Csnb 143 discrete mathematical structures
Download
1 / 19

CSNB 143 Discrete Mathematical Structures - PowerPoint PPT Presentation


  • 123 Views
  • Uploaded on

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.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'CSNB 143 Discrete Mathematical Structures' - jin-mcguire


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
Csnb 143 discrete mathematical structures

CSNB 143 Discrete Mathematical Structures

Chapter 3 – Sequence and String


Csnb 143 discrete mathematical structures

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.


What which where when
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 )


Csnb 143 discrete mathematical structures


Sequence
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.


Csnb 143 discrete mathematical structures

Ex 1:

  • S = 2, 4, 6, …, 2n

  • S = S1, S2, S3, … Sn

    where 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


Csnb 143 discrete mathematical structures


Csnb 143 discrete mathematical structures

Ex 3: called

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.


Csnb 143 discrete mathematical structures


Csnb 143 discrete mathematical structures

  • Both sequences also can have an infinite sequence. 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


Csnb 143 discrete mathematical structures

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


String
String decreasing.

  • 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}


Csnb 143 discrete mathematical structures

  • A string over A set is a finite sequence of elements from A. decreasing.

  • 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.


Csnb 143 discrete mathematical structures

  • 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


Csnb 143 discrete mathematical structures


Concatenation
Concatenation contains all words either it has any meaning or not, regardless its length.

  • 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


Csnb 143 discrete mathematical structures

Ex 12: Let say R = aabc, S = dacb contains all words either it has any meaning or not, regardless its length.

  • So, R.S = aabcdacb S.R = dacbaabc

  • R. = aabc .R = aabc


Subsequence
Subsequence contains all words either it has any meaning or not, regardless its length.

  • 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.


Csnb 143 discrete mathematical structures

Exercise contains all words either it has any meaning or not, regardless its length.

  • 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?