1. Chapter 3 – Sequence and String CSNB 143 Discrete Mathematical Structures

2. 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. 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 )

4. String(Clear / Not Clear ) Concatenation(Clear / Not Clear ) Subsequence(Clear / Not Clear )

5. 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.

6. 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

7. 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.

8. 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.

9. 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

10. 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

11. 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

12. 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}

13. 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.

14. 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

15. 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.

16. 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

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

18. 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.

19. 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?