Chapter 5

1
Chapter 5 Relations and Functions
2
5.1 Cartesian Products and Relations

• Definition 5.1 For sets A, B ? U, the Cartesian
  product, or cross product, of A and B is denoted
  by A ? B and equals (a, b) a ? A, b ? B.
• We say that the elements of A ? B are ordered
  pairs. The following properties hold
• For (a, b), (c, d) ? A ? B, we have (a, b) (c,
  d) if and only if a c and b d.
• If A, B are finite, it follows from the rule of
  product that A ? B A ? B. We will have
  A ? B B ? A, but not have A ? B B ? A.
• Although A, B ? U, it is not necessary that A ? B
  ? U.
• If n ? Z, n ? 3, and A1, A2, , An ? U, then the
  (n-fold) product of A1, A2, , An is denoted by
  A1?A2??An and equals (a1,a2,,an) ai ?Ai ,
  1?i?n.
• The elements of A1?A2??An are called ordered
  n-tuples.
• If (a1,a2,,an), (b1,b2,,bn) ? A1?A2??An, then
  (a1,a2,,an) (b1,b2,,bn) if and only if ai
  bi, for all 1?i?n.

3
Example 5.1 and 5.2 page 218.
4
Definition 5.2

• For sets A, B ? U, any subset of A ? B is called
  a relation from A to B. Any subset of A ? A is
  called a binary relation on A.

5
Example 5.5 page 220.
6
Lemma

• For finite sets A, B with A m and B n,
  there are 2mn relations from A to B, including
  the empty relation and A ? B itself. There are
  also 2nm (2mn) relations from B to A, one of
  which is also ? and another of which is B ? A.

7
Example 5.6 page 220
8
Example 5.7 page 220221. (note on infix
notation for a relation!)
9
Observations

• For any set A ? U, A?? ?. (If A?? ??, let (a, b)
  ? A ? ?. Then a?A and b??. Impossible!) Likewise,
  ? ?A?.

10
Theorem 5.1

• For any sets A, B, C ? U 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)

11
5.2 Functions Plain and One-to-One

• Definition 5.3 For nonempty sets A, B, a
  function, or mapping, f from A to B, denoted fA
  ? 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.
• 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
  of b.
• The definition suggests that f is a method for
  associating with each a ? A the unique element
  f(a)b?B. Consequently, (a,b), (a,c)?f implies
  bc.

12
Example 5.9 page 223.
13
Definition 5.4

• For the function fA ? 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. (See Fig 5.4)This diagram
  suggests that a may be regarded as an input that
  is transformed by f into the corresponding
  output, f(a).)

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15
Example 5.10 page 223224.
16
Example 5.10
17
Example 5.10
18
Lemma

• If A a1,a2,,am and B b1,b2,,bn, then a
  typical function fA ? B can be described by
  (a1,x1), (a2,x2), , (am,xm). We can select any
  of the n elements of B for x1 and then do the
  same for x2 (so that the same element of B may be
  selected for both x1 and x2). We continue this
  selection process until one of the n elements of
  B is finally selected for xm. In this way, using
  the rule of product, there are nm BA
  functions from A to B.

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Definition

• Definition 5.5 A function fA ? B is called
  one-to-one, or injective, if each element of B
  appears at most once as the image of an element
  of A.
• If fA ? B is one-to-one, with A, B finite, we
  must have A ? B.
• For arbitrary sets A, B, fA ? B is one-to-one if
  and only if for all a1,a2 ? A, f(a1) f(a2) ?
  a1a2.

20
Example 5.13 page 226.
21
Example 5.14 page 226.
22
Lemma

• With A a1,a2,,am, B b1,b2,,bn, and m?n,
  a one-to-one function fA ? B has the form
  (a1,x1), (a2,x2), , (am,xm), where there are n
  choices for x1 (that is, any element of B), n 1
  choices for x2 (that is, any element of B except
  the one chosen for x1), n 2 choices for x3, and
  so on, finishing with n (m 1) n m 1
  choices for xm.
• By the rule of product, 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).

23
Definition 5.6

• If fA ? B and A1 ? A, then f(A1)
  b?Bbf(a), for some a?A1, and f(A1) is
  called the image of A1 under f.

