2. 강원대학교 컴퓨터과학전공 문양세 이산수학 (Discrete Mathematics) 수열과 합 (Sequences and Summations)

3. Discrete Mathematics by Yang-Sae Moon Page 2 Introduction 3.2 Sequences and Summations A sequence or series is just like an ordered n-tuple (a 1, a 2, …, a n ), except: Each element in the sequences has an associated index number. ( 각 element 는 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. (  )

4. Discrete Mathematics by Yang-Sae Moon Page 3 Sequences 3.2 Sequences and Summations Formally: A sequence {a n } 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 {a n } 은 은 자연수 집합으로부터 A 로의 함수 …) If f is a generating function for a sequence {a n }, then for n  S, the symbol a n denotes f(n). The index of a n is n. (Or, often i is used.) SA 12341234 a 1 = f(1) a 2 = f(2) a 3 = f(3) a 4 = f(4)  f

5. Discrete Mathematics by Yang-Sae Moon Page 4 Sequence Examples 3.2 Sequences and Summations Example of an infinite series ( 무한 수열 ) Consider the series {a n } = a 1, a 2, …, where (  n  1) a n = f(n) = 1/n. Then, {a n } = 1, 1/2, 1/3, 1/4, … Example with repetitions ( 반복 수열 ) Consider the sequence {b n } = b 0, b 1, … (note 0 is an index) where b n = (  1) n. {b n } = 1,  1, 1,  1, … Note repetitions! {b n } denotes an infinite sequence of 1’s and  1’s, not the 2-element set {1,  1}.

6. Discrete Mathematics by Yang-Sae Moon Page 5 Recognizing Sequences (1/2) 3.2 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) = 2n − 1

7. Discrete Mathematics by Yang-Sae Moon Page 6 Recognizing Sequences (2/2) 3.2 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.)

8. Discrete Mathematics by Yang-Sae Moon Page 7 What are Strings? (1/2) - skip 3.2 Sequences and Summations Strings are often restricted to sequences composed of symbols drawn from a finite alphabet, and may be indexed from 0 or 1. ( 스트링은 유한한 알파벳으로 구성된 심볼의 시퀀스이고, 0(or 1) 부터 색인될 수 있다.) More formally, Let  be a finite set of symbols, i.e. an alphabet. A string s over alphabet  is any sequence {s i } of symbols, s i , indexed by N or N  {0}. If a, b, c, … are symbols, the string s = a, b, c, … can also be written abc …(i.e., without commas). If s is a finite string and t is a string, the concatenation of s with t, written st, is the string consisting of the symbols in s followed by the symbols in t.

9. Discrete Mathematics by Yang-Sae Moon Page 8 What are Strings? (2/2) - skip 3.2 Sequences and Summations More string notation The length |s| of a finite string s is its number of positions (i.e., its number of index values i). If s is a finite string and n  N, s n denotes the concatenation of n copies of s. ( 스트링 s 를 n 번 concatenation 하는 표현 )  denotes the empty string, the string of length 0. If  is an alphabet and n  N, −  n  {s | s is a string over  of length n} ( 길이 n 인 스트링 ) −  *  {s | s is a finite string over  } (  상에서 구현 가능한 유한 스트링 )

10. Discrete Mathematics by Yang-Sae Moon Page 9 Summation Notation 3.2 Sequences and Summations Given a sequence {a n }, an integer lower bound j  0, and an integer upper bound k  j, then the summation of {a n } from j to k is written and defined as follows: ({a n } 의 i 번째에서 j 번째까지의 합, 즉, a j 로부터 a k 까지의 합 ) Here, i is called the index of summation.

11. Discrete Mathematics by Yang-Sae Moon Page 10 Generalized Summations 3.2 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 X 의 모든 원소 x 에 대해서 ) Or, if X={x|P(x)}, we may just write: (P(x) 를 true 로 하는 모든 x 에 대해서 )

12. Discrete Mathematics by Yang-Sae Moon Page 11 Summation Examples 3.2 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:

13. Discrete Mathematics by Yang-Sae Moon Page 12 Summation Manipulations (1/2) 3.2 Sequences and Summations Some useful identities for summations: (Distributive law) (Application of commutativity) (Index shifting)

14. Discrete Mathematics by Yang-Sae Moon Page 13 Summation Manipulations (2/2) 3.2 Sequences and Summations Some more useful identities for summations: (Grouping) (Order reversal) (Series splitting)

15. Discrete Mathematics by Yang-Sae Moon Page 14 An Interesting Example 3.2 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.” (1 에서 n 까지의 합을 수초 내에 계산하겠다 !) I.e., Evaluate the summation: There is a simple formula for the result, discovered by Euler at age 12!

