1. MSU/CSE 260 Fall 20091 1 Functions Read Section 1.8

2. MSU/CSE 260 Fall 2009 2 2 Outline Introduction: definitions, example and terminology, image of a subset of domain, sum and product of functions One-to-one functions: strictly increasing, decreasing functions Onto functions Bijections: identity functions Graphs of Functions Inverse Functions Compositions of Functions Some Important Functions: Floor, Ceiling and Factorial functions

3. MSU/CSE 260 Fall 2009 3 3 Function Example  Consider your final grades in CSE 260. Your grades will be one of the values from the set {4, 3.5, 3, 2.5, 2, 1.5, 1, 0} What kind of properties does this assignment have? Consider the courses you are taking this semester. { (A123, CSE 260), (A123, CSE 232), (A123, MTH 234), (A123, DYLANTHOMAS 111)} What kind of properties does this assignment have?

4. MSU/CSE 260 Fall 2009 4 4 Introduction Definition: Let A, B be sets. A function f from A to B, denoted f: A → B, is an assignment where each element of A is assigned exactly one element of B. Notation:  f: A → B  We write f (a) = b, if b is the element of B assigned under f to the element a of A.  We also say f maps A to B Formally, f is a function from A to B if and only if  x  A  !y  B f(x) = y. where  ! is the uniqueness quantifier.

5. MSU/CSE 260 Fall 2009 5 5 Example Stevens Sara 4 0 Adams 2 3 1 Marie f: A → B GradesStudents f(Adams)=1 f(Marie)=4 f(Stevens)=3 f(Sara)=3 A B

6. MSU/CSE 260 Fall 2009 6 6 Domain, Co-domain, … Definition: Let f be a function from A to B, that is, f: A → B. Then A is called the domain of f, and B is the codomain of f. If f(a) = b, then  b is called the image of a, and  a is a pre-image of b. The range of f is the set of all images of elements of A. How are codomain and range related?  Range is a subset of the codomain

7. MSU/CSE 260 Fall 2009 7 7 Example Stevens Sara 4 0 Adams 2 3 1 Marie GradesStudents f (Adams)=1 f (Marie)=4 f (Stevens)=3 f (Sara)=3 The image of ‘Marie’ is 4; the pre-images of 3 are ‘Stevens’ and ‘Sara’; the range of f is {1, 3, 4}. A B f: A → B

8. MSU/CSE 260 Fall 2009 8 8 Sum, Products of real-valued Functions Definition: Let f 1 and f 2 be functions from A to R. Then f 1 + f 2 and f 1  f 2 are also functions from A to R, defined as:  (f 1 + f 2 )(x) = f 1 (x) + f 2 (x)  (f 1  f 2 )(x) = f 1 (x)  f 2 (x) Note that if f 1 and f 2 do not have the same domain, the above operations do not make sense.

9. MSU/CSE 260 Fall 2009 9 9 Image of a subset of a domain Definition: Let f be a function from A to B, and let S be a subset of A. The image of S, denoted f (S), is the subset of B consisting of the images of the elements of S.  Formally: f (S) = { f (s) | s  S}.  Note that f (A) is the range of f. In the previous Example,  f ({Adams, Sara}) = {3, 1}

10. MSU/CSE 260 Fall 2009 10 One-to-one Functions Definition: A function f from A to B is said to be one-to-one, or injective, if and only if distinct elements of the domain have distinct images. That is,  x  A  y  A f (x) = f (y) → x = y. 1 b a c 2 3 4

11. MSU/CSE 260 Fall 2009 11 Onto Functions Definition: A function f from A to B is said to be onto, or surjective, if and only if its range and codomain are the same. That is,  y  B  x  A f(x) = y. 1 c b d 2 3 a

12. MSU/CSE 260 Fall 2009 12 Bijections Definition: A function f: A → B is a bijection, or one-to-one correspondence, if it is both one-to-one and onto.  Note that the cardinalities (when dealing with finite sets) of domain and codomain of a bijection are equal. c b d 3 4 a 2 1

13. MSU/CSE 260 Fall 2009 13 Summary of Function Types b a c 2 3 4 2 3 c b d a c b d 3 4 a 1 1 2 1 1 2 c b d 3 4 a c b 3 4 a 2 1 onto but not 1-to-1 both 1-to-1 and onto; bijection Neither 1-to-1 nor onto Not a function 1-to1 but not onto

14. MSU/CSE 260 Fall 2009 14 Monotonic Functions Definition: Let A and B be subsets of R, the set of real numbers. A function f: A → B is strictly increasing if  x  A  y  A x < y → f (x) < f (y). f: A → B is strictly decreasing if  x  A  y  A x f (y). Note that strictly increasing, or strictly decreasing (strictly monotone) functions must be one-to-one.

