Discrete Structures Chapter 5 Relations and Functions

1. Discrete StructuresChapter 5 Relations and Functions Nurul Amelina Nasharuddin Multimedia Department

2. Objectives • On completion of this chapter, student should be able to: • Define a relation and function • Determine the type of function (one-to-one, onto, one-to-one correspondence) • Find a composite function • Find an inverse function

3. Outline • Cartesian products and relations • Functions: Plain, one-to-one, onto • Function composition and inverse functions • Functions for computer science • Properties of relations • Computer recognition: Zero-one matrices and directed graphs • Use in database example

4. Relationship

5. Recall: Cartesian Products • For sets A, B, the Cartesian product, or cross product, of A and B is denoted by A × B and equals {(a, b) | a  A, b  B} • Elements of A × B are ordered pairs. For (a, b), (c, d)  A × B , (a, b) = (c, d) if and only if a = c and b = d

6. Recall: Cartesian Products Properties: If A, B are finite, it follows from the rule of product that |A × B| = |A||B| Although we generally will not have A × B = B × A, we will have |A×B|=|B×A|

7. Example (1) Let A = {2, 3, 4}, B = {4, 5}. Then a) A × B = {(2, 4), (2, 5), (3, 4), (3, 5), (4, 4), (4, 5)} b) B × A = {(4, 2), (4, 3), (4, 4), (5, 2), (5, 3), (5, 4)} c) B2 = B × B = {(4, 4), (4, 5), (5, 4), (5, 5)} d) B3 = B × B × B = {(a, b, c) | a, b, c  B}; for instance, (4, 5, 5)  B3

8. Example (2) An experiment E is conducted as follows: A single dice is rolled and its outcome noted, and then a coin is flipped and its outcome noted. Determine a sample space S for E S1={1, 2, 3, 4, 5, 6} be a sample space dice. S2= {H, T} be a sample space coin. Then S = S1 × S2 is a sample space for E.

9. Example (2)

10. Example (3) At the Wimbledon Tennis Championships, women play at most three sets in a match The winner is the first to win two sets. If we let N and E denote the two players, the tree diagram indicates the six ways in which this match can be won For example, the starred line segment (edge) indicates that player E won the first set The double starred edge indicates that player N has won the match by winning the first and third sets

11. Example (3)

12. Relations • Let A = {0,1,2}, B = {1,2,3}. A x B = {(0,1), (0,2), (0,3), (1,1), (1,2), (1,3), (2,1), (2,2), (2,3)} • Let say an element x in A is related to an element y in B iff x is less than y. x R y: x is related to y • 0 R 1, 0 R 2, 0 R 3, 1 R 2, 1 R 3, 2 R 3 • The set of all ordered pair in A x B where elements are related {(0,1), (0,2), (0,3), (1,2), (1,3), (2,3)}

13. Relations • For sets A, B, a (binary) relation R from A to B is a subset of A × B. Any subset of A × A is called a (binary) relation on A • Given an ordered pair (a, b) in A x B, x is related to y by R (x R y) iff (x, y) is in R • In general, for finite sets A, B with |A| = m and |B|= n, there are 2mn relations from A to B, including the empty relation as well as the relation A × B itself

14. Example (1) Let A = {2, 3, 4}, B = {4, 5}. Then A × B = {(2, 4), (2, 5), (3, 4), (3, 5), (4, 4), (4, 5)}. The following are some of the relations from A to B.  {(2, 4)} {(2, 4), (2, 5)} {(2, 4), (3, 4), (4, 4)} {(2, 4), (3, 4), (4, 5)} A × B Since |A × B| = 6, there are 26 possible relations from A to Β (for there are 26 possible subsets of A × B )

15. Example (2) Let A = {1,2}, B = {1,2,3} and define a binary relation from A to be as follows: Given any (x,y)  A x B, (x,y)  R  x – y is even State explicitly which ordered pairs are in A x B and which are in R Is 1 R 3? Is 2 R 3? Is 2 R 2?

