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Chapter 5 Relations and Functions

Chapter 5 Relations and Functions. Yen-Liang Chen Dept of Information Management National Central University. 5.1. Cartesian products and relations.

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Chapter 5 Relations and Functions

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  1. Chapter 5 Relations and Functions Yen-Liang Chen Dept of Information Management National Central University

  2. 5.1. Cartesian products and relations • Definition 5.1. The Cartesian product of A and B is denoted by AB and equals {(a, b)aA and bB}. The elements of AB are ordered pairs. The elements of A1A2…An are ordered n-tuples. • AB=AB • Ex 5.1. A={2, 3, 4}, B={4, 5}. • What are AB, BA, B2 and B3? • Ex 5.2, What are RR, R+R+ and R3?

  3. Tree diagrams for the Cartesian product

  4. Relations • Definition 5.2. Any subsets of AB is called a relation from A to B. Any subset of AA is called a binary relation on A. • Ex 5.5. The following are some of relations from A={2,3,4} to B={4,5}: (a) , (b) {(2, 4)}, (c) {(2, 4), (2, 5)}, (d) {(2, 4), (3, 4), (4, 4)}, (e) {(2, 4), (3, 4), (4, 5)}, (f) AB. • For finite sets A and B with A=m and B=n, there are 2mn relations from A to B. There are also 2mn relations from B to A.

  5. Examples • Ex 5.6. R is the subset relation where (C, D)R if and only if C, DB and CD. • Ex 5.7. We may define R on set A as {(x, y)xy}. • Ex 5.8. Let R be the subset of NN where R={(m, n)n=7m} • For any set A, A=. Likewise,  A =.

  6. Theorem 5.1. • A(BC)=(AB)(AC) • A(BC)=(AB)(AC) • (AB)C=(AC)(BC) • (AB)C=(AC)(BC) • Why?

  7. 5.2. Functions: Plain and one-to-one • Definition 5.3. f: AB, A is called domain and B is codomain. f(A) is called the range of f. • For (a, b)f, b is called image of a under f whereas a is a pre-image of b. • Ex 5.10. • Greatest integer function, floor function • Ceiling function • Truncate function • Row-major order mapping function • Ex 5.12. a sequence of real numbers r1, r2,… can be thought of as a function f: Z+R and a sequence of integers can be thought of as f: Z+Z

  8. properties • For finite sets A and B with A=m and B=n, there are nm functions from A to B. • Definition 5.5. f: AB, is one-to-one or injective, if each element of B appears at most once as the image of an element of A. If so, we must have AB. Stated in another way, f: AB, is one-to-one if and only if for all a1, a2A, f(a1)=f(a2) a1=a2. • Ex 5.13. f(x)=3x+7 for xR is one-to-one. But g(x)=x4-x is not. (Why?)

  9. Number of one-to-one functions • Ex 5.14. A={1, 2, 3}, B={1, 2, 3, 4, 5}, there are 215 relations from A to B and 53 functions from A to B. • In the above example, we have P(5, 3) one-to-one functions. • Given finite sets A and B with A=m and B=n, there are P(n, m) one-to-one functions from A to B.

  10. Theorem 5.2. • Let f: AB with A1, A2A. Then (a) f(A1A2)=f(A1)f(A2), (b) f(A1A2)f(A1)f(A2), (c) f(A1A2)=f(A1)f(A2) when f is one-to-one. A1={2,3,4}, A2={3,4,5} f(2)=b, f(3)=a, f(4)=a, f(5)=b

  11. Restriction and Extension • Definition 5.7. If f: AB and A1A, then fA1: AB is called the restriction of f to A1 if fA1(a)=f(a) for all aA1. • Definition 5.8. Let A1A and f: A1B. If g: AB and g(a)=f(a) for all aA1, then we call g an extension of f to A. • Ex 5.17.Let f: AR be defined by {(1, 10), (2, 13), (3, 16), (4, 19), (5, 22)}. Let g: QR where g(q)= 3q+7 for all qQ. Let h: RR where h(r)= 3r+7 for all rR. • g is an extension of f, f is the restriction of g • h is an extension of f, f is the restriction of h • h is an extension of g, g is the restriction of h • Ex 5.18. g and f are shown in Fig 5.5. f is an extension of g.

