1 CMSC 250 Chapter 7, Functions

2 CMSC 250 Function terminology l A relationship between elements of two sets such that no element of the first set is related to more than one element of the second set l Domain: the set which contains the values to which the function is applied l Codomain: the set which contains the possible values (results) of the function l Range (or image): the set of actual values produced when applying the function to the values of the domain

3 CMSC 250 More function terminology l f: X  Y –f is the function name –X is the domain –Y is the co-domain –x  X y  Y f sends x to y –f(x) = y f of x; the value of f at x ; the image of x under f l A total function is a relationship between elements of the domain and elements of the co-domain where each and every element of the domain relates to one and only one value in the co-domain l A partial function does not need to map every element of the domain

4 CMSC 250 Formal definitions l The range of f is {y  Y | (  x  X)[f(x) = y]} –where X is the domain and Y is the co-domain l The inverse image of y  Y is {x  X | f(x) = y} –the set of things in the domain X that map to y l Arrow diagrams Determining if something is a function using an arrow diagram l Equality of functions (  functions f,g with the same domain X and codomain Y) [f = g iff (  x  X)[f(x) = g(x)] ]

5 CMSC 250 Discrete Structures CMSC 250 Lecture 38 April 30, 2008

6 CMSC 250 Types of functions l F:X  Y is a one-to-one (or injective) function iff (  x 1,x 2  X)[F(x 1 ) = F(x 2 )  x 1 = x 2 ], or alternatively (  x 1,x 2  X)[x 1  x 2  F(x 1 )  F(x 2 )] l F: X  Y is not a one-to-one function iff (  x 1,x 2  X)[(F(x 1 ) = F(x 2 )) ^ (x 1  x 2 )] l F: X  Y is an onto (or surjective) function iff (  y  Y)(  x  X)[F(x) = y] l F: X  Y is not an onto function iff (  y  Y)(  x  X)[F(x)  y]

7 CMSC 250 Proving functions one-to-one and onto f: R  R f(x) = 3x  4 l Prove or give a counterexample that f is one-to-one –recall the definition (one of two definitions) of one-to-one is l Prove or give a counterexample that f is onto –recall the definition of onto is

8 CMSC 250 One-to-one correspondence or bijection F: X  Y is bijective iff F: X  Y is one-to-one and onto If F: X  Y is bijective then it has an inverse function

9 CMSC 250 Proving something is a bijection l F: Q  Q F(x) = 5x + 1/2 –prove it is one-to-one –prove it is onto –then it is a bijection –so it has an inverse function find F  1

10 CMSC 250 The pigeonhole principle       l Basic form: A function from one finite set to a smaller finite set cannot be one-to-one; there must be at least two elements in the domain that have the same image in the codomain.

11 CMSC 250 Examples l Using this class as the domain: –must two people share a birth month? –must two people share a birthday? l Let A = {1,2,3,4,5,6,7,8} –if I select 5 different integers at random from this set, must two of the numbers sum exactly to 9? –if I select 4 integers? l There exist two people in New York City who have the same number of hairs on their heads. l There exist two subsets of {1,…,10} with three elements which sum to the same value.

12 CMSC 250 Discrete Structures CMSC 250 Lecture 39 May 2, 2008

13 CMSC 250 Another (more useful) form of the pigeonhole principle l The generalized pigeonhole principle: –For any function f from a finite set X to a finite set Y and for any positive integer k, if n(X) > k * n(Y), then there is some y  Y such that y is the image of at least k+1 distinct elements of X. l Contrapositive form: –For any function f from a finite set X to a finite set Y and for any positive integer k, if for each y  Y, f –1 (y) has at most k elements, then X has at most k  n(y) elements.

14 CMSC 250 Examples l Using the generalized form: –assume 50 people in the room, how many must share the same birth month? –n(A)=5 n(B)=3 F: P (A)  P (B) how many elements of P (A) must map to a single element of P (B)?

15 CMSC 250 Composition of functions l f: X  Y 1 and g: Y  Z where Y 1  Y –g ○ f: X  Z where (  x  X)[g(f(x)) = g ○ f(x)] f(x)g(y) g(f(x)) y Y1Y1 Y Z X x z

16 CMSC 250 Composition on finite sets- example l Example X = {1,2,3}, Y 1 = {a,b,c,d}, Y = {a,b,c,d,e}, Z = {x,y,z} f(1) = cg(a) = yg○f(1) = g(f(1)) = z f(2) = bg(b) = yg○f(2) = g(f(2)) = y f(3) = ag(c) = zg○f(3) = g(f(3)) = y g(d) = x g(e) = x

17 CMSC 250 Composition for infinite sets- example f: Z  Z f(n) = n + 1 g: Z  Z g(n) = n 2 g ○ f(n) = g(f(n)) = g(n+1) = (n+1) 2 f ○ g(n) = f(g(n)) = f(n 2 ) = n Note: g ○ f  f ○ g

18 CMSC 250 Identity function l i X the identity function for the domain X i X : X  X (  x  X) [i X (x) = x] l i Y the identity function for the domain Y i Y : Y  Y (  y  Y) [i Y (y) = y] l composition with the identity functions

19 CMSC 250 Composition with inverse l Recall: if f is a bijection then f  1 exists. l Let f: X  Y be a bijection. l What is f ○ f  1 ? l What is f  1 ○ f?

20 CMSC 250 One-to-one in composition l If f: X  Y and g: Y  Z are both one-to-one, then g ○ f: X  Z is one-to-one. l If f: X  Y and g: Y  Z are both onto, then g ○ f: X  Z is onto.

21 CMSC 250 Cardinality l Comparing the “sizes” of sets: –finite sets (  or there is a positive integer n such that there is a bijective function from the set to {1,2,…,n}) –infinite sets (there is no such n such that there is a bijective function from the set to {1,2,…,n}) l  sets A,B, A and B have the same cardinality iff there is a one-to-one correspondence from A to B In other words, Cardinality(A) = Cardinality(B)  (  a function f ) [f: A  B  f is a bijection]

22 CMSC 250 Countable sets l A set S is called countably infinite iff Cardinalit(S) = Cardinality(Z + ). l A set is called countable iff it is finite or countably infinite. l A set which is not countable is called uncountable.

23 CMSC 250 Discrete Structures CMSC 250 Lecture 40 May 5, 2008

24 CMSC 250 Countability of sets of integers and the rationals l N is this a countably infinite set? l Z is this countably infinite set? l N even is this a countably infinite set? l Card(Q + ) =?= Card(Z)

25 CMSC 250 Real numbers l We’ll take just a part of this infinite set l Reals between 0 and 1 (noninclusive) X = {x  R | 0 < x < 1} l All elements of X can be written as 0.a 1 a 2 a 3 … a n …

26 CMSC 250 Cantor’s proof l Assume the set X = {x  R | 0 < x < 1} is countable l Then the elements in the set can be listed 0.a 11 a 12 a 13 a 14 …a 1n … 0.a 21 a 22 a 23 a 24 …a 2n … 0.a 31 a 32 a 33 a 34 …a 3n … … … … … l Select the digits on the diagonal Build a number d, such that d differs in its n th position from the n th number in the list

27 CMSC 250 All reals Cardinality({x  R | 0 < x < 1}) = Cardinality(R)