Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
UCSD CSE 21, Spring 2014 Mathematics for Algorithm and System Analysis Week2 Class URL: http://vlsicad.ucsd.edu/courses/cse21-s14/ 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. 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 Administrivia • Personnel changes in CSE21 – I am now covering both Monday sections – Jay Dessai is no longer a TA for this class – TUTORS!!!! • Kacy Raye Espinoza – [email protected] • Tracy Nham – [email protected] • Hours TBD 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 ith choice can be made in ci ways. Total number of structures is c1 x … x ck 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 𝑛+𝑘−1 𝑘 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 𝑛+𝑘−1 𝑘 – The things we’re actually counting are not actually {1,2,…,9} – They’re stars and bars! Review (Technique) • Stars and Bars ( Combinatoric counting method ) – Number 8 from HW1: – Answer is 9+6−1 6 – Why is k = 6 ? – k = 6 because there are 6 – 1 = 5 divisions between the digits – n = 9 because we have 9 possible items 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 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 Subsets • Example: Consider set S = { x, y, z } – How many 2-lists without repetitions? • A: 3 • B: 6 • C: 9 • D: 8 • E: 0 Subsets • Example: Consider set S = { x, y, z } – How many 2-lists without repetitions? • A: 3 • B: 6 • C: 9 • D: 8 • E: 0 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 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 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 } Theorem 7: k-subsets of an n-set • Proof: Each k-subset is the set of elements of k! klists without repetitions! Up Next: Probability! • Counting and Probability go hand in hand • Here is a game that demonstrates this