16. Example (2) A x B= {(1,1), (1,2), (1,3), (2,1), (2,2), (2,3)} and R, when x – y is even = {(1,1), (1,3), (2,2)}  A x B Is 1 R 3? Yes Is 2 R 3? No Is 2 R 2? Yes (1,1)  R because 1 – 1= 0 is even (1,2)  R because 2 – 1 = 1 is not even

17. Example (3) Let B={1,2} and A=P(B) = {,{1},{2},{1,2}} |A×A| = 4.4 = 16 A×A = {(∅,∅),(∅,{1}),(∅,{2}),(∅,{1,2}), ({1},∅), ({1},{1}), ({1},{2}), ({1},{1,2}) ({2},∅),({2},{1}), ({2},{2}),({2},{1,2}) ({1,2},∅),({1,2},{1}),({1,2},{2}, ({1,2},{1,2})} The following is an example of a relation on A: R = {(∅, ∅), (∅, {1}), (∅, {2}), (∅, {1, 2}), ({1}, {1}), ({1}, {1, 2}), ({2}, {2}), ({2}, {1, 2}), ({1, 2}, {1, 2})}

18. Example (4) • With A = Z+ (set of positive integers), we may define a relation R on set A as {(x, y) | x ≤ y} • This is the familiar “is less than or equal to” relation for the set of positive integers • It can be represented graphically as the set of points, with positive integer components, located on or above the line y = x in the Euclidean plane, as partially shown in the figure below

19. (7, 7), (7, 11)  R (8, 2)  R (7, 11)  R or 7 R 11 (infix notation)

20. Arrow Diagrams of Relations • Let A = {1,2,3}, B = {1,3,5} • For all x A and y B, relations S and T • (x,y)  S  x < y • T = {(2,1), (2,5)}

21. Functions • For nonempty sets A and B, • A function, or mapping, f from A to B, denoted f: A  B, is a relation from A to B in which every element of A appears exactly once as the first component of an ordered pair in the relation • Sample functions: • f : R  R, f(x) = x2 • f : Z  Z, f(x) = x + 1 • f : Q  Z, f(x) = 2

22. Functions • A function f from a non-empty set A to a set B is a relation from A to B satisfying the following two properties: • x  A, y  B such that (x,y)  f • (x, y), (x, y’)  f, y = y’ • The 1st property says every x  A is related to at least one y  B • The 2nd property says each x  A is related to at most one y  B • That is, a relation from A to B is a function from A to B if and only if every x  A is related to exactly one y  B

23. Example (1) Let A = {1,2,3}, B = {7,8,9} g = {(1,8), (2,9), (3,9), (3,10)}  A x B is not a function from A to B: (3,9), (3,10)  g but 9  10. Relation g fails to be a function because 3  A is related to two (distinct) elements 9, 10  B h = {(1,9), (2,10), (3,9)}  A x B is a function from A to B. Relation h is a function because each element of A is related to exactly one element in B

24. Arrow Diagram • We often write f(a) = b when (a, b) is an ordered pair in the function f. For (a, b)  f, b is called the image of a under f, whereas a is a preimage (inverse image) of b

25. Arrow Diagram The arrow diagram of a function from A to B has the characteristic that there is exactly one arrow shooting out from every element of A However, a element of B can be hit by no arrows, one arrow, or many arrows

26. Domain and Codomain • For the function f: A → B, A is called the domain of f and B the codomain of f • The subset of B consisting of those elements that appear as second components in the ordered pairs of f is called the range of f and is also denoted by f (A) because it is the set of images (of the elements of A) under f • Eg: Let A = {1, 2, 3}, B = {w, x, y, z }, f={(1, w), (2, x), (3, x)} • Domain of f = {1,2,3}, the codomain of f = {w, x, y, z}, and the range of f = f (A) = {w, x}

