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CSNB 143 Discrete Mathematical Structures

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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.

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 )

- 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
whereS1=2, S2= 4, S3=6, … Sn = 2n

Ex 2:

- T = a, a, b, a, b
whereT1=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

- 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.

- 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 = aabcdacbS.R = dacbaabc
- R. = aabc.R = aabc

- 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?