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Functions

Functions. Read Section 1.8. 1. 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

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Functions

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  1. Functions Read Section 1.8 MSU/CSE 260 Fall 2009 1

  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 2 MSU/CSE 260 Fall 2009

  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? 3 MSU/CSE 260 Fall 2009

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

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

  6. Domain, Co-domain, … • Definition:Let fbe a function from A to B,that is, f: A→ B. Then A is called the domain of f, and B is the codomainof f. • If f(a) = b, then • bis called the image of a, and • a is apre-imageof 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 6 MSU/CSE 260 Fall 2009

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

  8. Sum, Products of real-valued Functions • Definition:Let f1 and f2be functions from A to R. Then f1 + f2 and f1 f2 are also functions from A to R,defined as: • (f1 + f2)(x) = f1(x) + f2(x) • (f1  f2)(x) = f1(x) f2(x) • Note that if f1 and f2 do not have the same domain, the above operations do not make sense. 8 MSU/CSE 260 Fall 2009

  9. Image of a subset of a domain • Definition:Let f be a function from A to B, and letSbe a subset of A. Theimage of S, denotedf (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} 9 MSU/CSE 260 Fall 2009

  10. a 2 b 3 c 4 One-to-one Functions • Definition: A function f from Ato 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Af (x) = f (y) →x = y. 1 10 MSU/CSE 260 Fall 2009

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

  12. a b c 3 d 4 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. 1 2 12 MSU/CSE 260 Fall 2009

  13. a a b b c c 3 d d 4 1 a a b b 2 c 3 c 3 d 4 4 Summary of Function Types 1-to1 but not onto onto but not 1-to-1 both 1-to-1 and onto; bijection 1 1 a 2 1 2 b 3 2 c 4 3 1 Not a function 2 Neither 1-to-1 nor onto 13 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 strictlyincreasing if xA yAx < y→f (x) < f (y). • f: A→ B is strictly decreasing ifxA yAx < y→f (x) > f (y). • Note that strictly increasing, or strictly decreasing (strictly monotone) functions must be one-to-one. 14 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Bf(a)= b →f -1(b) = a. • f is called invertible. a 1 f -1(1) = d f -1(2) = b f -1(3) = a f -1(4) = c b 2 f c 3 d 4 15 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. 16 MSU/CSE 260 Fall 2009

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

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

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

  20. Composition of Functions f o g g f B B B B B B c c c c c c c c A A A A 20 MSU/CSE 260 Fall 2009

  21. Example • Consider the two functions f :Z→Z,where f (x) = 2x + 3 g:Z→Z, whereg (x) = 3x + 2. • What are f o g, andg 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 21 MSU/CSE 260 Fall 2009

  22. Example 22 MSU/CSE 260 Fall 2009

  23. Example 23 MSU/CSE 260 Fall 2009

  24. Example…. 24 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 Gf = {(x, f (x))| x  A}. 25 MSU/CSE 260 Fall 2009

  26. Graph of a Function…. • The graph of the functionf :Z→Z, wheref (n) = 2n + 1, is Gf = {(n, 2n + 1) | n Z} (2,5) f (n) (1,3) (0,1) n (-1,-1) 26 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, • Themod function. 27 MSU/CSE 260 Fall 2009

  28. Floor Function • Definition:The floorfunction from R to Zassigns 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 28 MSU/CSE 260 Fall 2009

  29. Ceiling Function • The ceiling function from R to Zassigns 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Rx– 1 <xx  x< x + 1. • Examples: • 18= 18 • 3.75 = 4 • – 4.5 = – 4 29 MSU/CSE 260 Fall 2009

  30. Floor and Ceiling Functions, recap * * x x 0 x x 30 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Rx– 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 * * x x 0 x x 31 MSU/CSE 260 Fall 2009

  32. Example 32 MSU/CSE 260 Fall 2009

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

  34. Integer Functions MSU/CSE 260 Fall 2009 34

  35. 35 MSU/CSE 260 Fall 2009

  36. Example 36 MSU/CSE 260 Fall 2009

  37. Example: Solution 37 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 modfunction is about: • n mod m • m is called modulus • n = m n/m + n mod m quotient remainder 38 MSU/CSE 260 Fall 2009

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

  40. Example • nmodm = 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. 40 MSU/CSE 260 Fall 2009

  41. mod Function m 0 + – 1 m-1 2 41 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 42 MSU/CSE 260 Fall 2009

  43. Hash Functions MSU/CSE 260 Fall 2009 43

  44. Hash Functions • A hash function h: keys→ integersmaps “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 shouldnot be “easy” to find its pre-image • Applications • Searching/indexing • Information hiding • File signature 44 MSU/CSE 260 Fall 2009

  45. Hashing for Indexing • A hash function h: keys→ integersmaps “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. 45 MSU/CSE 260 Fall 2009

  46. Example • Storing CSE 260, both sections, PIDs\A • Using Hash Function h(PID)=kmod31 46 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 47 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 48 MSU/CSE 260 Fall 2009

  49. How password is checked. 49 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. 50 MSU/CSE 260 Fall 2009

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