Chapter 3-Sequence and Strings

8/12/2019 Chapter 3-Sequence and Strings

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Chapter 3 Sequence and String

CSNB 143Discrete Mathematical Structures

Revised on 2011


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OBJECTIVESStudents should be able to differentiate few characteristics ofsequence.

Students should be able to use sequence and strings.Students should be able to concatenate string and know howto use them.


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What, which, where, whenKnowledge 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 )


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


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SequenceA list of objects in its order. That is, taking order as animportant thing.A list in which the first one should be in front, followed bythe second element, third element and so on.List might be ended aftern , n N and it is named asFiniteSequence . We called n as an index for that sequence.List might have no ending value, and this is called asInfinite

Sequence. Elements might be redundancy.


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Ex 1: S = 2, 4, 6, , 2 n S = S1, S2, S3, Sn

where S1=2, S2= 4, S3=6, Sn = 2 n

Ex 2:

T = a, a, b, a, bwhere T1=a, T 2=a, T 3=b, T 4=a, T 5=b


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If the sequence is depending on the previous value, it is calledRecursive Sequence.

If the sequence is not depending on the previous value, inwhich the value can be directly retrieved, it is calledExplicit Sequence.


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Ex 3:An = An-1 + 5; A1 = 1, 2 n < , this is a recursive sequence

where: A2 = A1 + 5A3 = A2 + 5

Ex 4:An = n 2 + 1; 1 n < , this is an explicit sequencewhere: A1 = 1 + 1 = 2

A2 = 4 + 1 = 5A3 = 9 + 1 = 10That is, we can get the value directly, without any dependency toprevious value.


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Both recursive and explicit formula can have both finite andinfinite sequence.Ex 5: Consider all the sequences below, and identifywhich sequence isrecursive/explicit andfinite/infinite.

a) C1 = 5, C n = 2Cn-1, 2 n 6

b) D1 = 3, D n = Dn-1 + 4c) Sn = (-4)n, 1 n d) Tn = 92 5n , 1 n 5


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Both sequences also can have anIncreasing or Decreasing sequence.A sequence is said to be increased if for each Sn, the value isless than Sn + 1 for alln ,

Sn Sn + 1 ; all n

A sequence is said to be decreased if for each Sn the value is

bigger than Sn + 1 for alln, Sn Sn + 1 ; all n


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Ex 6: Determine either this sequence in increasing ordecreasing.

Sn = 2(n

+ 1),n

1Xn = () n, n 1S = 3, 5, 5, 7, 8, 8, 13


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StringSequences or letters or other symbols that is written withoutcommas are also referred as strings.

An infinite string such as abababa may be regarded asinfinite 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}


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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. Forexample 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.


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The string with no element is call the null string and isdenoted as . We let setA* denote the set of all strings overA, including the null string.

Ex 10: Let sayA = {a, b, c, , z} Then

A* = {aaaa, computer, denda, pqr, sysrq, }

Or let X = {a, b }. Some elements of X* are:a, b, baba, , b2a29 ba


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That is, all finite sequence that can be build from A, containsall words either it has any meaning or not, regardless itslength.

The number of elements in any string A is called ElementsLength, denoted as |A|.

Ex 11:

If A = abcdez, then |A| = 26.


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ConcatenationIf W1 and W2 are two strings, the string consisting of W1 followed by W2 written W1. W2 is called concatenation ofW1 and W2 :

W1.W2 =A1A2A3AnB1B2B3Bm where W1.W2

And it is known thatW1. = W 1 and .W1 = W 1


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Ex 12: Let say R = aabc, S = dacbSo, R.S = aabcdacb S.R = dacbaabc R. = aabc .R = aabc


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SubsequenceIt is quite different from what we have learn in subsetA new sequence can be build from original sequence, but theorder of elements must remains.

Ex 13: T = a, a, b, c, q

where T1=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 thesame with the original sequence.


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Exercise1. 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?

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Chapter 3-Sequence and Strings