이산수학(Discrete Mathematics) 수열과 합 (Sequences and Summations) 2013년 봄학기 강원대학교 컴퓨터과학전공 문양세

Introduction 3.2 Sequences and Summations A sequence or series is just like an ordered n-tuple (a1, a2, …, 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. ()

The index of an is n. (Or, often i is used.) Sequences 3.2 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 1 2 3 4  a1 = f(1) a2 = f(2) a3 = f(3) a4 = f(4) 

Example of an infinite series (무한 수열) Sequence Examples 3.2 Sequences and Summations Example of an infinite series (무한 수열) Consider the series {an} = a1, a2, …, 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} = b0, b1, … (note 0 is an index) where bn = (1)n. {bn} = 1, 1, 1, 1, … Note repetitions! {bn} denotes an infinite sequence of 1’s and 1’s, not the 2-element set {1, 1}.

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

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.)

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 {si} of symbols, si, 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.

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, sn denotes the concatenation of n copies of s. (스트링 s를 n번 concatenation하는 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 n인 스트링) *  {s | s is a finite string over } (상에서 구현 가능한 유한 스트링)

Here, i is called the index of summation. Summation Notation 3.2 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.

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={x1, x2, …}: (집합 X의 모든 원소 x에 대해서) Or, if X={x|P(x)}, we may just write: (P(x)를 true로 하는 모든 x에 대해서)

An infinite sequence with a finite sum: 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:

Summation Manipulations (1/2) 3.2 Sequences and Summations Some useful identities for summations: (Distributive law) (Application of commutativity) (Index shifting)

Summation Manipulations (2/2) 3.2 Sequences and Summations Some more useful identities for summations: (Series splitting) (Order reversal) (Grouping)

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.” (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!

… Euler’s Trick, Illustrated Consider the sum: 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). (합이 n+1인 두 쌍의 element가 n/2개 있다.) n+1 … n+1 n+1

Symbolic Derivation of Trick (1/2) - skip 3.2 Sequences and Summations

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).

Geometric Progression (등비수열) 3.2 Sequences and Summations A geometric progression is a series of the form a, ar, ar2, ar3, …, 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...

Derivation of Geometric Sum (1/3) - skip 3.2 Sequences and Summations

Derivation of Geometric Sum (2/3) - skip 3.2 Sequences and Summations

Derivation of Geometric Sum (3/3) - skip 3.2 Sequences and Summations

These have the meaning you’d expect. Nested Summations 3.2 Sequences and Summations These have the meaning you’d expect.

Some Shortcut Expressions 3.2 Sequences and Summations Sum Closed Form Infinite series (무한급수)

Infinite Series (무한급수) (1/2) - skip 3.2 Sequences and Summations Let a = 1 and r = x, then If k  , then xk+1  0 Therefore,

Infinite Series (무한급수) (2/2) - skip 3.2 Sequences and Summations

Using the Shortcuts Example: Evaluate . Use series splitting. 3.2 Sequences and Summations Example: Evaluate . Use series splitting. Solve for desired summation. Apply quadratic series rule. Evaluate.

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 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 크기임을 보이는 것은 간단하다.)

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 countable하며, 그렇지 않으면 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, R2 (어떠한 나열 방법도 집합 S의 모든 원소를 포함할 수 없다. 즉, 집합 S의 원소에 번호를 매길 수 있는 방법이 없다.)

Countable Sets: Examples 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. 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 … (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) … Note a set of rational numbers is countable! (1,5) (2,5) (3,5) (4,5) (5,5) … … … … … … …

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 {ri} = r1, r2, ... containing all elements r[0,1). Consider listing the elements of {ri} in decimal notation in order of increasing index: r1 = 0.d1,1 d1,2 d1,3 d1,4 d1,5 d1,6 d1,7 d1,8… r2 = 0.d2,1 d2,2 d2,3 d2,4 d2,5 d2,6 d2,7 d2,8… r3 = 0.d3,1 d3,2 d3,3 d3,4 d3,5 d3,6 d3,7 d3,8… r4 = 0.d4,1 d4,2 d4,3 d4,4 d4,5 d4,6 d4,7 d4,8… … Now, consider r’ = 0.d1 d2 d3 d4 … where di = 4 if dii  4 and di = 5 if dii = 4.

Uncountable Sets: Example (2/2) - skip 3.2 Sequences and Summations E.g., a postulated enumeration of the reals: r1 = 0.3 0 1 9 4 8 5 7 1 … r2 = 0.1 0 3 9 1 8 4 8 1 … r3 = 0.0 3 4 1 9 4 1 9 3 … r4 = 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.d1 d2 d3 d4 … 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.