27. Interesting Functions in Computer Science Greatest integer function, or floor function: This function f: R → Z, is given by f(x) =  x  = the greatest integer n less than or equal to x, n  x  n + 1 Consequently, if x is a real number and n is an integer, then f(x) =  x  = is the integer to the immediate left of x on the real number line. For this function, we find that 1)  3.8  = 3,  3  = 3, –3.8  = –4, –3  = –3; 2)  7.1 + 8.2  =  15.3  = 15 = 7 + 8 =  7.1  +  8.2  3)  7.7 + 8.4  =  16.1  = 16 ≠ 15 = 7 + 8 =  7.7  +  8.4 

28. Interesting Functions in Computer Science Ceiling function: This function g: R → Z, is given by g(x) =  x  = the least integer greater than or equal to x, n  x  n + 1 Consequently, if x is a real number and n is an integer, then g(x) =  x = is the integer to the immediate right of x on the real number line. For this function, we find that 1)  3 = 3,  3.01 =  3.7 = 4 =  4 , –3.01 = –3.7 = –3; 2)  3.6 + 4.5 =  8.1 = 9 = 4 + 5 =  3.6 +  4.5 3)  3.3 + 4.2 =  7.5 = 8 ≠ 9 = 4 + 5 =  3.3 +  4.2

29. Interesting Functions in Computer Science Trunc function (for truncation): valued function defined on R. This function deletes the fractional part of a real number For example, trunc(3.78) = 3, trunc(5) = 5, trunc(–7.22) = –7 Note that trunc(3.78) = 3.78 = 3 while trunc(–3.78) = –3.78 = –3

30. Total Number of Functions For general case, let A, B be nonempty sets with |A| = m, |B| = n. Consequently, If A = {a1, a2, …, am} and B={b1,b2,…,bn}, then a typical function f: A → B can be described by {(a1, x1), (a2, x2), (a3, x3), …, (am, xm)} – m ordered pairs. x1 can selected from any of the n elements of B x2 “ ………………….. xm “ In this way, using the rule of product, there are nm = |B||A| functions from A to B 30

31. Total Number of Functions • Let A = {1, 2, 3}, B = {w, x, y, z}, f = {(1, w), (2, x), (3, x)} • There are 24.3=212 = 4096 relations from A to B • We have examined one function among these relations, and now we wish to count the total number of functions from A to B • Therefore, there are 43 = |B||A| = 64 functions from A to B, and 34 = |A||B| = 81 functions from B to A 31

32. Properties of Functions • Two important properties that functions may satisfy: • The property of being one-to-one and • The property of being onto • Functions that satisfy both properties are called one-to-one correspondences or one-to-one onto functions 32

33. One-to-one Function • Let f be a function from A to B. f is called one-to-one, or injective, iff for all elements x1 and x2 in A • If f(x1) = f(x2), then x1 = x2 • or, equivalently if x1 x2, then f(x1)  f(x2) • Each element of B appears atmost once as the image of an element of A 33

34. One-to-one Function 34

35. Not One-to-one Function 35

36. One-to-one Function • If f: A → B is one-to-one, with A, B finite, we must have |A|≤|B| • For arbitrary sets A, B, f: A → B is one-to-one if and only if for all, a1, a2 A, • f (a1) = f (a2)  a1 = a2 36

37. Identifying One-to-one Functions Defined on Finite Sets • Let X = {1,2,3} and Y = {a,b,c,d} • Define H: X  Y as follows: H(1) = c, H(2) = a, H(3) = d. Is H one-to-one? • Define K: X  Y as follows: K(1) = d, K(2) = b, K(3) = d. Is K one-to-one? 37

38. Identifying One-to-one Functions Defined on Infinite Sets • Suppose f is a function defined on an infinite set X. By definition, f is one-to-one iff the following is true: • x1, x2  X, if f(x1) = f(x2), then x1 = x2 • Suppose x1 and x2 are elements of X such that • f(x1) = f(x2) • (2) Show that x1 = x2 38

