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Title: Chapter 5 Relations and Functions


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. 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?

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
    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.

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)?n7m
  • 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)? a1a2.
  • Ex 5.13. f(x)3x7 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. 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.

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.
  • A12,3,4, A23,4,5
  • 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) 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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13
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.

14
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.

15
The number of onto functions
  • For finite sets A and B with ?A?m and ?B?n, the
    number of onto functions is

16
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

17
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

18
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

19
  • It is the number of possible ways to distribute m
    distinct objects into n identical containers with
    empty containers allowed.

20
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.

21
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

22
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.

23
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))

24
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.

27
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.

30
The projection of a database
31
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.

32
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.

33
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?.

34
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.

35
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.

36
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.

37
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.

38
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.

39
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.

40
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

41
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.

42
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.

43
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.

44
Ex 5.64
  • Table 5.9 for fZ?R with f(x)x25
  • Table 5.10 for gR?R with g(x) x25

45
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)

46
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.

47
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.

48
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)

52
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.

53
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
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