Discrete Mathematics Sequences and Summations 2015 Introduction Sequences
- Slides: 22
이산수학(Discrete Mathematics) 수열과 합 (Sequences and Summations) 2015년 봄학기 강원대학교 컴퓨터과학전공 문양세
Introduction Sequences and Summations A sequence or series is just like an ordered n-tuple (a 1, a 2, …, an), except: • Each element in the sequences has an associated index number. (각 element는 색인(index) 번호와 결합되는 특성을 가진다. ) • A sequence or series may be infinite. (무한할 수 있다. ) • Example: 1, 1/2, 1/3, 1/4, … A summation is a compact notation for the sum of all terms in a (possibly infinite) series. ( ) Page 2 Discrete Mathematics by Yang-Sae Moon
Sequences and Summations Formally: A sequence {an} is identified with a generating function f: S A for some subset S N (S=N or S=N {0}) and for some set A. (수열 {an}은 자연수 집합으로부터 A로의 함수…) If f is a generating function for a sequence {an}, then for n S, the symbol an denotes f(n). The index of an is n. (Or, often i is used. ) S f A a 1 = f(1) a 2 = f(2) a 3 = f(3) a 4 = f(4) 1 2 3 4 Page 3 Discrete Mathematics by Yang-Sae Moon
Sequence Examples Sequences and Summations Example of an infinite series (무한 수열) • Consider the series {an} = a 1, a 2, …, where ( n 1) an= f(n) = 1/n. • Then, {an} = 1, 1/2, 1/3, 1/4, … Example with repetitions (반복 수열) • Consider the sequence {bn} = b 0, b 1, … (note 0 is an index) where bn = ( 1)n. • {bn} = 1, 1, … • Note repetitions! {bn} denotes an infinite sequence of 1’s and 1’s, not the 2 -element set {1, 1}. Page 4 Discrete Mathematics by Yang-Sae Moon
Recognizing Sequences (1/2) Sequences and Summations Sometimes, you’re given the first few terms of a sequence, and you are asked to find the sequence’s generating function, or a procedure to enumerate the sequence. (순열의 몇몇 값들에 기반하여 f(n)을 발견하는 문제에 자주 직면하게 된다. ) Examples: What’s the next number and f(n)? • 1, 2, 3, 4, … (the next number is 5. f(n) = n • 1, 3, 5, 7, … (the next number is 9. f(n) = 2 n − 1 Page 5 Discrete Mathematics by Yang-Sae Moon
Recognizing Sequences (2/2) Sequences and Summations Trouble with recognition (of generating functions) • The problem of finding “the” generating function given just an initial subsequence is not well defined. (잘 정의된 방법이 없음) • This is because there are infinitely many computable functions that will generate any given initial subsequence. (세상에는 시퀀스를 생성하는 셀 수 없이 많은 함수가 존재한다. ) Page 6 Discrete Mathematics by Yang-Sae Moon
Summation Notation Sequences and Summations Given a sequence {an}, an integer lower bound j 0, and an integer upper bound k j, then the summation of {an} from j to k is written and defined as follows: ({an}의 j번째에서 k번째까지의 합, 즉, aj로부터 ak까지의 합) Here, i is called the index of summation. Page 7 Discrete Mathematics by Yang-Sae Moon
Generalized Summations Sequences and Summations For an infinite series, we may write: To sum a function over all members of a set X={x 1, x 2, …}: (집합 X의 모든 원소 x에 대해서) Or, if X={x|P(x)}, we may just write: (P(x)를 true로 하는 모든 x에 대해서) Page 8 Discrete Mathematics by Yang-Sae Moon
Summation Examples Sequences and Summations A simple example An infinite sequence with a finite sum: Using a predicate to define a set of elements to sum over: Page 9 Discrete Mathematics by Yang-Sae Moon
Summation Manipulations (1/2) Sequences and Summations Some useful identities for summations: (Distributive law) (Application of commutativity) (Index shifting) Page 10 Discrete Mathematics by Yang-Sae Moon
Summation Manipulations (2/2) Sequences and Summations Some more useful identities for summations: (Series splitting) (Order reversal) (Grouping) Page 11 Discrete Mathematics by Yang-Sae Moon
An Interesting Example Sequences and Summations “I’m so smart; give me any 2 -digit number n, and I’ll add all the numbers from 1 to n in my head in just a few seconds. ” (1에서 n까지의 합을 수초 내에 계산하겠다!) I. e. , Evaluate the summation: There is a simple formula for the result, discovered by Euler at age 12! Page 12 Discrete Mathematics by Yang-Sae Moon
