heeren@cs.uiuc.edu Rm 2213 Siebel Center Office Hours: W 9:30-11:30a

  2. CS 173 Announcements • Midterm 1: 10/05/06, 7-9p, location on web. • Class on Thursday cancelled. • Sections this week will be exam review. Section that meets Thur, 6-7p is cancelled. Cs173 - Spring 2004

  3. CS 173 Infinite Cardinality Are there more evens than odds? Are there more natural numbers than evens? Are there more evens than multiples of 3? Cs173 - Spring 2004

  4. 1 2 3 4 Hotel Infinity Vacancy CS 173 Hotel Infinity Did I tell you about the summer I worked in a hotel? This was not just any old hotel… 0 When I arrived, the previous assistant was leaving in haste. For you see, the job of the assistant is to make sure everyone has a bed, and there were no keys left on the rack. Cs173 - Spring 2004

  5. 1 2 3 4 Hotel Infinity Vacancy Whew, disaster averted! CS 173 Hotel Infinity …when along came a sleepy new client. Gulp. 0 I announced, “If you are in room k, kindly move to room k + 1.” Everyone did so and I handed the key to room 0 to the new client. Cs173 - Spring 2004

  6. 1 2 3 4 Hotel Infinity Vacancy Whew, disaster averted again! CS 173 Hotel Infinity …when along came a sleepy bunch of students on a ski trip. They needed 10 rooms. Gulp. 0 I announced, “If you are in room k, kindly move to room k + 10.” Cs173 - Spring 2004

  7. 1 2 3 4 Hotel Infinity Vacancy Whew, disaster averted again! CS 173 Hotel Infinity …when along came a infinitely big bus full of students on spring break. They needed infinitely many rooms!! Gulp. 0 I announced, “If you are in room k, kindly move to room 2k.” Cs173 - Spring 2004

  8. 1 2 3 4 Hotel Infinity Vacancy Whew, disaster averted again! CS 173 Hotel Infinity …when along came 2 infinitely big busses full of students on spring break. They needed infinitely many rooms!! Gulp. 0 I announced, “If you are in room k, kindly move to room 3k.” I then gave keys to the first people on each bus, the 2nd, etc. Cs173 - Spring 2004

  9. 1 2 3 4 Whew, disaster averted again! CS 173 Hotel Infinity …when along came infinitely many infinitely big busses full of students on spring break. Is there space? 0 I announced, “If you are in room k, kindly move to room 2k.” I then gave a key to person #1 on bus 1, then #2 on 1 and #1 on 2, then #3 on 1, #2 on 2 and #1 on 3, then #4 on 1, #3 on 2, #2 on 3, and #1 on 4, etc. … Cs173 - Spring 2004

  10. An infinite set is “countably infinite” if it can be put into one-to-one correspondence with the set of natural numbers. A set is “countable” if it is either finite or countably infinite. CS 173 Infinite Cardinality Two sets A and B have the same cardinality if and only if there exists a bijection between them, A ~ B. In the hotel infinity, we were creating bijections between the various descriptions of the sets of people and the set of natural numbers (the hotel room #s). Cs173 - Spring 2004

  11. CS 173 Infinite Cardinality Two sets A and B have the same cardinality if and only if there exists a bijection between them, A ~ B. Cs173 - Spring 2004

  12. {0,2,4,6,8,…} ~ {1,3,5,7,9,…}, f(x) = x-1 N ~ {0,2,4,6,8,…}, f(x) = 2x {0,2,4,6,8,…} ~ {0,3,6,9,12,…}, f(x) = 3x/2 {0,2,4,6,8,…} ~ {2,4,6,8,…}, f(x) = x-2 Perfect squares ~ N, f(x) = x CS 173 Infinite Cardinality Are there more evens than odds? Are there more natural numbers than evens? Are there more evens than multiples of 3? Cs173 - Spring 2004

  13. … 1/1, 1/2, 1/3, 1/4, … 2/1, 2/2, 2/3, 2/4, … 3/1, 3/2, 3/3, 3/4, … 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, … CS 173 Infinite Cardinality How many rational numbers are there? Cs173 - Spring 2004

  14. … 0.4 3 2 9 0 1 3 2 9 8 4 2 0 3 9 … 0.8 2 5 9 9 1 3 2 7 2 5 8 9 2 5 … 0.9 2 5 3 9 1 5 9 7 4 5 0 6 2 1 … “Countably many! There’s the list!” “Are you sure they’re all there?” Counterexample: 0.5 3 6 … So we say the reals are “uncountable.” CS 173 Infinite Cardinality How many real numbers are in interval [0, 1]? Cs173 - Spring 2004

  15. Countably many, we can list them by length. CS 173 Can we write programs for all functions? How many different programs can we write? Depends on the size of the alphabet we use. Let’s say our alphabet is size 100. How many 1 character programs are there? How many 2 character programs are there? How many n character programs are there? How many programs are there? Cs173 - Spring 2004

  16. 0.1415926… Real, irrational. CS 173 Can we write programs for all functions? How many different functions are there? Suppose domain is N, and codomain is {0,1,…9}, so we’re really only asking “how many functions f:N{0,1,2,…9} are there?” Can we even write programs for all of these? Cs173 - Spring 2004

  17. Punchline: there are MANY more functions than programs. 0.1415926… Real, irrational. CS 173 Can we write programs for all functions? How many different functions are there? We can make a function for each of the real numbers in [0,1]. This means there are uncountably many functions. Cs173 - Spring 2004

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