  12. 5.3. Onto Functions: Stirling numbers of the second kind • Definition 5.9. f: AB, is onto, or surjective, if f(A)=B-that is, for all bB there is at least one aA with f(a)=B. If so, we must have AB. • Ex 5.19. The function f: RR defined by f(x)=x3 is an onto function. But the function g: RR defined by f(x)=x2 is not an onto function. • Ex 5.20. The function f: ZZ defined by f(x)=3x+1 is not an onto function. But the function g: QQ defined by g(x)=3x+1 is an onto function. The function h: RR defined by h(x)=3x+1 is an onto function.

  13. The number of onto functions • Ex 5.22. If A={x, y, z} and B={1,2}, there are 23-2=6 onto functions. In general, if A=m and B=2, then there are 2m-2 onto functions. • Ex 5.23. If A={w, x, y, x} and B={1,2, 3}. • There are C(3, 3)34 functions from A to B. • Consider subset B of size 2, such as {1, 2}, {1, 3}, {2, 3}, there are C(3, 2)24 functions from A to B. • Consider subset B of size 1, such as {1}, {3}, {2}, there are C(3, 1)14 functions from A to B. • Totally, there are C(3, 3)34- C(3, 2)24+ C(3, 1)14 onto functions from A to B.

  14. The number of onto functions • For finite sets A and B with A=m and B=n, the number of onto functions is:

  15. Examples • Ex 5.24. Let A={1, 2, 3, 4, 5, 6, 7} and B={w, x, y, z}. So, m=7 and n=4. There are 8400 onto functions. • C(4, 4)47-C(4, 3)37+C(4, 2)27-C(4, 1)17=8400 • Ex 5.26. Let A={a, b, c, d} and B={1, 2, 3}. So, m=4 and n=3. There are 36 onto functions, or equivalently, 36 ways to distribute four distinct objects into three distinguishable containers, with no container empty. • For mn, the number of ways to distribute m distinct objects into n numbered containers with no container left empty is :

  16. Distinguishable and identical • distribute m distinct (identical) objects into n numbered (identical) containers • {a, b} in container 1, {c} in container 2, {d} in container 3 • {a, b} in one container, {c} in the other container, {d} in another container • 2 objects in container 1, 1 object in container 2, 1 object in container 3 • 2 objects in one container, 1 object in the other container, 1 object in another container

  17. Stirling number of the second kind • The stirling number of the second kind is the number of ways to distribute m distinct objects into n identical containers, with no container left empty, denoted S(m,n), which is

  18. It is the number of possible ways to distribute m distinct objects into n identical containers with empty containers allowed.

  19. Theorem 5.3. • S(m+1, n)=S(m, n-1) + n S(m, n). • am+1 is in a container by itself. Objects a1, a2, …, am will be distributed to the first n-1 containers, with none left empty. • am+1 is in the same container as another object. Objects a1, a2, …, am will be distributed to the n containers, with none left empty.

  20. Ex 5.28. • 30030=23571113. How many ways can we factorize the number into two factors? The answer is S(6, 2)=31. • How many ways can we factorize the number into three factors? The answer is S(6, 3)=90. • If we want at least two factors in each of these unordered factorization, then there are 202=

  21. 5.4. Special functions • Definition 5.10. f: AAB is called a binary operation. If BA, then it is closed on A. • Definition 5.11. A function g:AA is called unary, or monary, operation on A. • Ex: • the function f: ZZZ, defined by f(a, b)=a-b, is a closed binary operation. • The function g: Z+Z+Z, defined by g(a, b)=a-b, is a binary operation on Z+, but it is not closed. • The function h: R+R+, defined by h(a)=1/a, is a unary operation.

  22. Commutative and associative • Definition 5.12. f is commutative if f(a, b)= f(b, a) for all a, b. • When BA, f is said to be associative if for all a, b, c we have f(f(a, b), c)=f(a, f(b, c))

  23. Commutative and associative • Ex 5.32. • The function f: ZZZ, defined by f(a, b) = a+b-3ab is commutative and associative. • The function f: ZZZ, defined by h(a, b) = ab is not commutative but is associative. • Ex 5.33. Assume A={a, b, c, d} and f: AAA. There are 416 closed binary operations on A. Determine the number of commutative and closed operations g. • there are four choices for g(a, a), g(b, b), g(c, c) and g(d, d). • The other 12 ordered pairs can be classified into 6 groups because of the commutative property. • The total number of binary and commutative operations is 4446.