15. MSU/CSE 260 Fall 2009 15 Inverse Function Definition: Let function f: A → B be a bijection. The inverse function of f, denoted f -1, is the function, f -1 : B → A, that assigns to each element b of B the element a of A such that f (a) = b.   a  A  b  B f(a) = b → f -1 (b) = a.  f is called invertible. c b d 3 4 a1 2 f -1 (1) = d f -1 (2) = b f -1 (3) = a f -1 (4) = c f

16. MSU/CSE 260 Fall 2009 16 Inverse Function… Example: Let f : Z → Z, where f (x) = x + 1.  f is a bijection; what is f -1 ?  Suppose f (x) = y; then x + 1= y; so x = y - 1= f -1 (y) f -1 (x) = x ­ 1.

17. MSU/CSE 260 Fall 2009 17 Identity Function Let A be a set. The identity function on A is the function ι A : A → A, where  x  A ι A (x) = x. Notes:  ι A assigns each element of A to itself.  ι A is a bijection.

18. MSU/CSE 260 Fall 2009 18 Characteristic and Constant Functions Let A be a subset of universe U. The characteristic function f A : U → {0, 1}, is such that f A (x) = 1 if x  A and f A (x) = 0 if ¬ (x  A ) Let A be a set. The constant function f : A → {t} maps each element of A to the same value t.

19. MSU/CSE 260 Fall 2009 19 Compositions of Functions Definition: Let g be a function from A to B and f a function from B to C, that is, g: A → B f: B → C The composition of f and g, denoted f o g, is function from A to C, defined as follows  x  A ( f o g)(x) = f (g(x)). v k 3 4 p 2 1 c b d a (f o g)(a) = 2 (f o g)(b) = 3 (f o g)(c) = 1 (f o g)(d) = 2 g f q A={a, b, c, d} B={q, p, k, v} C={1, 2, 3, 4}

20. MSU/CSE 260 Fall 2009 20 Composition of Functions A Bc gf f o g A Bc A BcccBcBc A Bc

21. MSU/CSE 260 Fall 2009 21 Example Consider the two functions f : Z → Z, where f (x) = 2x + 3 g: Z → Z, where g (x) = 3x + 2.  What are f o g, and g o f ?  f o g: Z → Z, where (f o g)(x) = f (g(x)) = f (3x + 2) = 2(3x + 2) + 3 = 6x +7  g o f: Z → Z, where (g o f )(x) = g( f (x)) = g (2x + 3) = 3(2x + 3) + 2 = 6x + 11

22. MSU/CSE 260 Fall 2009 22 Example

23. MSU/CSE 260 Fall 2009 23 Example

24. MSU/CSE 260 Fall 2009 24 Example….

25. MSU/CSE 260 Fall 2009 25 Graph of a Function Definition: Let f: A → B. The graph of f is the set of ordered pairs G f = {(x, f (x)) | x  A}.

26. MSU/CSE 260 Fall 2009 26 Graph of a Function…. The graph of the function f : Z → Z, where f (n) = 2n + 1, is G f = {(n, 2n + 1) | n  Z} n f (n) (0,1) (1,3) (2,5) (-1,-1)

27. MSU/CSE 260 Fall 2009 27 Important Integer Functions Whole numbers constitute the backbone of discrete mathematics. We often need to convert fractions or arbitrary real numbers to integers. These integer functions will help us do that. Besides the identity function, some important functions are:  The floor function,  The ceiling function,  The mod function.

28. MSU/CSE 260 Fall 2009 28 Floor Function Definition: The floor function from R to Z assigns to the real number x, the largest integer ≤ x. The value of the floor function at x is denoted by  x .   x  R  n  Z  x  = n  n  x < n + 1. Examples:   18  = 18   3.75  = 3   – 4.5  = – 5

29. MSU/CSE 260 Fall 2009 29 Ceiling Function The ceiling function from R to Z assigns to the real number x the smallest integer ≥ x. The value of the ceiling function at x is denoted by  x .   x  R  n  Z  x  = n  n – 1 < x  n.   x  R x – 1 <  x   x   x  < x + 1. Examples:   18  = 18   3.75  = 4   – 4.5  = – 4

