Lecture 4 Sequences CSCI – 1900 Mathematics for Computer Science Fall 2014 Bill Pine

Lecture 4Sequences

CSCI – 1900 Mathematics for Computer Science

Fall 2014

Bill Pine

CSCI 1900 Lecture 4 - 2

Lecture Introduction

• Reading– Rosen - Section 2.4

• Sequences– Definition– Properties

• Recursion• Characteristic Function


Sequence

• A sequence is an ordered list of objects• It can be finite

– 1,2,3,2,1,0,1,2,3,1,2

Or infinite– 3,1,4,1,5,9,2,6,5,…

• In contrast to sets, with sequences– Duplicates are significant– Order is significant


Sequence Examples

• 1, 2, 3, 2, 2, 3,1 is a sequence, but not a set• The sequences 1, 2, 3, 2, 2, 3, 1 and 2, 2, 1,

3 are made from elements of the set {1, 2, 3}

• The sequences 1, 2, 3, 2, 2, 3, 1 and 2, 1, 3, 2, 2, 3, 1 only switch two elements, but that is sufficient to make them unequal


Alphabetic Sequences

• Finite list of characters– D,i,s,c,r,e,t,e– d,a,b,d,c,a

• Infinite lists of characters– a,b,a,b,a,b,a,b…– This,is,the,song,that,never,ends,

It,goes,on,and,on,my,friends,

Someone,started,singing,it,not,knowing,what,it,was,and,they’ll,continue,singing,it,forever,just,because…

• String : a sequence of letters or symbols written without commas


Linear Array

• Principles of sequences can be applied to computers, specifically, arrays

• There are some differences though– Sequence

• Well-defined• Modification of any element or its order results in a new sequence

– Array• May not have all elements initialized• Modification of array by software may occur• Even if the array has variable length, we’re ultimately limited to finite

length


Set Corresponding to a Sequence

• Set corresponding to a sequence - the set of all distinct elements of the sequence– {a,b} is the set corresponding to sequence

abababab…– {1,2,3} is the set corresponding to the

sequences• 1,2,3• 1,2,3,3,3,3,2,1,2,1• 1,2,3,3,2,1,1,2,3,3,2,1….


Defining Infinite Sequences

• Explicit Formula,– The value for the nth item – an – is determined by a

formula based solely on n

• Recursive Formula– The value for the nth item is determined by a

formula based on n and the previous value(s) – an ,

an-1 , … – in the sequence


Explicit Sequences

• Easy to calculate any specific element• By convention a sequence starts with n = 1• an = 2 * n

– 2, 4, 6, 8, 10, 12, 14, 16, …– a3 = 2 * 3 = 6 – a5 = ???

• an = 3 * n + 6– 9, 12, 15, 18, 21, 24, …– a5 = 3 * 5 + 6 = 21– a100 = ???

• Defines the values a variable is assigned in a program loop when adding/subtracting/multiplying/dividing by a constant


Explicit Sequences and a Program

for n = 1 thru 100 by 1y = n * 5display “y sub” n “is equal to” y

next n

yn = 5 *n in this sequence


Recursive Sequences

• Used when the nth element depends on some calculation on the prior element(s)

• You must specify the values of the first term(s)• Example:

a1 = 1a2 = 2an = an-1 + an-2

• Defines the values a variable is assigned in a program loop when the current value depends upon its previous value(s)


Recursive Sequences and a Program

yMinus2 = 1print “y sub 1 is equal to” yMinus2yMinus1 = 2print “y sub 2 is equal to” yMinus1;for n = 3 thru 100 by 1

y = yMinus1 + yMinus2 print “y sub” n “ is equal to” y yMinus2 = yMinus1 yMinus1 = y

next n

yn = yn-1+yn-2 in this sequence


Characteristic Functions

• Characteristic function – function defining membership in a set, over the universal set

• fA(x) =

• Example A = {1, 4, 6} Where U = Z+

– fA(1) = 1, fA(2) = 0, fA(4) = 1, fA(5) = 0, fA(6) = 1, fA(7) = 0, …

– fA(0) = ???

1 if xA

0 if xA


Representing Sets with a Computer

• Recall: sets have no order and no duplicated elements

• The need to assign each element in a set to a memory location – Gives the set order in a computer

• We can use the characteristic function to define a set with a computer program


Characteristic Function in Programming

• To see a simplistic implementation, consider the following example with A ={1,4,6}

function isInA( x )

if( (x == 1) OR (x == 4) OR (x == 6) ) return ( 1 )

else return ( 0 )

• Might be used to validate user input


Properties of Characteristic Functions

Characteristic functions of subsets satisfy the following properties (proofs are on page 15 of textbook)– fAB(x) = fA(x)fB(x)

– fAB(x) = fA(x) + fB(x) – fA(x)fB(x)

– fAB(x) = fA(x) + fB(x) – 2fA(x)fB(x)


Proving Characteristic Function Properties

• A way to prove these is by enumeration of cases

• Example: Prove fAB = fAfB

fA(a) = 0, fB(a) = 0, fA(a) fB(a) = 0 0 = 0 = fAB(a) fA(b) = 0, fB(b) = 1, fA(b) fB(b) = 0 1 = 0 = fAB(b) fA(c) = 1, fB(c) = 0, fA(c) fB(c) = 1 0 = 0 = fAB(c) fA(d) = 1, fB(d) = 1, fA(d) fB(d) = 1 1 = 1 = fAB(d)

A Babc d


More Properties of Sequences

• Any set with n elements can be arranged as a sequence of length n, but not vice versa– Sets have no order and duplicates don’t matter– Each subset can be identified with its characteristic

function – A sequence of 1’s and 0’s

• Characteristic function for universal set, U, is

fU(x) = 1


Countable and Uncountable

• A set is countable if it corresponds to some sequence– Members of set can be arranged in a list– Elements have position

• All finite sets are countable• Some, but not all infinite sets are countable,

otherwise are said to be uncountable– An example of an uncountable set is set of real numbers– E.g., what comes after 1.23534 ?


Countable Sets

• A finite set S is always countable – It can correspond the sequence

• an = n for all n N where n |S|

• Countable { x | x = 5 * n and n Z+ }– Corresponds to the sequence an = n for all

Set 5 10 15 20 25 30

Seq 1 2 3 4 5 6


Strings

• Sequences can be made up of characters• Example: W, a, k, e, , u, p• Removing the commas and you get a string

“Wake up”• Strings can illustrate the difference between

sequences and sets more clearly– a, b, a, b, a, b, a, … is a sequence, i.e.,

“abababa…” is a string– The corresponding set is {a, b}


Key Concepts Summary

• Sequences– Integers– Strings

• Recursion• Characteristic Function

Documents

Lecture 4 Sequences CSCI – 1900 Mathematics for Computer Science Fall 2014 Bill Pine