CSE 21 Math for Algorithms and Systems Analysis. Lecture 7 Func=ons Lecture 2

Transcription

1 CSE 21 Math for Algorithms and Systems Analysis Lecture 7 Func=ons Lecture 2

2 Outline For Today Quick Review of Func=ons Permuta=ons Cycle form for Permuta=ons Func=on Composi=on Compu=ng the order of a permuta=on

3 Func=ons Formally A func=on from set A to be B specifies a mapping between each element in the set A and a unique element in B We write f : A à B, to indicate that f is a func=on from A to B Some terminology A is called the domain of the func=on B is called the co- domain or the range of the func=on

4 Func=ons Graphically How about a func=on from the numbers 1 through 10 to their parity (even or odd) even odd 9 10

5 Func=ons in Two Line Form Given a domain S = {a,b,c,d,e} and a range T = {b,f,h} we can write a par=cular func=on from S to T in two line form as: ( a b c d e ) b f h h h f(a) = b f(b) = f f(c) = h f(d) = h f(e) = h

6 Types of Func=ons Injec=ve A func=on is injec=ve if each element in the range is mapped to by no more than one element in the domain The number of func=ons from an n element set to an m element set that are injec=ve is m! (m n)!

7 Types of Func=ons Surjec=ve a func=on is surjec=ve if each element in the range is mapped to by at least one element in the domain The number of func=ons from an n element set to an m element set that are surjec=ve is S(n, m)m!

8 Types of Func=ons Bijec=ons A func=on is a bijec=on if it is both surjec=ve and injec=ve The number of func=ons from an n element set to an n element set that are bijec=ve is n!

9 Permuta=ons Recall from last =me that a permuta=on is a special case of a bijec=on where the domain and range are the same set Here is an example of a permuta=on on the set {1,2,3,4,5} wriaen in two line form

10 Which of the following are Permuta=ons of the set {1,2,3,4,5}?

11 Another way to Look at Permuta=ons There is another way to write a permuta=on func=on that reveals the structure of how the permuta=on func=on transforms its input Cycle form for a permuta=on. Suppose f is permuta=on on the elements of the set S = {1,2,3,4,5} In Two line form f is wriaen as In cycle form f can be wriaen as f =(1, 2), (3, 4, 5)

12 What is Cycle Form? Here is another way to visualize what our func=on f is doing (we will use the graphical form of func=ons)

13 What does our func=on do? f(1) =

14 What does our func=on do? f(2) =

15 What does our func=on do? f(3) =

16 Cycle Form Given a func=on f we write it in cycle form by crea=ng a list of the elements in the domain such that the element to the right of it in the list gives the output of the func=on applied to that value. Addi=onally, we enclose a series of elements of the domain in parentheses to indicate that they belong to a cycle. In order to determine the output of the func=on at the end of a cycle we return to the first element in the cycle.

17 f = (1,3,5,7),(2,4,6) What is f(1)? What is f(2)? What is f(3)? What is f(4)? What is f(5)? What is f(6)? What is f(7)? Example Problem

18 Conver=ng between Cycle form and Two Line Form Convert the permuta=on f = (3)(4,6),(1,2,5) wriaen in cycle form to two line form

19 Conver=ng between Two Line Form Write the func=on in cycle form and Cycle Form

20 The Iden=ty Permuta=on The iden=ty permuta=on is the permuta=on that does not change its inputs f(x) = x We refer to the iden=ty permuta=on as id for short

21 Some Ques=ons About Permuta=ons Is there more than one way to write a func=on in cycle form?

22 Some Ques=ons About Permuta=ons Can the same element appear in more than 1 cycle when a permuta=on is wriaen in cycle form? Why or why not?

23 Func=on Composi=on We can write the composi=on of two func=ons f, and g as g f or gf The result of composing two func=ons g and f is a new func=on defined as: (g f)(x) =g(f(x)) In order for the composi=on of g and f to make sense what are the restric=ons on the domain and range of f and g?

24 Func=on Composi=on Graphically Here is a func=on g: A 1 B 2 C D 3

25 Func=on Composi=on Graphically Here is a func=on f: 1 E 2 F 3

26 Func=on Composi=on Graphically (fg) g f A B C D E F

27 Func=on Composi=on Graphically What is (fg)(a)? g f A B C D E F

28 Func=on Composi=on We will also use the nota=on to denote the func=on f composed with itself n =mes f n

29 Composing Permuta=ons Consider the permuta=on f = (1,2),(3,4,5) Please write the following func=ons in two line form f f 2 f 3 f 4 f 5 f 6

30 Workspace

31 Workspace

32 Another Ques=on about Permuta=ons Given an arbitrary permuta=on f does there always exist a posi=ve integer n such that f n = id?

33 The Order of a Permuta=on The order of a permuta=on f is the smallest integer such that f n = id Derive an expression for the order of a permuta=on given that it has m cycles with cycle lengths given by: c 1, c 2, c m

34 Workspace

35 Just in case you were wondering (not officially part of this class) The S=rling number of the first kind counts the number of permuta=ons with k disjoint (and non- empty) cycles of an n- element