  24. Identity • Definition 5.13. Let f: AAB be a binary operation on A. An element x in A is called an identity for f if f(a, x)= f(x, a)=a for all a in A. • Ex 5.34. • If f(a, b)=a+b, then 0 is the identity. • If f(a, b)=ab, then 1 is the identity. • If f(a, b)=a-b, then there is no identity.

  25. Identity • Theorem 5.4. Let f: AAB be a binary operation on A. If f has an identity, then that identity is unique. • Ex 5.35. If A={x, a, b, c, d}, how many closed binary operations on A have x as the identity? • Because x is the identity, we have Table 5.2, where there are 16 cells left unfilled. • There are 516 closed binary operations on A, where x is the identity. • Of these, 510=5456 are commutative. • If every element can be used as the identity, we have 511 closed binary operations that are commutative.

  26. Projection • Definition 5.14. if DAB, then A: DA, defined by A(a, b)=a is called the projection on the first coordinate. The function B: DB, defined by B(a, b)=b is called the projection on the second coordinate. • if DA1 A2…An, then A: DAi1 Ai2 Ai3,,,, Aim, defined by (a1, a2, …, an)= ai1, ai2, ai3, …, aim is called the projection on the i1, i2, …, im coordinates.

  27. The projection of a database

  28. 5.5. Pigeonhole principle • The pigeonhole principle: If m pigeons occupy n pigeonholes and m>n, then at least one pigeonhole has two or pigeons roosting in it. • Ex 5.39: among 13 people, at least two of them have birthdays during the same month. • Ex 5.40. In a laundry bag, there are 12 pairs of socks. Drawing the socks from the bag randomly, we will draw at most 13 of them to get a matched pair. • Ex 5.42. Let SZ+ and S=37. Then S contains two elements that have the same remainder upon division by 36.

  29. Examples • Ex 5.43. If 101 integers are selected from the set S={1, 2, …, 200}, then there are two integers such that one divide the other. • For each xS, we may write x=2ky, with k0 and gcd(2,y)=1. Then yT={1, 3, 5, …, 199}, where T =100. By the principle, there are two distinct integers of the form a=2my and b=2ny for some y in T. • Ex 5.44. Any subset of size 6 from the set S={1, 2, …, 9} must contain two elements whose sum is 10.

  30. Examples • Ex 5.45. Triangle ACE is equilateral with AC=1. If five points are selected from the interior of the triangle, there are at least two whose distance apart is less than 1/2. • Ex 5.46. Let S be a set of six positive integers whose maximum is at most 14. The sums of the elements in all the nonempty subsets of S cannot be all distinct. • There are 26-1=63 subsets of S. • 1SA 9+10+…+14=69 • If A=5, then 1SA 10+…+14=60 • There are 62 nonempty subsets A of A with 5A.

  31. Ex 5.47 • Let m in Z+ and m is odd. There exists a positive integer n such that m divides 2n-1. • Consider the m+1 positive integers 21-1, 22-1,…, 2m-1, 2m+1-1. By the principle, we have 1s<tm+1, where 2s-1 and 2t-1 have the same remainder upon division by m. • Hence 2s-1=q1m+r and 2t-1=q2m+r. • (2t-1)-(2s-1)= 2t-2s=2s(2t-s-1)=(q2-q1)m. • Since m is odd, gcd(m, 2s)=1. • Hence, m2t-s-1, and the result follows with n=t-s.

  32. Ex 5.49 • For each nZ+, a sequence of n2+1 distinct real numbers contains a decreasing or increasing subsequence of length n+1. • Let the sequence be a1, a2,…,an2+1. For 1k n2+1 • xk= the maximum length of a decreasing subsequence that ends with ak. • yk= the maximum length of an increasing subsequence that ends with ak. • If there is no such sequence, then 1xkn and 1ykn for 1k n2+1. • Consequently, there are at most n2 distinct ordered pairs of xk and yk. • But we have n2+1 ordered pairs of xk and yk. • Thus, there are two identical (xi, yi) and (xj, yj). • But since every real number is distinct from one another, this is a contradiction.

  33. 5.6. Function composition and inverse function • For each integer c there is a second integer d where c+d = d+c=0, and we call d the additive inverse of c. Similarly, for each real number c there is a second real number d where cd = dc=1, and we call d the multiplicative inverse of c. • Definition 5.15. If f: AB, then f is said to be bijective, or to be one-to-one correspondence, if f is both onto and one-to-one. • Definition 5.16. The function 1A: AA, defined by 1A(a)=a for all aA, is called the identity function for A.

