Week 2 Discussion

1. UCSD CSE 21, Spring 2014Mathematics for Algorithm and System AnalysisWeek2Class URL: http://vlsicad.ucsd.edu/courses/cse21-s14/

2. Week 2 Discussion • UCSD CSE 21, Spring 2014 • Administrivia • From now on attendance at this discussion section is counted via clicker questions • A: I understand. • B: I understand. • C: I understand. • D: I understand. • E: I understand.

3. Administrivia • From now on attendance in this discussion is counted via clicker questions • Homework Two is due 4/13/2014 • Midterm In-class on May 1 (ABK) and May 2 (RRR) • 30% of final grade • This week: • Lists without repetitions • Sets

4. Administrivia • Personnel changes in CSE21 • I am now covering both Monday sections • Jay Dessai is no longer a TA for this class • TUTORS!!!! • KacyRaye Espinoza • krespinoza@ucsd.edu • Tracy Nham • tnham@ucsd.edu • Hours TBD

5. Review (Theroems / Def’s) • Cartesian Product: Generalization of Cartesian plane (RxR) • Lexicographic Order: Generalization of alphabetical order • Rule of Sum: Size of disjoint union is sum of size of components • Rule of Product: Sequence of k choices. The ithchoice can be made in ci ways. Total number of structures is c1 x … x ck

6. Review (Technique) • Stars and Bars ( Combinatoric counting method ) • Number 8 from HW1: • “A monotone increasing number consists of digits taken from the set {1, 2, …, 9}, with each digit greater than or equal to its neighbor digit to the left (if that digit exists). E.g., 1112256888899 is a monotone increasing number with 13 digits. How many 6-digit monotone increasing numbers are there? ” • Applicable Theorem: • For any pair of natural numbers n and k, the number of distinct n-tuples of non-negative integers whose sum is k is given by the binomial coefficient

7. Review (Technique) • Stars and Bars ( Combinatoric counting method ) • Number 8 from HW1: • Applicable Theorem: • For any pair of natural numbers n and k, the number of distinct n-tuples of non-negative integers whose sum is k is given by the binomial coefficient • The things we’re actually counting are not actually {1,2,…,9} • They’re stars and bars!

8. Review (Technique) • Stars and Bars ( Combinatoric counting method ) • Number 8 from HW1: • Answer is • Why is k = 6 ? • k = 6 because there are 6 – 1 = 5 divisions between the digits • n = 9 because we have 9 possible items

9. Subsets • Example: Consider set S = { x, y, z } • How many 2-lists does S generate? • A: 3 • B: 6 • C: 9 • D: 8 • E: 0

10. Subsets • Example: Consider set S = { x, y, z } • How many 2-lists does S generate? • A: 3 • B: 6 • C: 9 • D: 8 • E: 0

11. Subsets • Example: Consider set S = { x, y, z } • How many 2-lists without repetitions? • A: 3 • B: 6 • C: 9 • D: 8 • E: 0

12. Subsets • Example: Consider set S = { x, y, z } • How many 2-lists without repetitions? • A: 3 • B: 6 • C: 9 • D: 8 • E: 0

13. Subsets • Example: Consider set S = { x, y, z } • How many 2-sets which are subsets? • A: 3 • B: 6 • C: 9 • D: 8 • E: 0

14. Subsets • Example: Consider set S = { x, y, z } • How many 2-sets which are subsets? • A: 3 • B: 6 • C: 9 • D: 8 • E: 0

15. Subsets • Example: Consider set S = { x, y, z } • 2-lists: there are 32 = 9 • 2-lists without repetitions: 3*2 = 6 • 2-sets which are subsets: ??? How many??? • { x, y } { x, z } { y, z }

16. Theorem 7: k-subsets of an n-set • Proof: Each k-subset is the set of elements of k! k-lists without repetitions!

17. Up Next: Probability! • Counting and Probability go hand in hand • Here is a game that demonstrates this