Title: Chapter 5 Relations and Functions
1Chapter 5 Relations and Functions
- Yen-Liang Chen
- Dept of Information Management
- National Central University
25.1. 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?
3Tree diagrams for the Cartesian product
4Relations
- 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.
5Examples
- 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 ?.
6Theorem 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?
75.2. 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
8properties
- 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?)
9Number 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.
10Theorem 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
11Restriction 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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135.3. 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.
14The 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.
15The number of onto functions
- For finite sets A and B with ?A?m and ?B?n, the
number of onto functions is
16Examples
- 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
17Distinguishable 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
18Stirling 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.
20Theorem 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.
21Ex 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
225.4. 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.
23Commutative 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))
24Commutative 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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26Identity
- 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.
27Identity
- 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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29Projection
- 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.
30The projection of a database
315.5. 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.
32Examples
- 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.
33Examples
- 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?.
34Ex 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.
35Ex 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.
365.6. 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.
37Equal 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.
38Composite 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.
39Is 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.
40Definitions
- 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
41Invertible 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.
42Invertible 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.
43Preimage
- 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.
44Ex 5.64
- Table 5.9 for fZ?R with f(x)x25
- Table 5.10 for gR?R with g(x) x25
45Theorems
- 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)
46Theorem 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.
475.7. 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.
48Order
- 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.
49Examples
- 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.
50some order functions that are commonly seen
515.8. Analysis of algorithms
- Ex 5.69. The time complexity is f(n)7n5?O(n)
52Ex 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.
53Ex 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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55The growth of complexities