Euler’s Trick, Illustrated Sequences and Summations Consider the sum: 1 + 2 + … + (n/2) + ((n/2)+1) + … + (n-1) + n n+1 … n+1 n/2 pairs of elements, each pair summing to n+1, for a total of (n/2)(n+1). (합이 n+1인 두 쌍의 element가 n/2개 있다. ) Page 13 Discrete Mathematics by Yang-Sae Moon
Geometric Progression (등비수열) Sequences and Summations A geometric progression is a series of the form a, ar 2, ar 3, …, ark, where a, r R. The sum of such a sequence is given by: We can reduce this to closed form via clever manipulation of summations. . . Page 14 Discrete Mathematics by Yang-Sae Moon
Nested Summations Sequences and Summations These have the meaning you’d expect. Page 15 Discrete Mathematics by Yang-Sae Moon
Some Shortcut Expressions Sum Sequences and Summations Closed Form Infinite series (무한급수) Page 16 Discrete Mathematics by Yang-Sae Moon
Using the Shortcuts Example: Evaluate Sequences and Summations . • Use series splitting. • Solve for desired summation. • Apply quadratic series rule. • Evaluate. Page 17 Discrete Mathematics by Yang-Sae Moon
Cardinality: Formal Definition Sequences and Summations For any two (possibly infinite) sets A and B, we say that A and B have the same cardinality (written |A|=|B|) iff there exists a bijection (bijective function) from A to B. (집합 A에서 집합 B로의 전단사함수가 존재하면, A와 B의 크기는 동일하다. ) When A and B are finite, it is easy to see that such a function exists iff A and B have the same number of elements n N. (집합 A, B가 유한집합이고 동일한 개수의 원소를 가지면, A와 B가 동일한 크기 임을 보이는 것은 간단하다. ) Page 18 Discrete Mathematics by Yang-Sae Moon
Countable versus Uncountable Sequences and Summations For any set S, if S is finite or if |S|=|N|, we say S is countable. Else, S is uncountable. (유한집합이거나, 자연수 집합과 크기가 동일하면 countable하며, 그렇지 않으 면 uncountable하다. ) Intuition behind “countable: ” we can enumerate (sequentially list) elements of S. Examples: N, Z. (집합 S의 원소에 번호를 매길 수(순차적으로 나열할 수) 있다. ) Uncountable means: No series of elements of S (even an infinite series) can include all of S’s elements. Examples: R, R 2 (어떠한 나열 방법도 집합 S의 모든 원소를 포함할 수 없다. 즉, 집합 S의 원소에 번호를 매길 수 있는 방법이 없다. ) Page 19 Discrete Mathematics by Yang-Sae Moon
Countable Sets: Examples Sequences and Summations Theorem: The set Z is countable. • Proof: Consider f: Z N where f(i)=2 i for i 0 and f(i) = 2 i 1 for i<0. Note f is bijective. (…, f( 2)=3, f( 1)=1, f(0)=0, f(1)=2, f(2)=4, …) Theorem: The set of all ordered pairs of natural numbers (n, m) is countable. (1, 1) (2, 1) (3, 1) (4, 1) (5, 1) … (1, 2) (2, 2) (3, 2) (4, 2) (5, 2) … (1, 3) (2, 3) (3, 3) (4, 3) (5, 3) … (1, 4) (2, 4) (3, 4) (4, 4) (5, 4) … (1, 5) (2, 5) (3, 5) (4, 5) (5, 5) … … … … Page 20 consider consider sum sum sum … is is is 2, 3, 4, 5, 6, then then Note a set of rational numbers is countable! Discrete Mathematics by Yang-Sae Moon
Uncountable Sets: Example (1/2) - skip Sequences and Summations Theorem: The open interval [0, 1) : {r R| 0 r < 1} is uncountable. ([0, 1)의 실수는 uncountable) Proof by Cantor • Assume there is a series {ri} = r 1, r 2, . . . containing all elements r [0, 1). • Consider listing the elements of {ri} in decimal notation in order of increasing index: r 1 = 0. d 1, 1 d 1, 2 d 1, 3 d 1, 4 d 1, 5 d 1, 6 d 1, 7 d 1, 8… r 2 = 0. d 2, 1 d 2, 2 d 2, 3 d 2, 4 d 2, 5 d 2, 6 d 2, 7 d 2, 8… r 3 = 0. d 3, 1 d 3, 2 d 3, 3 d 3, 4 d 3, 5 d 3, 6 d 3, 7 d 3, 8… r 4 = 0. d 4, 1 d 4, 2 d 4, 3 d 4, 4 d 4, 5 d 4, 6 d 4, 7 d 4, 8… … • Now, consider r’ = 0. d 1 d 2 d 3 d 4 … where di = 4 if dii 4 and di = 5 if dii = 4. Page 21 Discrete Mathematics by Yang-Sae Moon
Uncountable Sets: Example (2/2) - skip • Sequences and Summations E. g. , a postulated enumeration of the reals: r 1 = 0. 3 0 1 9 4 8 5 7 1 … r 2 = 0. 1 0 3 9 1 8 4 8 1 … r 3 = 0. 0 3 4 1 9 3 … r 4 = 0. 9 1 8 2 3 7 4 6 1 … … • OK, now let’s make r’ by replacing dii by the rule. (Rule: r’ = 0. d 1 d 2 d 3 d 4 … where di = 4 if dii 4 and di = 5 if dii = 4) • r’ = 0. 4454… can’t be on the list anywhere! (왜냐면, 4가 아니면 4로, 4이면 5로 바꾸었기 때문에) • This means that the assumption({ri} is countable) is wrong, and thus, [0, 1), {ri}, is uncountable. Page 22 Discrete Mathematics by Yang-Sae Moon
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