Download Ch2 Basic Structures : Sets, functions, sequences, and

Discrete
Mathematics
Chapter 2
Basic Structures : Sets, Functions,
Sequences, and Sums
大葉大學 資訊工程系 黃鈴玲(Lingling Huang)
2-1 Sets
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Def 1 : A set is an unordered collection of objects.
Def 2 : The objects in a set are called the elements, or
members of the set.
Example 5 : 常見的重要集合
= { 0,1,2,3,…} , the set of natural number (自然數)
 Z = { …,-2,-1,0,1,2,…} , the set of integers (整數)
 Z+ = { 1,2,3,…} , the set of positive integers (正整數)
 Q = { p / q | p ∈ Z , q ∈ Z , q≠0 } , the set of rational
numbers (有理數)
 R = the set of real numbers (實數)
(元素可表示成1.234等小數形式)
N
Ch2-2
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Def 4 : A ⊆ B iff ∀x , x ∈ A → x ∈ B
補充： A ⊂ B 表示A ⊆ B 但 A ≠ B
Def 5 : S : a finite set
The cardinality of S , denoted by |S|, is the number of
elements in S.
Def 7 : S : a set
The power set of S , denoted by P(S), is the set of all
subsets of S.
Example 13 : S = {0,1,2}
P(S) = {, {0} , {1} , {2} , {0,1} , {0,2} , {1,2} , {0,1,2} }
Def 8 : A , B : sets The Cartesian Product of A and B,
denoted by A x B, is the set
A x B = { (a,b) | a ∈ A and b ∈ B }
Ch2-3
Note. |A x B| = |A|．|B|
 Example 16 :
A = {1,2} , B = {a, b, c}
A x B = {(1,a), (1,b), (1,c), (2,a), (2,b), (2,c)}
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Exercise : 5, 7, 8, 17, 21, 23
Ch2-4
2-2 Set Operations
Def 1,2,4 : A,B : sets
 A∪B = { x | x  A or x  B }
(union)
 A∩B = { x | x  A and x  B }
(intersection)
 A – B = { x | x  A and x  B } (也常寫成A \ B)
 Def 3 : Two sets A,B are disjoint if A∩B = 
 Def 5 : Let U be the universal set.
The complement of the set A, denoted by A, is the set
U – A.
 Example 10 : Prove that A∩B = A∪B
 pf :
稱為 Venn Diagram
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Ch2-5
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Def 6 : A
n 1 , A2 , … , An : sets
 Ai  A1  A2  ...  An
i 1
n
A  A A
i
1
2
 ...  An
i 1
Let I = {1,3,5} ,
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A  A A A
i
1
3
5
iI
Def : (p.131右邊) A,B : sets
The symmetric difference of A and B, denoted by A⊕B,
is the set
{ x | x  A - B or x  B - A } = ( A∪B ) - ( A ∩B )
※Inclusion – Exclusion Principle (排容原理)
|A ∪ B| = |A| + |B| - |A ∩ B|
Exercise : 14, 45
Ch2-6
2-3 Functions
Def 1 : A,B : sets
A function f : A → B is an assignment of exactly
one element of B to each element of A.
We write f(a) = b if b is the unique element of B
assigned by f to a ∈ A.
 eg.
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A
B
A
α
1
α
β
2
β
γ
3
γ
Not a function
B
1
2
Not a function
Ch2-7
A
B
α
A
1
β
2
γ
B
1
α
2
β
3
γ
a function
a function
4
Def : (以 f : A→B 為例，右上圖)
f (α) = 1, f (β) = 4, f (γ) = 2
1 稱為α的image (unique) , α稱為1的pre-image(not unique)
A : domain of f , B : codomain of f
range of f = {f (a) | a ∈ A} = f (A) = {1,2,4} (未必=B)
Example 4 : f : Z → Z , f (x) = x2 , 則 f 的domain, codomain
及range?
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Ch2-8
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Example 6 : Let f1 : R → R and f2 : R → R s.t.
f1(x) = x2, f2(x) = x - x2, What are the function f1 + f2 and
f1 f2 ?
Sol :
( f1 + f2 )(x) = f1(x) + f2(x) = x2 + ( x – x2 ) = x
(f1 f2)(x) = f1(x)．f2(x) = x2( x – x2 ) = x3 – x4
Def 5: A function f is said to be one-to-one, or injective,
iff f (x) ≠ f (y) whenever x ≠ y.
Example 8 :
A
f