16. Discrete Mathematics by Yang-Sae Moon Page 15 Euler’s Trick, Illustrated 3.2 Sequences and Summations Consider the sum: 1 + 2 + … + (n/2) + ((n/2)+1) + … + (n-1) + n n/2 pairs of elements, each pair summing to n+1, for a total of (n/2)(n+1). n/2 pairs of elements, each pair summing to n+1, for a total of (n/2)(n+1). ( 합이 n+1 인 두 쌍의 element 가 n/2 개 있다.) … n+1

17. Discrete Mathematics by Yang-Sae Moon Page 16 Symbolic Derivation of Trick (1/2) - skip 3.2 Sequences and Summations

18. Discrete Mathematics by Yang-Sae Moon Page 17 Symbolic Derivation of Trick (2/2) - skip 3.2 Sequences and Summations So, you only have to do 1 easy multiplication in your head, then cut in half. Also works for odd n (prove it by yourself).

19. Discrete Mathematics by Yang-Sae Moon Page 18 Geometric Progression ( 등비수열 ) 3.2 Sequences and Summations A geometric progression is a series of the form a, ar, ar 2, ar 3, …, ar k, 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...

20. Discrete Mathematics by Yang-Sae Moon Page 19 Derivation of Geometric Sum (1/3) - skip 3.2 Sequences and Summations

21. Discrete Mathematics by Yang-Sae Moon Page 20 Derivation of Geometric Sum (2/3) - skip 3.2 Sequences and Summations

22. Discrete Mathematics by Yang-Sae Moon Page 21 Derivation of Geometric Sum (3/3) - skip 3.2 Sequences and Summations

23. Discrete Mathematics by Yang-Sae Moon Page 22 Nested Summations 3.2 Sequences and Summations These have the meaning you’d expect.

24. Discrete Mathematics by Yang-Sae Moon Page 23 Some Shortcut Expressions 3.2 Sequences and Summations SumClosed Form Infinite series ( 무한급수 )

25. Discrete Mathematics by Yang-Sae Moon Page 24 Infinite Series ( 무한급수 ) (1/2) - skip 3.2 Sequences and Summations Let a = 1 and r = x, then If k  , then x k+1  0 Therefore,

26. Discrete Mathematics by Yang-Sae Moon Page 25 Infinite Series ( 무한급수 ) (2/2) - skip 3.2 Sequences and Summations

27. Discrete Mathematics by Yang-Sae Moon Page 26 Using the Shortcuts 3.2 Sequences and Summations Example: Evaluate. Use series splitting. Solve for desired summation. Apply quadratic series rule. Evaluate.

28. Discrete Mathematics by Yang-Sae Moon Page 27 Cardinality: Formal Definition 3.2 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 A 에서 집합 B 로의 전단사함수가 존재하면, A 와 B 의 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, B 가 유한집합이고 동일한 개수의 원소를 가지면, A 와 B 가 동일한 크기 임을 보이는 것은 간단하다.)

29. Discrete Mathematics by Yang-Sae Moon Page 28 Countable versus Uncountable 3.2 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 countable 하며, 그렇지 않으 면 uncountable 하다.) Intuition behind “countable:” we can enumerate (sequentially list) elements of S. Examples: N, Z. ( 집합 S 의 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 S 의 모든 원소를 포함할 수 없다. 즉, 집합 S 의 원소에 번호를 매길 수 있는 방법이 없다.)

30. Discrete Mathematics by Yang-Sae Moon Page 29 Countable Sets: Examples – skip 3.2 Sequences and Summations Theorem: The set Z is countable. Proof: Consider f:Z  N where f(i)=2i for i  0 and f(i) =  2i  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) (1,2) (1,3) (1,4) (1,5) (2,1) (2,2) (2,3) (2,4) (2,5) (3,1) (3,2) (3,3) (3,4) (3,5) (4,1) (4,2) (4,3) (4,4) (4,5) (5,1) (5,2) (5,3) (5,4) (5,5) …………… … … … … … … consider sum is 2, then consider sum is 3, then consider sum is 4, then consider sum is 5, then consider sum is 6, then consider … Note a set of rational numbers is countable!

31. Discrete Mathematics by Yang-Sae Moon Page 30 Uncountable Sets: Example (1/2) – skip 3.2 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 {r i } = r 1, r 2,... containing all elements r  [0,1). Consider listing the elements of {r i } 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 d i = 4 if d ii  4 and d i = 5 if d ii = 4.

32. Discrete Mathematics by Yang-Sae Moon Page 31 Uncountable Sets: Example (2/2) – skip 3.2 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 4 1 9 3 … r 4 = 0.9 1 8 2 3 7 4 6 1 … … OK, now let’s make r’ by replacing d ii by the rule. (Rule: r’ = 0.d 1 d 2 d 3 d 4 … where d i = 4 if d ii  4 and d i = 5 if d ii = 4) r’ = 0.4454… can’t be on the list anywhere! This means that the assumption({r i } is countable) is wrong, and thus, [0,1), {r i }, is uncountable.