1. CMPD133 – DISCRETE STRUCTURE Chapter 1 - Topic 2 – Sequence & Strings

2. Topic 2 – Sequence & Strings 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.

3. 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.

4. Ex 1:  S = 2, 4, 6, …, 2n  S = S 1, S 2, S 3, … S n whereS 1 =2, S 2 = 4, S 3 =6, … S n = 2n Ex 2:  T = a, a, b, a, b whereT 1 =a, T 2 =a, T 3 =b, T 4 =a, T 5 =b Sequence

5.  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. Sequence

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

7. 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. a) C 1 = 5, C n = 2C n-1,2  n  6 b) D 1 = 3, D n = D n-1 + 4 c) S n = (-4) n, 1  n   d) T n = 92 – 5n, 1  n  5 Sequence

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

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

10. 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}

11.  A string over A set is a finite sequence of elements from A.  Let A = {a, b, c}. If we let A 1 = b, A 2 = a, A 3 = a, A 4 = 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 b 2 a 3 c. String

12.  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, , b 2 a 29 ba String

13.  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. String

14. Concatenation  If W 1 and W 2 are two strings, the string consisting of W 1 followed by W 2 written W 1. W 2 is called concatenation of W 1 and W 2 : W 1.W 2 =A 1 A 2 A 3 …A n B 1 B 2 B 3 …B m where W 1.W 2 And it is known that W 1.  = W 1 and .W 1 = W 1

15. Ex 12:Let say R = aabc, S = dacb  So,R.S = aabcdacbS.R = dacbaabc  R.  = aabc .R = aabc Concatenation

16. 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 T 1 =a, T 2 =a, T 3 =b, T 4 =c, T 5 =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.

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