30. MSU/CSE 260 Fall 2009 30 Floor and Ceiling Functions, recap 0 ** xxxx xxxx xxxx xxxx

31. MSU/CSE 260 Fall 2009 31 Properties of  x  and  x   x  R  n  Z  x  = n  n  x < n + 1.  x  R  n  Z  x  = n  n – 1 < x  n.  x  R  n  Z  x  = n  x – 1 < n  x.  x  R  n  Z  x  = n  x  n < x +1.  x  R x – 1 <  x   x   x  < x + 1.  x  R  – x  = –  x   x  R  – x  = –  x   x  R  m  Z  x + m  =  x  + m  x  R  m  Z  x + m  =  x  + m 0 ** xxxx xxxx xxxx xxxx

32. MSU/CSE 260 Fall 2009 32 Example

33. MSU/CSE 260 Fall 2009 33 Example: Solution

34. MSU/CSE 260 Fall 200934 Integer Functions

35. MSU/CSE 260 Fall 2009 35

36. MSU/CSE 260 Fall 2009 36 Example

37. MSU/CSE 260 Fall 2009 37 Example: Solution

38. MSU/CSE 260 Fall 2009 38 The mod Function When dividing an integer n by a number m, the quotient of the division is  n/m . What about a simple notation for the remainder of this division? That’s what the mod function is about: n mod m m is called modulus n = m   n/m  + n mod m quotient remainder

39. MSU/CSE 260 Fall 2009 39 Example Formally, the mod function is a mapping: mod : Z  Z + → N where n mod m = n – m   n/m  Examples : 5 mod 3 = 5 – (3   5/3  ) = 5 – (3   1.6  ) = 5 – (3  1 ) = 2

40. MSU/CSE 260 Fall 2009 40 Example n mod m = n – (m   n/m  ) Examples: - 5 mod 3 = -5 – (3   -5/3  ) = -5 – (3   -1.6  ) = -5 – (3  (-2)) = 1. We also write: 5  2 mod 3, 9  0 mod 3, -5  1 mod 3.

41. MSU/CSE 260 Fall 2009 41 mod Function m 0 1 2 m-1 + –

42. MSU/CSE 260 Fall 2009 42 List Search Methods Problem: Given a list of elements, how fast can we decide whether or not a given input element belongs to the list?  Linear search  Binary search; need to sort the list first  Hash table

43. MSU/CSE 260 Fall 200943 Hash Functions

44. MSU/CSE 260 Fall 2009 44 Hash Functions A hash function h: keys → integers maps “keys” to “small” integers (buckets) Ideal features:  The function should be easy to compute  The range values should be “evenly” distributed  Given an image, it should not be “easy” to find its pre- image Applications  Searching/indexing  Information hiding  File signature

45. MSU/CSE 260 Fall 2009 45 Hashing for Indexing A hash function h: keys → integers maps “keys” to “small” integers (buckets) Ideally this mapping is done in a “random” manner so that the bucket values are evenly distributed despite irregularities in the keys. For simplicity, we will assume that the keys are also integers, denoted by k, and the number of buckets is demoted by m. Note that the buckets are indexed 0 through m - 1.

46. MSU/CSE 260 Fall 2009 46 Example Storing CSE 260, both sections, PIDs\A  Using Hash Function h(PID) = k mod 31

47. MSU/CSE 260 Fall 2009 47 Simple Hash Functions h(k) = k mod m  Suggestion: Choose m to be a prime number that isn’t close to a power of 2. h(k) = k(k + 3) mod m

48. MSU/CSE 260 Fall 2009 48 Hashing for Hiding Information Here, the hash function maps a string to another string with the property of being very difficult to reverse the result of the hash. Used in hiding user’s password

49. MSU/CSE 260 Fall 2009 49 How password is checked.

50. MSU/CSE 260 Fall 2009 50 Hashing for file signature The hash function maps a large string (e.g., a file) to a fixed size string called digest Examples:  MD5 (Message-Digest algorithm 5), gives a 128-bit hash (digest)  SHA-1 (Secure Hash Algorithm) is a most commonly used from SHA series of cryptographic hash functions, designed by the National Security Agency  SHA-1 produces the 160-bit hash value. Original SHA (or SHA- 0) also produce 160-bit hash value, but SHA-0 has been withdrawn by the NSA shortly after publication and was superseded by the revised version commonly referred to as SHA-1. The other functions of SHA series produce 224-, 256-, 384- and 512-bit hash values.

51. MSU/CSE 260 Fall 2009 51 Secure Hashes in Python >>> from hashlib import md5 >>> md5("cse201").hexdigest() 'fa8190eb6032a99f61d822c2474980bf‘ >>> from hashlib import sha1 >>> sha1("cse201").hexdigest() '44dd02666ee30406837b6b5897c6d013fdf41dc1'