CSNB 143 Discrete Mathematical Structures

CSNB 143 Discrete Mathematical Structures Chapter 3 – Sequence and String

OBJECTIVES • Students should be able to differentiate few characteristics of sequence. • Students should be able to use sequence and strings. • Students should be able to concatenate string and know how to use them.

What, which, where, when Knowledge about sequence • Finite (Clear / Not Clear ) • Infinite (Clear / Not Clear ) • Recursive (Clear / Not Clear ) • Explicit (Clear / Not Clear ) • Increasing (Clear / Not Clear ) • Decreasing (Clear / Not Clear )

String (Clear / Not Clear ) • Concatenation (Clear / Not Clear ) • Subsequence (Clear / Not Clear )

Sequence • A list of objects in its order. That is, taking order as an important thing. • A list in which the first one should be in front, followed by the second element, third element and so on. • List might be ended after n, n N and it is named as Finite Sequence. We called n as an index for that sequence. • List might have no ending value, and this is called as Infinite Sequence. • Elements might be redundancy.

Ex 1: • S = 2, 4, 6, …, 2n • S = S1, S2, S3, … Sn where S1=2, S2= 4, S3=6, … Sn = 2n Ex 2: • T = a, a, b, a, b where T1=a, T2=a, T3=b, T4=a, T5=b

If the sequence is depending on the previous value, it is called Recursive Sequence. • If the sequence is not depending on the previous value, in which the value can be directly retrieved, it is called Explicit Sequence.

Ex 3: An = An-1 + 5; A1 = 1, 2 n < , this is a recursive sequence where: A2 = A1 + 5 A3 = A2 + 5 Ex 4: An = n2 + 1; 1 n < , this is an explicit sequence where: A1 = 1 + 1 = 2 A2 = 4 + 1 = 5 A3 = 9 + 1 = 10 • That is, we can get the value directly, without any dependency to previous value.

Both recursive and explicit formula can have both finite and infinite sequence. • Ex 5: Consider all the sequences below, and identify which sequence is recursive/explicit and finite/infinite. • C1 = 5, Cn = 2Cn-1, 2 n 6 • D1 = 3, Dn = Dn-1 + 4 • Sn = (-4)n, 1 n • Tn = 92 – 5n, 1 n 5

Both sequences also can have an Increasing or Decreasing sequence. • A sequence is said to be increased if for each Sn, the value is less than Sn + 1 for all n, Sn Sn + 1 ; all n • A sequence is said to be decreased if for each Sn the value is bigger than Sn + 1 for all n, Sn Sn + 1 ; all n

Ex 6: Determine either this sequence in increasing or decreasing. • Sn = 2(n + 1), n 1 • Xn = (½)n, n 1 • S = 3, 5, 5, 7, 8, 8, 13

String • Sequences or letters or other symbols that is written without commas are also referred as strings. • An infinite string such as abababa… may be regarded as infinite sequence of a,b,a,b,a,b,a… • The set corresponding to sequence is simply the set of all distinct elements in the sequence. • E.g 1: 1,4,8,9,2… is {1,4,8,9,2…} • E.g 2 : a,b,a,b,a,b,a… is simply {a, b}

A string over A set is a finite sequence of elements from A. • Let A = {a, b, c}. If we let A1 = b, A2 = a, A3 = a, A4 = c Then we obtain a string over A. The string is written baac. • Since a string is a sequence, order is taken into account. For example the string baac is different from acab. • Repetition in a string can be specified by superscript. For example the string bbaaac may be written b2a3c.

The string with no element is call the null string and is denoted as . We let set A* denote the set of all strings over A, including the null string. Ex 10: • Let say A = {a, b, c, …, z} • Then A* = {aaaa, computer, denda, pqr, sysrq,… } • Or let X = {a, b }. Some elements of X* are: • a, b, baba, , b2a29ba

That is, all finite sequence that can be build from A, contains all words either it has any meaning or not, regardless its length. • The number of elements in any string A is called Elements’ Length, denoted as |A|. Ex 11: • If A = abcde…z, then |A| = 26.

Concatenation • If W1 and W2 are two strings, the string consisting of W1 followed by W2 written W1. W2 is called concatenation of W1 and W2 : W1.W2 =A1A2A3…AnB1B2B3…Bm where W1.W2 And it is known that W1. = W1 and .W1 = W1

Ex 12: Let say R = aabc, S = dacb • So, R.S = aabcdacb S.R = dacbaabc • R. = aabc .R = aabc

Subsequence • It is quite different from what we have learn in subset • A new sequence can be build from original sequence, but the order of elements must remains. Ex 13: • T = a, a, b, c, q where T1=a, T2=a, T 3=b, T4=c, T5=q S = b, c is a subsequence of T but R = c, b is not a subsequence of T • *Take note that the order in which b and c appears must be the same with the original sequence.

Exercise • List all string on X = {0, 1}, with length 2. • With your own words, explain the meaning of sequence. What is the basic difference between sequence and set?

CSNB 143 Discrete Mathematical Structures