39. Example (1) Consider the function f: R→ R where f (x) = 3x + 7 for all x  R Then for all x1 , x2 ,  R, we find that f (x1) = f (x2)  3x1 + 7 = 3x2 + 7  3x1 = 3x2 (minus both side with 7)  x1 = x2 , (dividing both side with 3) so the given function f is one-to-one 39

40. Example (2) On the other hand, suppose that g: R → R is the function defined by g (x) = x4 – x for each real number x Let x1 = 0 and x2 =1.Then g(x1) = g(0) = (0)4 – 0 = 0 g(x2) = g(1) = (1)4 – (1) = 1 – 1 = 0 Hence g(x1) = g(x2) but x1  x2 (0 ≠ 1) – that is, g is not one to-one because there exist real numbers x1, x2 where g (x1) = g (x2) but x1  x2 40

41. Example (3) Let A = {1, 2, 3} and B = {1, 2, 3, 4, 5} The function f = {(1, 1), (2, 3), (3, 4)} is a one-to-one function from A to B; g = {(1, 1), (2, 3), (3, 3)} is a function from A to B, but fails to be one-to-one because g(2) = g(3) = 3 but 2 ≠ 3 For A, B in the above example, there are 215 relations from A to B and 53 of these are functions from A to B. The next question we want to answer is how many functions f: A → B are one-to-one 41

42. Calculate Total No of One-to-one Functions • With • A = {a1, a2, a3, …, am}, • B = {b1, b2, b3, …, bn}, and m ≤ n , • a one-to-one function f: A → B has the form • {(a1, x1), (a2, x2), (a3, x3), …, (am, xm)}, • Where there are • n choices for x1 • n – 1 choices for x2 • n – 2 choices for x3 • ……….. • n – m+1 choices for xm., • The number of one-to-one functions from A to B is • n(n-1)(n-2)…(n-m+1)= n!/(n-m)! = P(n,m)= P(|B|,|A|) 42

43. Example (1) • Consequently, for A, B where A = {1, 2, 3} and B = {1, 2, 3, 4, 5}, there are • P(5,3) = P(|B|,|A|) • =5 . 4 . 3 • = 60 one-to-one functions f: A → B. 43

44. Onto Function • A function f: A→ B is called onto, or surjective, if • f (A) = B – that is, if for all b  B there is at least one a  A with f (a) = b 44

45. Not Onto Function 45

46. Identifying Onto Functions Defined on Finite Sets • Let X = {1,2,3,4} and Y = {a,b,c} • Define H: X  Y as follows: H(1) = c, H(2) = a, H(3) = c, H(4) = b. Is H onto? • Define K: X  Y as follows: K(1) = c, K(2) = b, K(3) = b, K(4) = c. Is K onto? 46

47. Identifying Onto Functions Defined on Infinite Sets • Suppose f is a function from a set X to a set Y, and suppose Y is infinite. By definition, f is onto iff the following is true: • y  Y, x  X such that f(x) = y • Suppose that y is any element of Y • (2) Show that there is an element of X with f(x) = y 47

48. Example (1) The function f: R → R defined by f(x) = x3 is an onto function If r is any real number in the codomain of f, then the real number 3√r is in the domain of f and f(3√r) = (3√r)3 = r E.g. f(3) = 27, f(-3) = -27 Hence the codomain of f = R = range of f, and the function f is onto 48

49. Example (2) • The function g: R → R, where g(x) = x2 for each real number x, is not an onto function • In this case, no negative real number appears in the range of g • For example, for –9 to be in the range of g, we would have to be able to find a real number r with g(r) = r2 = –9 • Note, however, that the function h: R → [ 0, +∞ ) defined by h(x) = x2 is an onto function 49

50. Example (3) • Consider the function f: Z → Z, where f(x) = 3x + 1 for each x  Z • Here the range of f = {…, –8, –5, –2, 1, 4, 7, …}  Z, so f is not an onto function • E.g. f(x) = 3x + 1 = 8 then x = 7/3 • Rational number 7/3 is not an integer –so there is no x in the domain Z with f(x) = 8 50