9 14 10 Great Theoretical Ideas In Computer Science Anupam Gupta Danny Sleator Lecture 7 CS 15 251 Sep 14 2010 Fall 2010 Carnegie Mellon University Counting I Choice Trees and Correspondences If I have 14 teeth on the top and 12 teeth on the bottom how many teeth do I have in all In the next few lectures we will learn some fundamental counting methods Addition and Product Rules The Inclusion Exclusion Principal Choice Tree Permutations and Combinations The Binomial Theorem The Pigeonhole Principal Diophantine Equations Generating Functions Addition Rule Let A and B be two disjoint finite sets Addition of Multiple Disjoint Sets Let A1 A2 A3 An be disjoint finite sets Addition Rule 2 Possibly Overlapping Sets Let A and B be two finite sets A B A B A B 1 9 14 10 Inclusion Exclusion Inclusion Exclusion If A B C are three finite sets what is the size of A B C If A1 A2 An are n finite sets what is the size of A1 A2 An i Ai i j Ai Aj i j k Ai Aj Ak 1 n 1 A1 A2 An A B C A B A C B C A B C A school has 100 students 50 take French 40 take Latin and 20 take both How many students take neither language French F French students L Latin students Latin 20 100 F 50 L 40 French AND Latin students F L 20 French OR Latin students F L F L F L 50 40 20 70 Neither language 100 70 30 How many positive integers 70 are relatively prime to 70 U 1 70 70 2x5x7 A1 integers in U divisible by 2 A2 integers in U divisible by 5 A3 integers in U divisible by 7 A1 35 A2 14 A3 10 A1 A2 A3 A1 A2 A3 A1 A2 A1 A3 A2 A3 A1 A2 A3 How many positive integers less than 70 are relatively prime to 70 U A2 A1 A3 A1 A2 7 A1 A3 5 A2 A3 2 A1 A2 A3 1 A1 A2 A3 A1 A2 A3 A1 A2 A1 A3 A2 A3 A1 A2 A3 A1 A2 A3 35 14 10 7 5 2 1 46 Thus the number of relatively prime to 70 is 70 46 24 2 9 14 10 The Principle of Inclusion and Exclusion Let Sk be the sum of the sizes of All k tuple intersections of the Ai s U A2 A1 S1 A1 A2 A3 S2 A1 A2 A1 A3 A2 A3 S3 A1 A2 A3 A3 Partition Method To count the elements of a finite set S partition the elements into non overlapping subsets A1 A2 A3 An A1 A2 A3 S1 S2 S3 Partition Method Partition Method S all possible outcomes of one white die and one black die S all possible outcomes of one white die and one black die Partition S into 6 sets A1 the set of A2 the set of A3 the set of A4 the set of A5 the set of A6 the set of outcomes where the white die is 1 outcomes where the white die is 2 outcomes where the white die is 3 outcomes where the white die is 4 outcomes where the white die is 5 outcomes where the white die is 6 Each of 6 disjoint set have size 6 36 outcomes Partition Method S all possible outcomes where the white die and the black die have different values S Set of all outcomes where the dice show different values S Ai set of outcomes where black die says i and the white die says something else 6 S i 1 6 Ai 5 30 i 1 3 9 14 10 S Set of all outcomes where the dice show different values S B set of outcomes where dice agree S B of outcomes 36 S B 36 B 6 Difference Method To count the elements of a finite set S find two sets A and B such that S and B are disjoint and S B A then S A B S 36 6 30 S Set of all outcomes where the black die shows a smaller number than the white die S Ai set of outcomes where the black die says i and the white die says something larger S A1 A2 A3 A4 A5 A6 S 5 4 3 2 1 0 15 It is clear by symmetry that S L S Set of all outcomes where the black die shows a smaller number than the white die S L set of all outcomes where the black die shows a larger number than the white die S L 30 It is clear by symmetry that S L Therefore S 15 Pinning Down the Idea of Symmetry by Exhibiting a Correspondence Put each outcome in S in correspondence with an outcome in L by swapping color of the dice S L Each outcome in S gets matched with exactly one outcome in L with none left over Thus S L 4 9 14 10 Let s Restrict Our Attention to Finite Sets Let f A B Be a Function From a Set A to a Set B A f is injective if and only if x y A x y f x f y B f is surjective if and only if z B x A f x z There Exists For Every A injective 1 1 f A B A B B surjective onto f A B A B A bijective f means the inverse f 1 is well defined bijective f A B A B B A B bijective f A B A B Correspondence Principle If two finite sets can be placed into bijection then they have the same size It s one of the most important mathematical ideas of all time 5 9 14 10 Question How many n bit sequences are there 000000 000001 000010 000011 111111 0 1 2 3 2n 1 Each sequence corresponds to a unique number from 0 to 2n 1 Hence 2n sequences Question How Many Subsets Can Be Made From The Elements of a 5 Element Set a b c d e 0 1 1 0 1 b c e S a b c d e has Many Subsets a a b a d e a b c d e e The entire set and the empty set are subsets with all the rights and privileges pertaining thereto S a1 a2 a3 an T all subsets of S B set of all n bit strings a1 a2 a3 a4 a5 b1 b2 b3 b4 b5 For bit string b b1b2b3 bn let f b ai bi 1 1 means TAKE IT 0 means LEAVE IT Each subset corresponds to a 5 bit sequence using the take it or leave it code S a1 a2 a3 an T all subsets of S B set of all n bit strings a1 a2 a3 a4 a5 b1 b2 b3 b4 b5 For bit string b b1b2b3 bn let f b ai bi 1 Claim f is injective Any two distinct …
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