Test Reminder. CSE21: Lecture 7 1

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1 Test Reminder Quiz 2 will be in the first 15 minutes of Tomorrow s class. You can use any resources you have during the quiz. It covers the first three sections of Unit 2. It has four questions, two of them will be choices. Midterm will be at this Monday s class. One hour and twenty minutes long. It covers all four sections of Unit 1 and the first three sections of Unit 2. You can bring one letter size paper (8.5 by 11 ) for notes (front and back, hand written). Write down your name on the paper, and turn this in with your test. No other resources are allowed. Extra question: we will have one question in the end with extra points. It will be added to your score without exceeding the maximal score. Midterm practice is online now. TA will discuss it on Thursday. CSE21: Lecture 7 1

2 Quiz 1 Result The papers for Quiz 1 will handed out at the end of this class. I ll curve scores up if they are final scores. Scores of each test will be at GradeSource The first quiz s score information will be sent out today. If you are not happy with the score, Try to read ahead notes/textbook for the next class. Try to solve more exercise questions after class. Ask help at Piazza or office hour. CSE21: Lecture 7 2

3 Piazza Advertisement for Piazza forum the last time Over 30 questions so far. The average response time is 24 minutes. Announcements will be posted there. Some students help others on piazza. Thanks! CSE21: Lecture 7 3

4 CSE 21 Mathematics for Algorithm and System Analysis Unit 2: Functions Section 3: Other Combinatorial Aspects of Functions CSE21: Lecture 7 4

5 Review: Subset and Set Partition (CL-25) The set of all subsets of A: P(A). P(A) =2 A A partition of a set B is a collection of nonempty subsets of B such that each element of B appears in exactly one subset. Each subset is called a block of the partition. Example: the 15 partitions of {1, 2, 3, 4} by number of blocks are 1 block: {{1, 2, 3, 4}} 2 blocks: {{1, 2, 3}, {4}}, {{1,2},{3,4}}, 4 blocks: {{1}, {2}, {3}, {4}} CSE21: Lecture 7 5

6 Review: Set Partitions (CL-25) S(n,k) is denoted to be the number of partitions of an n-set having exactly k blocks. S(n, k) = S(n 1, k 1) 1 + k S(n 1, k) Tabular representation k n S(n,k) CSE21: Lecture 7 6

7 Review : Function (Fn-2) The set of the first n positive integers, {1, 2,..., n} : n If A and B are sets, a function from A to B is a rule that tells us how to find a unique b Bfor each a A. f : A B means f is a function from A to B, which is the same with f B A. We call the set A the domain of f and the set B the range/codomain of f. One line form: when A is ordered by a 1, a 2,..., a A, a function can be written in one-line form: (f(a 1 ), f(a 2 ),..., f(a A )) or f(a 1 ), f(a 2 ),..., f(a A ) CSE21: Lecture 7 7

8 Learning Outcomes By the end of this lesson, you should be able to Understand coimage concept and use it to solve function counting problems. Understand the monotone functions and their relationship with sets/multisets. Use the above relationship to count sets/multisets and functions. CSE21: Lecture 7 8

9 Why do we need to learn them? Coimage and related concepts show how we can create functions and bijections. To organize and store objects in computer, we sometimes need to order them first. Monotone functions are good ways to order objects. CSE21: Lecture 7 9

10 Two parts in this section The Inverse of an Arbitrary Function Inverse Image, Coimage, etc. Monotonic Lists and Unordered Lists Monotone functions, counting sets/multisets, etc. CSE21: Lecture 7 10

11 Image and Inverse Image For function f : A B, Image of Function The image of f is the set of values f actually takes on: Image(f ) = { f (a) a A}. Image(f ) Range(f ) Inverse Image For each b B, the inverse image of b, written f 1 (b) is the set of those elements in A whose image is b; i.e., f 1 (b) = {a a A and f(a) = b}. Such an f 1 is not a function from B to A unless f is a bijection. Suppose f is given by the functional relation R A B. Then f 1 (b) is all those a such that (a,b) R. CSE21: Lecture 7 11

12 Definition 4 : Coimage For function f : A B, The coimage of f is the set of nonempty inverse images of elements of B. Coimage(f ) = {f 1 (b) b B, f 1 (b) } = {f 1 (b) b Image(f )}. Coimage(f ) = Image(f ) Coimage(f ), Coimage(f ) Examples: If f {a, b, c} 5 in one line form is (a,c,a,a,c), then f(1) = f(3) = f(4) = a, f(2) = f(5) = c Image(f) = {a, c} Range(f )={a, b, c} Coimage(f) = {f 1 (a), f 1 (c)} = {{1,3,4},{2,5}} CSE21: Lecture 7 12

13 Partition by Coimage The coimage of f is a partition of its domain. Example: If f {a, b, c} 5 in one line form is (a,c,a,a,c), then Coimage(f ) = {f 1 (a), f 1 (c)} = {{1,3,4},{2,5}} If f is constant, namely its value is the same for all domain elements, the partition by the coimage of f has only one block and the block is the domain set. Example: If f {a, b, c} 5 in one line form is (a,a,a,a,a), then f(i) =a for all i {1,2,3,4,5}, then Coimage(f ) = {f 1 (a)} = {{1,2,3,4,5}} CSE21: Lecture 7 13

14 Example 12 : Inverse Image as Function Recall inverse image f 1 is not a function from B to A unless f is a bijection. Can we make f 1 always a function by changing the range? Given function f : A B, for each b B, f 1 (b) = {a a Aand f(a) = b} P(A), which is the set of all subsets of A. So f 1 is a function from B to P(A). Example: f {a, b, c} 5 in one line form is (a,c,a,a,c), then f 1 is a b c {1,3,4} Φ {2,5} CSE21: Lecture 7 14

