Chapter 5 Relations and Functions

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Chapter 5 Relations and Functions

• Yen-Liang Chen
• Dept of Information Management
• National Central University

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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. A2, 3, 4, B4, 5.
• What are A?B, B?A, B2 and B3?
• Ex 5.2, What are R?R, R?R and R3?

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Tree diagrams for the Cartesian product
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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
  A2,3,4 to B4,5 (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.

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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)?n7m
• For any set A, A???. Likewise, ? ? A ?.

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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?

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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

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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)? a1a2.
• Ex 5.13. f(x)3x7 for x?R is one-to-one. But
  g(x)x4-x is not. (Why?)

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Number of one-to-one functions

• Ex 5.14. A1, 2, 3, B1, 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.

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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.
• A12,3,4, A23,4,5
• f(2)b, f(3)a, f(4)a, f(5)b

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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) 3q7 for all q?Q. Let h R?R where h(r)
  3r7 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.

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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)3x1
  is not an onto function. But the function g Q?Q
  defined by g(x)3x1 is an onto function. The
  function h R?R defined by h(x)3x1 is an onto
  function.

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The number of onto functions

• Ex 5.22. If Ax, y, z and B1,2, there are
  23-26 onto functions. In general, if ?A?m and
  ?B?2, then there are 2m-2 onto functions.
• Ex 5.23. If Aw, x, y, x and B1,2, 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.

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The number of onto functions

• For finite sets A and B with ?A?m and ?B?n, the
  number of onto functions is

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Examples

• Ex 5.24. Let A1, 2, 3, 4, 5, 6, 7 and Bw, x,
  y, z. So, m7 and n4. There are 8400 onto
  functions.
• C(4, 4)47-C(4, 3)37C(4, 2)27-C(4, 1)178400
• Ex 5.26. Let Aa, b, c, d and B1, 2, 3. So,
  m4 and n3. 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

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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

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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

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

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Theorem 5.3.

• S(m1, n)S(m, n-1) n S(m, n).
• am1 is in a container by itself. Objects a1, a2,
  , am will be distributed to the first n-1
  containers, with none left empty.
• am1 is in the same container as another object.
  Objects a1, a2, , am will be distributed to the
  n containers, with none left empty.

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Ex 5.28.

• 300302?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

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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 gA?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.

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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))

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Commutative and associative

• Ex 5.32.
• The function f Z?Z?Z, defined by f(a, b)
  ab-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 Aa, 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.

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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)ab, 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.

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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 Ax, 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, 51054?56 are commutative.
• If every element can be used as the identity, we
  have 511 closed binary operations that are
  commutative.

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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.

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The projection of a database
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5.5. Pigeonhole principle

• The pigeonhole principle If m pigeons occupy n
  pigeonholes and mgtn, 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.

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Examples

• Ex 5.43. If 101 integers are selected from the
  set S1, 2, , 200, then there are two integers
  such that one divide the other.
• For each x?S, we may write x2ky, with k?0 and
  gcd(2,y)1. Then y?T1, 3, 5, , 199, where ?T
  ?100. By the principle, there are two distinct
  integers of the form a2my and b2ny for some y
  in T.
• Ex 5.44. Any subset of size 6 from the set S1,
  2, , 9 must contain two elements whose sum is
  10.

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Examples

• Ex 5.45. Triangle ACE is equilateral with AC1.
  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 26-163 subsets of S.
• 1?SA ?9101469
• If ?A?5, then 1?SA ?101460
• There are 62 nonempty subsets A of A with 5??A?.

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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 m1 positive integers 21-1, 22-1,,
  2m-1, 2m1-1. By the principle, we have
  1?sltt?m1, where 2s-1 and 2t-1 have the same
  remainder upon division by m.
• Hence 2s-1q1mr and 2t-1q2mr.
• (2t-1)-(2s-1) 2t-2s2s(2t-s-1)(q2-q1)m.
• Since m is odd, gcd(m, 2s)1.
• Hence, m?2t-s-1, and the result follows with
  nt-s.

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Ex 5.49

• For each n?Z, a sequence of n21 distinct real
  numbers contains a decreasing or increasing
  subsequence of length n1.
• Let the sequence be a1, a2,,an21. For 1?k? n21
• 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? n21.
• Consequently, there are at most n2 distinct
  ordered pairs of xk and yk.
• But we have n21 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.

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5.6. Function composition and inverse function

• For each integer c there is a second integer d
  where cd dc0, and we call d the additive
  inverse of c. Similarly, for each real number c
  there is a second real number d where cd dc1,
  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.

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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.

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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.

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Is function composition associative?

• Theorem 5.6. If f A?B and g B?C and h C?D,
  then (h?g)?fh?(g?f).
• Ex 5.55.

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Definitions

• Definition 5.19. If f A?A, we define f1f and
  fn1f?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

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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?f1A and f?g1B. (Ex 5.58 )
• Theorem 5.7. If a function f A?B is invertible
  and a function g B?A satisfies g?f1A and
  f?g1B, then this function g is unique.

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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)-1f-1?g-1.
• Ex 5.60. fR?R is defined by f(x)mxb, and
  f-1R?R is defined by f-1(x)(1/m)(x-b).
• Ex 5.61. fR?R is defined by f(x)ex, and
  f-1R?R is defined by f-1(x)ln x.

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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 B16,
  8, B27, 8, B38, 9, B48, 9, 10, B58,
  10.

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Ex 5.64

• Table 5.9 for fZ?R with f(x)x25
• Table 5.10 for gR?R with g(x) x25

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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)

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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.

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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.

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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.

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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)atntat-1tt-1 a0, f?O(nt).
• Ex 5.68.
• f(n)12n?O(n2).
• f(n)1222n2?O(n3).
• f(n)1t2tnt?O(nt1).
• When dealing with the concept of function
  dominance, we seek the best ( or tightest) bound.

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some order functions that are commonly seen
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5.8. Analysis of algorithms

• Ex 5.69. The time complexity is f(n)7n5?O(n)

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Ex 5.70. Linear search.

• The best case is O(1).
• The worst case is O(n).
• The average case f(n)pn(n1)/2nq, where npq1.

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Ex 5.72, Ex 5.73. compute an.

• In Figure 5.14, the time complexity is f(n)
  ?O(n).
• In Figure 5.15, the complexity is ?O(log n).

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The growth of complexities