24
Example 5.15 page 227.
25
Example 5.16 page 227.
26
Theorem 5.2

• Let fA ? B, with A1, A2 ? A. Thenf(A1 ? A2)
  f(A1) ? f(A2) f(A1 ? A2) ? f(A1) ? f(A2) f(A1
  ? A2) f(A1) ? f(A2) when f is injective.
• Definition 5.7 If fA ? B and A1 ? A, then
  fA1A1 ? B is called the restriction of f to A1
  if fA1(a) f(a) for all a ? A1.
• Definition 5.8 Let A1 ? A and fA1 ? B. If gA ?
  B and g(a) f(a) for all a ? A1, then we call g
  an extension of f to A.

27
Example 5.17 page 228.
28
Example 5.18 page 228
29
5.3 Onto Functions Stirling Numbers of the
Second Kind

• Definition 5.9 A function fA ? 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.

30
Example 5.19 page 231.
31
Example 5.20
32
Example 5.21
33
Example 5.22 page 231232.
34
Example 5.23
35
Lemma

• For finite sets A, B with A m and B n,
  there are
• onto functions from A to B.

36
Example 5.24 page 232.

• ( calculate the distribution of 7 different
  objects into 4 distinct containers with no
  container left empty!)

37
Example 5.25 page 233
38
Lemma

• For m ? n there are ways
  to distribute m distinct objects into n numbered
  (but otherwise identical) containers with no
  container left empty.
• Removing the numbers on the containers, so that
  they are now identical in appearance, we find
  that one distribution into these n (nonempty)
  identical containers corresponds with n! such
  distributions into the numbered containers.

39

• So the number of ways in which it is possible to
  distribute the m distinct objects into n
  identical containers, with no container left
  empty, is
• This will be denoted by S(m, n) and is called a
  Stirling number of the second kind. We note that
  for A m ? n B, there are n!?S(m, n) onto
  functions from A to B.

40
Table 5.1
41
Theorem 5.3

• Let m, n be positive integers with 1 lt n ? m.
  ThenS(m1, n) S(m, n-1) nS(m, n).

42
Example 5.28 page 235.
43
5.4 Special Functions

• Definition 5.10 For any nonempty sets A, B, any
  function fA?A ? B is called a binary operation
  on A. If B?A, then the binary operation is said
  to be closed (on A). (When B?A we may also say
  that A is closed under f.)
• Definition 5.11 A function gA ? A is called a
  uniary, or monary, operation on A.

44
Example 5.29 page 238.
45
Definition 5.12

• Let fA?A ? B that is, f is a binary operation
  on A. f is said to be commutative if f(a, b)
  f(b, a) for all (a, b) ? A?A.
• When B?A (that is, when f is closed), f is said
  to be associative if for all a, b, c ? A, f(f(a,
  b), c) f(a, f(b, c)).

46
Example 5.32 page 239.
47
Example 5.32 page 239
48
Definition 5.13

• Let fA?A ? B be a binary operation on A. An
  element x?A is called an identity (or identity
  element) for f if f(a,x) f(x,a) a, for a?A.

49
Example 5.34 page 240.
50
Theorem 5.4

• Let fA?A ? B be a binary operation. If f has an
  identity, then that identity is unique.

51
Definition 5.14

• For sets A and B, if D ? A?B, then ?AD ? A,
  defined by ?A(a,b) a, is called the projection
  on the first coordinate.
• Then ?BD ? B, defined by ?B(a,b) b, is called
  the projection on the second coordinate.

52
Example 5.36 page 241.
53
Lemma

• Let A1, A2, , An be sets, and with and m?n. If
  D ? A1?A2??An , then the function defined by
  is the projection of D on the i1th, i2th,, imth
  coordinates. The elements of D are called
  (ordered) n-tuples an element in ?(D) is an
  (ordered) m-tuple. (These projections arise in a
  natural way in the study of relational
  databases.)

54
Example 5.38 page 242. (applications to
relational databases)
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5.5 The Pigeonhole Principle????

• In mathematics one sometimes finds that an almost
  obvious idea, when applied in a rather subtle
  manner, is the key needed to solve a troublesome
  problem.
• The Pigeonhole Principle If m pigeons occupy n
  pigeonholes and m gt n, then at least one
  pigeonhole has two or more pigeons roosting in
  it.

