Discrete Mathematics Chapter 2 Basic Structures Sets Functions

Discrete Mathematics Chapter 2 Basic Structures : Sets, Functions, Sequences, and Sums 大葉大學 資訊 程系 黃鈴玲(Lingling Huang)

2 -1 Sets n n n 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 u. N = { 0, 1, 2, 3, …} , the set of natural number (自然數) u Z = { …, -2, -1, 0, 1, 2, …} , the set of integers (整數) u Z+ = { 1, 2, 3, …} , the set of positive integers (正整數) u Q = { p / q | p ∈ Z , q≠ 0 } , the set of rational numbers (有理數) u R = the set of real numbers (實數) (元素可表示成 1. 234等小數形式) Ch 2 -2

n n n 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 } Ch 2 -3

Note. |A x B| = |A|．|B| n Example 16 : A = {1, 2} , B = {a, b, c} A x B = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)} n n Exercise : 5, 7, 8, 17, 21, 23 Ch 2 -4

2 -2 Set Operations Def 1, 2, 4 : A, B : sets u A∪B = { x | x A or x B } (union) u A∩B = { x | x A and x B } (intersection) u A – B = { x | x A and x B } (也常寫成A B) n Def 3 : Two sets A, B are disjoint if A∩B = n Def 5 : Let U be the universal set. The complement of the set A, denoted by A, is the set U – A. n Example 10 : Prove that A∩B = A∪B n pf : 稱為 Venn Diagram n Ch 2 -5

n Def 6 : A 1 , A 2 , … , An : sets Let I = {1, 3, 5} , n n n 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 Ch 2 -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. n eg. n A B A α 1 α β 2 β γ 3 γ Not a function B 1 2 Not a function Ch 2 -7

A B α β γ A 1 α 2 β B 1 2 3 γ 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) = x 2 , 則 f 的domain, codomain 及range? n Ch 2 -8

n n Example 6 : Let f 1 : R → R and f 2 : R → R s. t. f 1(x) = x 2, f 2(x) = x - x 2, What are the function f 1 + f 2 and f 1 f 2 ? Sol : ( f 1 + f 2 )(x) = f 1(x) + f 2(x) = x 2 + ( x – x 2 ) = x (f 1 f 2)(x) = f 1(x)．f 2(x) = x 2( x – x 2 ) = x 3 – x 4 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 d 3 4 c 5 is 1 -1 a b c d g B 1 2 3 4 5 not 1 -1 , 因 g(a) = g(d) = 4 Ch 2 -9

n n 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 Ch 2 -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|. Ch 2 -11

n n n ※Some important functions Def 12 : Ø floor function : x : ≤ x 的最大整數，即 [ x ] Ø ceiling function : x : ≥ x 的最小整數. Example 24 : ½ = -½ = 7 = Example 29 : Ø factorial function : f : N → Z+ , f (n) = n! = 1 x 2 x … x n Exercise : 1, 12, 17, 19 Ch 2 -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+ a 1 =1, a 2 =1/2 , a 3 =1/3, … Example 2. {bn} , where bn= (-1)n, n N b 0 = 1, b 1 = -1 , b 2 = 1, … Ch 2 -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 : a 1 = 5 a 2 =11 = 5 + 6 a 3 =17 = 11 + 6 = 5 + 6 2 : : an= 5 + 6 (n-1) = 6 n-1 Ch 2 -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 3 n … 比較： {3 n} : 3, 9, 27, 81, 243, 729, 2187, … {an} : 1, 7, 25, 79, 241, 727, 2185, … an = 3 n - 2 , n 1 Ch 2 -15

Summations Here, the variable j is call the index of summation, m is the lower limit, and n is the upper limit. Example 10. Example 13. (Double summation) Ch 2 -16

Example 14. Table 2. Some useful summation formulae Ch 2 -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. Ch 2 -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) = 2 n – 1 is 1 -1 & onto. Ch 2 -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 ∴ Z+ : 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 … Q+ : (注意，因 等於 ，故 不算) ※Note. R is uncountable. (Example 21) Exercise : 9, 13, 17, 42 Ch 2 -20