CSNB143 – Discrete Structure

1. CSNB143 – Discrete Structure Sequence and String

2. Sequence and String Learning Outcomes • 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 Introduction • 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, … Snwhere 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 • Note: If we change the position (e.g 3rd to 4th place and vice versa), we will think this is a difference sequence, S’.

4. Sequence Recursive and Explicit Sequence. • 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 arecursivesequence 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.

5. Sequence • 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. R E F I a) C1 = 5, Cn = 2Cn-1, 2 n 6 b) D1= 3, Dn = Dn-1 + 4 c) Sn= (-4)n, 1 n d) Tn= 92 – 5n, 1 n 5

6. Sequence Increasing/ Decreasing sequence. • 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

7. When we use sequence? • It is being used in running the computer code, where the code must be executed one line after another. The codes cannot jump from one line to the next 10 lines because of the sequences. All the codes are following the concept of First In First Out (FIFO). • It is also being used in array, where every single data can be reached as an individual because we always know the sequence. • Ex 7: Let say A[k] = 1 for all 1  k  8 Let say S(x) = 1, x = 1, 3, 6, 8 0, x = 2, 4, 5, 7 1 1 1 1 1 1 1 1 1 0 1 0 0 1 0 1

8. String • A sequence of characters without comma (,), will produce string. • This sequence can be finite or infinite. • Ex 8: A finite character sequence: s, c, h, o, o, l = character sequence school = string • Ex 9: infinite character sequence a, b, a, b, a, b, a, b,…. = sequence abababab… = string

9. String • Given a set A, we can build set A* where it will contains all finite sequence of all elements in A. If the elements in A are characters (a, b, c, …, z), then the product of A* is string or words. Sequence in A* is not written with comma (,). And each A* must contain an empty sequence that contains no characters at all, denoted as . • Ex 10: Let say A = {a, b, c, …, z} Then A* = {aaaa, computer, denda, pqr, sysrq,… } • 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.

10. String Concatenation • Let say W1 = A1A2A3…An and W2 = B1B2B3…Bm, and W1 and W2 are both elements in A* for set A. Concatenation for W1 and W2 is: W1.W2 =A1A2A3…AnB1B2B3…Bm where W1.W2 is also an element of A* • 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

11. 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.