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58
Example 5.39 page 245.
59
Example 5.41 page 245.
60
Example 5.43 page 246
61
Example 5.45 page 246.
62
Example 5.47 page 247.
63
Example 5.48 page 248
64
5.6 Function Composition and Inverse Functions

• In this section we study a method for combining
  two functions into a single function. Then we
  develop the concept of the inverse (of a
  function) for functions.
• Definition 5.15 If fA ? B, then f is said to be
  bijective, or to be a one-to-one correspondence,
  if f is both one-to-one and onto.

65
Example 5.50 page 250.
66
Definition

• Definition 5.16 The function IA A ? A, defined
  by IA(a) a for all a ? A, is called the
  identity function for A.
• 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.

67
Example 5.51 page 250.
68
Example 5.52 page 250251.
69
Definition 5.18

• If f A ? B and g B ? C, we define the composite
  function, which is denoted g?f A ? C, by g?f(a)
  g(f(a)), for each a ? A.

70
Example 5.53 page 251.
71
Example 5.54 page 251.

• For any f A ? B, we observe that f ? IA f IB
  ? f.

72
Theorem 5.5

• Let f A ? B and g B ? C. If f, g are
  one-to-one, then g?f is one-to-one. If f, g are
  onto, then g?f is onto.

73
Example 5.55 page 252.
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Theorem 5.6

• If f A ? B, g B ? C, and h C ? D, then (h?g)?f
  h?(g?f).

76
Definition 5.19

• If f A ? A, we define f1 f, and for n ? Z,
  fn1f?(fn).
• Example 5.56 page 253.

77
Definition 5.20

• For sets A, B ? U, if ? is a relation from A to
  B, then the converse of ?, denoted ?c, is the
  relation from B to A defined by ?c (b, a)(a,
  b) ? ?.

78
Example 5.57 page 254.
79
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 IA
  and f?g IB.

80
Example 5.58 page 254.
81
Theorem 5.7

• If a function f A ? B is invertible and a
  function g B ? A satisfies g?f IA and f?g
  IB, then this function g is unique. Note As a
  result of this theorem we shall call the function
  g the inverse of f and shall adopt the notation g
  f-1. Theorem 5.7 also implies that f-1 fc.
  Whenever f is an invertible function, so is the
  function f-1, and (f-1)-1 f.

82
Theorem 5.8

• A function f A ? B is invertible if and only if
  it is one-to-one and onto.

83
Example 5.59 page 255.
84
Theorem 5.9

• if f A ? B, g B ? C are invertible functions,
  then g?f A ? C is invertible and (g?f) -1
  f-1?g-1.

85
Example 5.60 page 255
86
Definition 5.22

• if f A ? B and B1 ? B, then f-1 (B1) x ?
  Af(x) ? B1. The set f-1 (B1) is called the
  preimage of B1 under f.

87
Example 5.62 page 256
88
Example 5.63 page 257.
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90
Example 2.63
91
Example 2.63
92
Example 2.64
93
Theorem 5.10

• If f A ? B and B1, B2 ? B, then f-1(B1 ? B2)
  f-1 (B1) ? f-1 (B2) f-1 (B1 ? B2) f-1 (B1) ?
  f-1 (B2) and

94
Theorem 5.11

• Let f A ? B for finite sets A and B, where A
  B. Then the following statements are
  equivalent (a) f is one-to-one (b) f is onto
  and (c) f is invertible.

95
5.7 Computational Complexity

• For a searching algorithm, how long does it take
  for a success or fail search? To measure this we
  seek a function f(n), called the time-complexity
  function of the algorithm. We expect that the
  value of f(n) will increase as n increases.

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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(n) ? mg(n) for all
  n ? Z, where n?k. Note When f is dominated by
  g we say that f is of order (at most) g and we
  use what is called bigOh notation to designate
  this. We write f ? O(g), where O(g) is read
  order g or bigOh of g. As suggested by the
  notation f ? O(g), O(g) represents the set of
  all functions with domain Z and codomain R that
  are dominated by g.

97
Example 5.65 page 262.
98
Example 5.67 page 263.
99
Example 5.68 page 264.
100
Complexity

• Some of the most important of these orders are
  listed in Table 5.11.

101
5.8 Analysis of Algorithms
102

• In Fig 5.17 we have graphed a log-linear plot for
  the functions associated with some of the orders
  given in Table 5.11.

103

• The data in Table 5.12 provide estimates of the
  running times of algorithms for certain orders of
  complexity.