15 Example 12 : Inverse Image as Function(2) Can we make inverse image f 1 always a bijection function by changing both domain and range? Given function f : A B, for each b Image(f ), f 1 (b) = {a a Aand f(a) = b} Coimage(f ). f 1 is a bijection: Image(f ) Coimage(f ). Last example: f {a, b, c} 5 in one line form is (a,c,a,a,c), then f 1 is a {1,3,4} c {2,5} CSE21: Lecture 7 15

16 Example 12 : Inverse Image as Function(3) If we know {{1, 3, 4}, {2, 5}} is the coimage of a function g with codomain/range {a, b, c}, how many g functions we can have? The domain of g is {1, 2, 3, 4, 5}, we need to know image and rule from domain to image. Image set: coimage block number is the size of image, 3 3! select 2-subset from 3-set {a, b, c}: = = 3 2 2!(3 2)! Rule from domain to image (select one element in image for{1,3,4}, and another one for {2, 5}, rule of product): 2! The results are 3 2!=6. We got the same result using a solution different from text book. CSE21: Lecture 7 16

17 Example 13 : Counting Functions with Specified Image Size How many functions in B A, namely f : A B, have an image with exactly k elements? B B! Get image set (k-subset) from B: = k k!( B k)! Get coimage set from A: If image has k elements, coimage has to be a partition of domain with k blocks. Get the number of partitions of set of size A into k blocks: S( A, k) Rule from k blocks in coimage to k elements in image: k! Results: B S( A, k) k k! = S( A, k) B! ( B k)! CSE21: Lecture 7 17

18 Second part in this section The Inverse of an Arbitrary Function Inverse Image, Coimage, etc. Monotonic Lists and Unordered Lists Monotone functions, counting sets/multisets, etc. CSE21: Lecture 7 18

19 Ordering in Computer In computers, data need to ordered for storage and retrieve. The most common orders are arrays and linked lists. Enable ordering in computer for mathematical objects Use the order of mathematical objects directly. Example: One-line notation for a function is an ordered list based on the order of domain set. So functions can be stored in computer as ordered lists. Add an order of mathematical objects in program, called canonical or unique ordering. Example: add order to sets and multisets that are unordered mathematical objects. CSE21: Lecture 7 19

20 Different Types of Order We need a unique way to order elements in a set or multiset to have an ordered list. Example: 3-element multisets whose elements are chosen from 5. An intuitive order in each 3-multiset:(1,1,1); (1,1,2); (1,1,3); An ordered list (b 1,b 2,...,b k ) is in weakly increasing order if b 1 b 2 b k weakly decreasing order if b 1 b 2 b k strictly increasing order if b 1 < b 2 < < b k strictly decreasing order if b 1 > b 2 > > b k CSE21: Lecture 7 20

21 Functions with Different Types of Orders Functions with domain k can be viewed as k-lists in one line form. Example: 3-list for f(1)=a, f(2)=b, f(3)=c is (a, b, c). f n k is a weakly increasing function if its one-line form is weakly increasing, i.e., f(1) f (2) f (k) Example: f(1)=1, f(2)=2, f(3)=2. Functions with other orders, e.g., weakly decreasing function, can be also defined. All these functions are called monotone functions. CSE21: Lecture 7 21

22 Relation between Sets/Multisets and Ordered Functions We have bijection from the k-multisets whose elements lie in n to the weakly increasing/decreasing functions in n k written in one-line form. Example: 3-element multisets from 5. Each multiset, e.g.,{2, 5, 5} corresponds to a weakly increasing function e.g., f = (2, 5, 5) and a weakly decreasing function (read in opposite direction) e.g., f = (5, 5, 2) in one line form. Replace the above multisets to sets, we will get strictly increasing/decreasing functions. Example: {1, 2, 3} corresponds to strictly increasing function f = (1, 2, 3) and strictly decreasing function f = (3, 2, 1) in one line form. CSE21: Lecture 7 22

23 Theorem 2 : Sets, Unordered Lists and Monotone Functions There are bijection between each of the following: { k-multisets} whose elements lie in n, k-sets { weakly} the increasing ordered k-lists made from n, strictly { weakly} the increasing functions in n k. strictly CSE21: Lecture 7 23

24 Example 14 : Counting Multisets How many different 3-element multisets have elements from 5? Convert the problem. First arrange the elements in multisets in weakly increasing order, e.g.: 1,1,1;1,1,2, then add 0 to the first item, add 1 to the second item, add 2 to the third item. 1,1,1 becomes 1,2,3; ; 1,2,5 becomes 1,3,7 It becomes listing strictly increasing k-lists from 7, or all subsets of 7 of size 3. The result is 7 3 = 7! 3! 4! = 35 CSE21: Lecture 7 24

25 Example 14 : Counting Multisets (2) How many different k-element multisets have elements from n? Arrange elements in multisets in weakly increasing order, then add 0 to the first item, add 1 to the second item, and add k-1 to the last item. It becomes listing all subsets of n of size k: Based on Theorem 2, we can get same number for: the weakly increasing ordered k-lists made from n, the weakly increasing functions in n k. n + k 1 k CSE21: Lecture 7 25

26 Homework and Pre-Reading Assignment Homework: Exercise 3.1, 3.4, 3.6, 3.7, 3.8 in page Fn-19 to Fn- 21 Prepare quiz 2 For next class, please read first a few pages of Section 1 in Unit 3, NOT Section 4 in Unit 2. Try to understand decision tree through examples. Try to understand different ways to traverse decision trees through examples. CSE21: Lecture 7 26