  34. Equal function • Definition 5.17. If f, g : AB, we say that f and g are equal and write f = g, if f(a)=g(a) for all aA. • A common pitfall may happen when f and g have a common domain A and f(a)=g(a) for all aA, but they are not equal. • Ex 5.51. f and g look similar but they are not equal. • Ex 5.52. f and g look different but they are indeed equal.

  35. Composite function • Definition 5.18. If f : AB and g : BC, we define the composition function, which is denoted by gf: AC, (gf) (a)=g(f(a)) for each aA. • Ex 5.53, Ex 5.54. • Properties • The codomain of f = domain of g • If range of f  domain of g, this will be enough to yield gf: AC. • For any f : AB, f1A = f = 1Bf.

  36. Is function composition associative? • Theorem 5.6. If f : AB and g : BC and h : CD, then (hg)f=h(gf). • Ex 5.55.

  37. Definitions • Definition 5.19. If f : AA, we define f1=f and fn+1=ffn. • Ex 5.56 • Definition 5.20. For sets A and B, if  is a relation from A to B, then the converse of , denoted by c, is the relation from B to A defined by c={(b, a) (a, b)}. • Ex 5.57

  38. Invertible function • Definition 5.21. If f : AB, then f is said to be invertible if there is a function g: BA such that gf=1A and fg=1B. (Ex 5.58 ) • Theorem 5.7. If a function f : AB is invertible and a function g : BA satisfies gf=1A and fg=1B, then this function g is unique.

  39. Invertible function • Theorem 5.8. A function f : AB is invertible if and only if it is one-to-one and onto. • Theorem 5.9. If f : AB and g : BC are invertible functions, then gf: AC is invertible and (gf)-1=f-1g-1. • Ex 5.60. f:RR is defined by f(x)=mx+b, and f-1:RR is defined by f-1(x)=(1/m)(x-b). • Ex 5.61. f:RR+ is defined by f(x)=ex, and f-1:R+R is defined by f-1(x)=ln x.

  40. Preimage • Definition 5.22. If f: AB and B1B, then f-1(B1)={xAf(x)B1}. The set f-1(B1) is called the pre-image of B1 under f. • Ex 5.62. If f={(1, 7), (2, 7), (3, 8), (4, 6), (5, 9), (6, 9)}, what are the preimage of B1={6, 8}, B2={7, 8}, B3={8, 9}, B4={8, 9, 10}, B5={8, 10}.

  41. Ex 5.64 • Table 5.9 for f:ZR with f(x)=x2+5 • Table 5.10 for g:RR with g(x)= x2+5

  42. Theorems • For aA, af-1(B1B2) f(a) B1B2 f(a) B1 or f(a)B2 af-1(B1) or af-1(B2) af-1(B1)f-1(B2)

  43. Theorem 5.11. • If f : AB and A=B. Then the following statements are equivalent: (a) f is one-to-one; (b) f is onto, (c) f is invertible.

  44. 5.7. Computational complexity • Can we measure how long it takes the algorithm to solve a problem of a certain size? To be independent of compliers used, machines used or other factors that may affect the execution, we want to develop a measure of the function, called time complexity function, of the algorithm. Let n be the input size. Then f(n) denotes the number of basic steps needed by the algorithm for input size n.

  45. Order • Definition 5.23. Let f, g: Z+R. we say that g dominates f (or f is dominated by g) if there exist constants mR+ and kZ+ such that fmg(n) for all nZ+, where nk. • When f is dominated by g we say that f is of order g and we use what is called “Big-Oh” notation to denote this. We write fO(g). • O(g) represents the set of all functions with domain Z+ and codomain R that are dominated by g.

  46. Examples • Ex 5.65, we observe that fO(g). • Ex 5.66. we observe that gO(f). • Ex 5.67. When f(n)=atnt+at-1tt-1 +…+a0, fO(nt). • Ex 5.68. • f(n)=1+2+…+nO(n2). • f(n)=12+22+…+n2O(n3). • f(n)=1t+2t+…+ntO(nt+1). • When dealing with the concept of function dominance, we seek the best ( or tightest) bound.

  47. some order functions that are commonly seen

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