B
A
1
2
a
b
a
b
d
3
4
c
c
5
d
is 1-1
g
B
1
2
3
4
5
not 1-1 , 因 g(a) = g(d) = 4
Ch2-9
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Example 10 : Determine whether the function f (x) = x + 1
is one-to-one ?
Sol : x ≠ y  x + 1 ≠ y + 1
 f (x) ≠ f (y)
∴ f is 1-1
Def 7 : A function f : A → B is called onto, or surjective, iff
for every element b ∈ B , ∃a ∈ A with f (a) = b. (即 B 的所有
元素都被 f 對應到)
Example 11 :
f
a
b
c
d
onto
A
1
a
2
b
3
c
f
B
1
2
3
4
Note :
當|A| < |B| 時，
必定不會onto.
not onto
Ch2-10
Def 8 : The function f is a one-to-one correspondence, or a
bijection, if it is both 1-1 and onto.
Examples in Fig 5
a
b
c
1
2
a
b
3
c
d
4
1-1 , not onto
1
2
3
not 1-1 , onto
a
1
2
3
4
b
c
d
1-1 and onto
※補充 : f : A →B
(1) If f is 1-1 , then |A| ≤ |B|
(2) If f is onto , then |A| ≥ |B|
(3) if f is 1-1 and onto , then |A| = |B|.
Ch2-11
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※Some important functions
Def 12 :
 floor function : x : ≤ x 的最大整數，即 [ x ]
 ceiling function : x : ≥ x 的最小整數.
Example 24 :
½ =
-½ =
7 =
½ =
-½ =
7 =
Example 29 :
 factorial function :
f : N → Z+ , f (n) = n! = 1 x 2 x … x n
Exercise : 1,12,17,19
Ch2-12
2.4 Sequences and Summations
※Sequence (數列)
Def 1. A sequence is a function f from A  Z+
(or A  N) to a set S. We use an to denote f(n),
and call an a term (項) of the sequence.
Example 1. {an} , where an = 1/n , n  Z+
 a1 =1, a2 =1/2 , a3 =1/3, …
Example 2. {bn} , where bn= (-1)n, n  N
 b0 = 1, b1 = -1 , b2 = 1, …
Ch2-13
Example 7. How can we produce the terms of a sequence
if the first 10 terms are
5, 11, 17, 23, 29, 35,41, 47, 53, 59?
Sol :
a1 = 5
a2 =11 = 5 + 6
a3 =17 = 11 + 6 = 5 + 6  2
:
:
 an= 5 + 6  (n-1) = 6n-1
Ch2-14
Example 8. Conjecture a simple formula for an if
the first 10 terms of the sequence {an} are
1, 7, 25, 79, 241, 727, 2185, 6559, 19681, 59047?
Sol：
顯然非等差數列
後項除以前項的值接近3
 猜測數列為 3n  …
比較：
{3n} : 3, 9, 27, 81, 243, 729, 2187,…
{an} : 1, 7, 25, 79, 241, 727, 2185,…
 an = 3n - 2 , n  1
Ch2-15
 Summations
n
a
j m
j
 am  am1    an
Here, the variable j is call the index of summation, m is the
lower limit, and n is the upper limit.
Example 10.
5
2
j
  1  4  9  16  25  55
j 1
Example 13. (Double summation)
4
3
4
4
i 1
i 1
 ij   (i  2i  3i)   6i 6(1  2  3  4)  60
i 1 j 1
Ch2-16
S  024  6
Example 14.
S{0 , 2 , 4}
Table 2. Some useful summation formulae
n 1
a
(
r
- 1)
k
(1)  ar 
, r 1
r -1
k 0
n
n(n  1)
(2)  k 
2
k 1
n
n(n  1)( 2n  1)
(3)  k 
6
k 1
n
2
Ch2-17
Cardinality
Def 4. The sets A and B have the same cardinality (size) if
and only if there is a one-to-one correspondence
(1-1 and onto function) from A to B.
Def 5. A set that is either finite or has the same
cardinality as Z+ (or N) is called countable (可數).
A set that is not countable is called uncountable.
Ch2-18
Example 18. Show that the set of odd positive
integers is a countable set.
Pf: (Figure 1)
Z+ :
1
2
3
4
5
6
{ 正奇數 } :
1
3
5
7
9
11
7
8 …
……
13 15 …
f : Z+  {all positive integers}
f (n) = 2n – 1
is 1-1 & onto.
Ch2-19
Example 19. Show that the set of positive rational number (Q+) is
countable.
Pf: Q+ = { a / b | a, b Z+ }
(Figure 2)
1
1
1
2
1
3
1
4
2
1
3
1
2
2
2
3
2
4
3
2
3
3
4
1
4
2
4
3
5
1

5
2
5
3



∴ Z+ : 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 …
Q+ :
1 1 2 3 1 1 2 3 4
1

5

1
2
1
2
,
2
,
1
,
1
,
3
(注意，因 2 等於 1 ，故 2 不算)
※Note. R is uncountable. (Example 21)
,
4
,
3
,
2
,
1
...
Exercise :
9,13,17,42
Ch2-20